Transcript x - School District 27J

6-5
with
Functions
6-5 Operations
Operations
with
Functions
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
Algebra2 Algebra 2
Holt
6-5
Operations with Functions
Warm Up
Simplify. Assume that all expressions
are defined.
1. (2x + 5) – (x2 + 3x – 2)
–x2 – x + 7
2. (x – 3)(x + 1)2
x3 – x2 – 5x – 3
3.
x–3
x–2
Holt McDougal Algebra 2
6-5
Operations with Functions
Objectives
Add, subtract, multiply, and divide
functions.
Write and evaluate composite
functions.
Holt McDougal Algebra 2
6-5
Operations with Functions
Vocabulary
composition of functions
Holt McDougal Algebra 2
6-5
Operations with Functions
You can perform operations on functions in
much the same way that you perform
operations on numbers or expressions. You can
add, subtract, multiply, or divide functions by
operating on their rules.
Holt McDougal Algebra 2
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Operations with Functions
Holt McDougal Algebra 2
6-5
Operations with Functions
Example 1A: Adding and Subtracting Functions
Given f(x) = 4x2 + 3x – 1 and g(x) = 6x + 2,
find each function.
(f + g)(x)
(f + g)(x) = f(x) + g(x)
= (4x2 + 3x – 1) + (6x + 2)
2
= 4x + 9x + 1
Holt McDougal Algebra 2
Substitute function rules.
Combine like terms.
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Operations with Functions
Example 1B: Adding and Subtracting Functions
Given f(x) = 4x2 + 3x – 1 and g(x) = 6x + 2,
find each function.
(f – g)(x)
(f – g)(x) = f(x) – g(x)
= (4x2 + 3x – 1) – (6x + 2)
Substitute function rules.
= 4x2 + 3x – 1 – 6x – 2
Distributive Property
= 4x2 – 3x – 3
Combine like terms.
Holt McDougal Algebra 2
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Operations with Functions
Check It Out! Example 1a
Given f(x) = 5x – 6 and g(x) = x2 – 5x + 6,
find each function.
(f + g)(x)
(f + g)(x) = f(x) + g(x)
= (5x – 6) + (x2 – 5x + 6)
2
=x
Holt McDougal Algebra 2
Substitute function rules.
Combine like terms.
6-5
Operations with Functions
Check It Out! Example 1b
Given f(x) = 5x – 6 and g(x) = x2 – 5x + 6,
find each function.
(f – g)(x)
(f – g)(x) = f(x) – g(x)
= (5x – 6) – (x2 – 5x + 6)
2
Substitute function rules.
= 5x – 6 – x + 5x – 6
Distributive Property
= –x2 + 10x – 12
Combine like terms.
Holt McDougal Algebra 2
6-5
Operations with Functions
When you divide functions, be sure to note any
domain restrictions that may arise.
Holt McDougal Algebra 2
6-5
Operations with Functions
Example 2A: Multiplying and Dividing Functions
Given f(x) = 6x2 – x – 12 and g(x) = 2x – 3,
find each function.
(fg)(x)
(fg)(x) = f(x) ● g(x)
= (6x2 – x – 12) (2x – 3)
Substitute function
rules.
= 6x2 (2x – 3) – x(2x – 3) – 12(2x – 3)
Distributive Property
= 12x3 – 18x2 – 2x2 + 3x – 24x + 36
Multiply.
= 12x3 – 20x2 – 21x + 36
Combine like terms.
Holt McDougal Algebra 2
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Operations with Functions
Example 2B: Multiplying and Dividing Functions
f
g
f
g
( )(x)
( )(x) =
f(x)
g(x)
6x2 – x –12
=
2x – 3
(2x – 3)(3x + 4)
=
2x – 3
(2x – 3)(3x +4)
=
(2x – 3)
= 3x + 4, where x ≠
Holt McDougal Algebra 2
Set up the division as a
rational expression.
Factor completely.
3
Note that x ≠ 2 .
Divide out common
factors.
3
2
Simplify.
6-5
Operations with Functions
Check It Out! Example 2a
Given f(x) = x + 2 and g(x) = x2 – 4, find each
function.
(fg)(x)
(fg)(x) = f(x) ● g(x)
= (x + 2)(x2 – 4)
Substitute function rules.
= x3 + 2x2 – 4x – 8
Multiply.
Holt McDougal Algebra 2
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Operations with Functions
Check It Out! Example 2b
g
(x)
f
g
g(x)
(x)
=
f
f(x)
x2 – 4
=
x+2
(x – 2)(x + 2)
=
x+2
(x – 2)(x + 2)
=
(x + 2)
( )
( )
= x – 2, where x ≠ –2
Holt McDougal Algebra 2
Set up the division as a
rational expression.
Factor completely.
Note that x ≠ –2.
Divide out common
factors.
Simplify.
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Operations with Functions
Another function operation uses the output from
one function as the input for a second function.
This operation is called the composition of
functions.
Holt McDougal Algebra 2
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Operations with Functions
The order of function operations is the same as the
order of operations for numbers and expressions. To
find f(g(3)), evaluate g(3) first and then substitute the
result into f.
Holt McDougal Algebra 2
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Operations with Functions
Reading Math
The composition (f
g of x.”
Holt McDougal Algebra 2
o
g)(x) or f(g(x)) is read “f of
6-5
Operations with Functions
Caution!
Be careful not to confuse the notation for
multiplication of functions with composition
fg(x) ≠ f(g(x))
Holt McDougal Algebra 2
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Operations with Functions
Example 3A: Evaluating Composite Functions
Given f(x) = 2x and g(x) = 7 – x, find each
value.
f(g(4))
Step 1 Find g(4)
g(4) = 7 – 4
g(x) = 7 – x
=3
Step 2 Find f(3)
3
f(3) = 2
=8
So f(g(4)) = 8.
Holt McDougal Algebra 2
f(x) = 2x
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Operations with Functions
Example 3B: Evaluating Composite Functions
Given f(x) = 2x and g(x) = 7 – x, find each
value.
g(f(4))
Step 1 Find f(4)
f(4) = 24
= 16
Step 2
f(x) = 2x
Find g(16)
g(16) = 7 – 16
= –9
So g(f(4)) = –9.
Holt McDougal Algebra 2
g(x) = 7 – x.
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Operations with Functions
Check It Out! Example 3a
Given f(x) = 2x – 3 and g(x) = x2, find each
value.
f(g(3))
Step 1 Find g(3)
g(3) = 32
g(x) = x2
=9
Step 2 Find f(9)
f(9) = 2(9) – 3
= 15
So f(g(3)) = 15.
Holt McDougal Algebra 2
f(x) = 2x – 3
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Operations with Functions
Check It Out! Example 3b
Given f(x) = 2x – 3 and g(x) = x2, find each
value.
g(f(3))
Step 1 Find f(3)
f(3) = 2(3) – 3
f(x) = 2x – 3
=3
Step 2 Find g(3)
2
g(3) = 3
=9
So g(f(3)) = 9.
Holt McDougal Algebra 2
g(x) = x2
6-5
Operations with Functions
You can use algebraic expressions as well as
numbers as inputs into functions. To find a rule
for f(g(x)), substitute the rule for g into f.
Holt McDougal Algebra 2
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Operations with Functions
Example 4A: Writing Composite Functions
x
2
Given f(x) = x – 1 and g(x) =
, write
1–x
each composite function. State the domain of
each.
f(g(x))
f(g(x)) = f(
=(
=
x
1–x
x
1–x
)
)2 – 1
–1 + 2x
(1 – x)2
Substitute the rule g into f.
Use the rule for f. Note that
x ≠ 1.
Simplify.
The domain of f(g(x)) is x ≠ 1 or {x|x ≠ 1} because
g(1) is undefined.
Holt McDougal Algebra 2
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Operations with Functions
Example 4B: Writing Composite Functions
x
2
Given f(x) = x – 1 and g(x) =
, write
1–x
each composite function. State the domain of
each.
g(f(x))
2
g(f(x)) = g(x – 1)
=
(x2 – 1)
2
1 – (x – 1)
x2 – 1
=
2 – x2
Substitute the rule f into g.
Use the rule for g.
Simplify. Note that x ≠
The domain of g(f(x)) is x ≠
or {x|x ≠
because f(
) = 1 and g(1) is undefined.
Holt McDougal Algebra 2
.
}
6-5
Operations with Functions
Check It Out! Example 4a
Given f(x) = 3x – 4 and g(x) =
+ 2 , write
each composite. State the domain of each.
f(g(x))
f(g(x)) = 3(
+ 2) – 4
Substitute the rule g into f.
=
+6–4
Distribute. Note that x ≥ 0.
=
+2
Simplify.
The domain of f(g(x)) is x ≥ 0 or {x|x ≥ 0}.
Holt McDougal Algebra 2
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Operations with Functions
Check It Out! Example 4b
Given f(x) = 3x – 4 and g(x) =
+ 2 , write
each composite. State the domain of each.
g(f(x))
g(f(x)) =
=
Substitute the rule f into g.
Note that x ≥
The domain of g(f(x)) is x ≥
Holt McDougal Algebra 2
4
3
4
3
.
or {x|x ≥
4
3
}.
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Operations with Functions
Composite functions can be used to simplify a
series of functions.
Holt McDougal Algebra 2
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Operations with Functions
Example 5: Business Application
Jake imports furniture from Mexico. The
exchange rate is 11.30 pesos per U.S. dollar.
The cost of each piece of furniture is given in
pesos. The total cost of each piece of furniture
includes a 15% service charge.
A. Write a composite function to represent the total
cost of a piece of furniture in dollars if the cost of
the item is c pesos.
Holt McDougal Algebra 2
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Operations with Functions
Example 5 Continued
Step 1 Write a function for the total cost in U.S.
dollars.
P(c) = c + 0.15c
= 1.15c
Step 2 Write a function for the cost in dollars based
on the cost in pesos.
D(c) =
Holt McDougal Algebra 2
c
11.30
Use the exchange rate.
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Operations with Functions
Example 5 Continued
Step 3 Find the composition D(P(c)).
D(P(c)) = 1.15P(c)
= 1.15 (
c
11.30
Substitute P(c) for c.
)
Replace P(c) with its rule.
B. Find the total cost of a table in dollars if it costs
1800 pesos.
Evaluate the composite function for c = 1800.
D(P(c) ) = 1.15 (
1800
11.30
)
≈ 183.19
The table would cost $183.19, including all charges.
Holt McDougal Algebra 2
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Operations with Functions
Check It Out! Example 5
During a sale, a music store is selling all drum
kits for 20% off. Preferred customers also
receive an additional 15% off.
a. Write a composite function to represent the final
cost of a kit for a preferred customer that
originally cost c dollars.
Step 1 Write a function for the final cost of a kit that
originally cost c dollars.
f(c) = 0.80c
Holt McDougal Algebra 2
Drum kits are sold at
80% of their cost.
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Operations with Functions
Check It Out! Example 5 Continued
Step 2 Write a function for the final cost if the
customer is a preferred customer.
g(c) = 0.85c
Holt McDougal Algebra 2
Preferred customers receive 15% off.
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Operations with Functions
Check It Out! Example 5 Continued
Step 3 Find the composition f(g(c)).
f(g(c)) = 0.80(g(c))
Substitute g(c) for c.
f(g(c)) = 0.80(0.85c)
Replace g(c) with its rule.
= 0.68c
b. Find the cost of a drum kit at $248 that a preferred
customer wants to buy.
Evaluate the composite function for c = 248.
f(g(c) ) = 0.68(248)
The drum kit would cost $168.64.
Holt McDougal Algebra 2
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Operations with Functions
Lesson Quiz: Part I
Given f(x) = 4x2 – 1 and g(x) = 2x – 1, find
each function or value.
1. (f + g)(x)
4x2 + 2x – 2
2. (fg)(x)
8x3 – 4x2 – 2x + 1
3.
f
g
( )(x)
4. g(f(2))
Holt McDougal Algebra 2
2x + 1
29
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Operations with Functions
Lesson Quiz: Part II
Given f(x) = x2 and g(x) =
, write each
composite function. State the domain of each.
5. f(g(x))
f(g(x)) = x – 1;
{x|x ≥ 1}
6. g(f(x))
{x|x ≤ – 1 or x ≥ 1}
Holt McDougal Algebra 2