§ 2.3 The Algebra of Functions Domain of a Function Finding a Function’s Domain If a function f does not model data or verbal.
Download ReportTranscript § 2.3 The Algebra of Functions Domain of a Function Finding a Function’s Domain If a function f does not model data or verbal.
Slide 1
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 2
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 3
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 4
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 5
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 6
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 7
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 8
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 9
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 10
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 11
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 12
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 13
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 14
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 15
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 2
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 3
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 4
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 5
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 6
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 7
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 8
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 9
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 10
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 11
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 12
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 13
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 14
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3
Slide 15
§ 2.3
The Algebra of Functions
Domain of a Function
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the
largest set of real numbers for which the value of f(x) is a real number.
Exclude from a function’s domain real numbers that cause division by zero.
Exclude from a function’s domain real numbers that result in a square root of
a negative number.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3
Domain of a Function
• Consider the function
1
f ( x)
x5
Because division by 0 is undefined (and not a
real number), the denominator, x – 5, cannot be
0. Then x cannot be 5, and 5 is not in the domain
of the function.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3
Domain of a Function
• Now consider the function:
g ( x) x 7
The equation tells us to take the square root of x – 7.
Because only nonnegative numbers have square roots that
are real numbers, the expression under the square root
must be nonnegative. Then x must be greater than or equal
to 7.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3
Domain of a Function
EXAMPLE
2
Find the domain of the function: f x 4x 2x 7 .
SOLUTION
Since the function f has no denominator or square root, there
are no real numbers that when plugged into the function for x
would cause the value of the function to yield something
other than a real number. Therefore, the domain is:
x | x is a real number
This is set notation and it is read: “the set of all x such that x is a real number.”.
Using this notation, the rule stating the conditions for x follows the vertical bar which
just means “such that.”
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3
Domain of a Function
EXAMPLE
Find the domain of the function: f x
2
6
x4 5 x
.
SOLUTION
The function has no square roots so we don’t have to worry
about pursuing that avenue. However the function does have
x in two different denominators. Therefore I do the
following:
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3
Domain of a Function
CONTINUED
x40
x4
Set a denominator equal to zero
Solve
5 x 0
x 5
Set a denominator equal to zero
Solve
Therefore, a denominator of the function is equal to zero
when x = 4 or x = -5. Then the domain is:
x | x is a real number and x 4 and x 5
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3
The Algebra of Functions
• We can combine functions using addition,
subtraction, multiplication and division by
performing operations with the algebraic
expression that appears on the right side of
the equations.
• For example, the functions f(x) = 2x-3 and
g(x) = 4x+5 can be combined to form the sum,
difference, product and quotient of f and g.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Sum
(f + g) = f(x) + g(x)
Difference
(f - g) = f(x) - g(x)
f g 2 f 2 g 2 3 22 2 4 2
3 4 2 4 2 12 2 4 2 12
g f 5 g 5 f 5 4 5 352 2
4 5 3 25 2 4 5 75 2 9 73 64
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3
The Algebra of Functions
The Algebra of Functions
Algebraic Rule
Example
Use f x 3x 2 2 and g x 4 x
Product
(f g) = f(x) g(x)
Quotient
f
f x
x
g x
g
fg 3 3 32 24 3 3 9 24 3
27 24 3 251 25
2
f
f 8 38 2 3 64 2 192 2
8
g 8
4 8
48
48
g
190 95
15.83
12
6
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.3
Applying the Algebra of Functions
EXAMPLE
The table shows the total number of births and the total number of
deaths in the United States from 1995 through 2002.
Births and Deaths in the U.S.A.
Year
Births
Deaths
1995
3,899,589
2,312,132
1996
3,891,494
2,314,690
1997
3,880,894
2,314,245
1998
3,941,553
2,337,256
1999
3,959,417
2,391,399
2000
4,058,814
2,403,351
2001
4,025,933
2,416,425
2002
4,022,000
2,436,000
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.3
Applying the Algebra of Functions
CONTINUED
The data can be modeled by the following functions:
Bx 24,770x 3,873,266
Dx 20,205x 2,294,970
In each function, x represents the number of years after 1999. Assume that the functions apply only
to the years shown in the table. Use these functions to solve the following exercises.
(a) Find the domain of B.
(b) Find the domain of D.
(c) Find (B-D)(x). What does this function represent?
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.3
Applying the Algebra of Functions
CONTINUED
SOLUTION
(a) Find the domain of B.
Since the year 1995 is year 0, the year 1996 is year 1, etc., then
we find that the year 2002 is year 7. Therefore,
x | x 0,1,2,3,4,5,6,7
(b) Find the domain of D.
x | x 0,1,2,3,4,5,6,7
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.3
Applying the Algebra of Functions
CONTINUED
(c) Find (B-D)(x). What does this function represent?
B Dx Bx Dx 24,770x 3,873,266 20,205x 2,294,970
24,770x 3,873,266 20,205x 2,294,970 4,565x 1,578,296
This function represents the difference between the
numbers of births and the numbers of deaths in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.3
Applying the Algebra of Functions
CONTINUED
(d) Use the function in part (c) to find (B-D)(6). What does this
mean in terms of the U.S. population and to which year does
this apply?
B Dx 4,565x 1,578,296
B D6 4,5656 1,578,296
B D6 27,390 1,578,296
B D6 1,605,686
Replace x with 6
Multiply
Subtract
The number 1,605,686 means that in year number 6 (2001),
approximately 1,605,686 more births took place in the U.S than the
number of deaths in the U.S. Had the answer been negative, that
would have suggested that more deaths had taken place in the U.S.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.3