Transcript Chapter 2 Section 8 - Canton Local Schools

Chapter 2
Graphs and
Functions
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All rights reserved
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1
SECTION 2.8 Combining Functions; Composite Functions
OBJECTIVES
1
2
3
4
5
Learn basic operations on functions.
Form composite functions.
Find the domain of a composite function.
Decompose a function.
Apply composition to practical problems.
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2
SUM, DIFFERENCE, PRODUCT,
AND QUOTIENT OF FUNCTIONS
Let f and g be two functions. The sum f + g,
the difference f – g, the product fg, and the
f
quotient
are functions whose domains
g
consist of those values of x that are common to
f
the domains of f and g (except for where all
g
values x where g(x) = 0 must be excluded).
These functions are defined as follows:
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SUM, DIFFERENCE, PRODUCT,
AND QUOTIENT OF FUNCTIONS
(i) Sum
f
 g  x   f  x   g  x 
(ii) Difference  f  g  x   f  x   g  x 
(iii) Product
 fg  x   f  x   g  x 
(iv) Quotient
f  x
 f 
 g   x   g x , g  x   0.
 
 
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EXAMPLE 1
Combining Functions
2
f
x

x
 6 x  8, and g  x   x  2.
Let  
Find each of the following functions.
 f 
a.  f  g  4  b.    3 c.  f  g  x 
g
d.  f  g  x 
e.  fg  x 
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 f 
f.    x 
g
5
EXAMPLE 1
Solution
Combining Functions
f x   x  6x  8, and g x   x  2.
2
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EXAMPLE 1
Combining Functions
Solution continued
f x   x 2  6x  8, and g x   x  2.
c.
f
 g  x   f  x   g  x 
  x  6 x  8   x  2 
2
 x  5x  6
2
d.
f
 g  x   f  x   g  x 
  x2  6 x  8   x  2 
 x 2  7 x  10
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EXAMPLE 1
Combining Functions
Solution continued
f x   x  6x  8 and g x   x  2
2
e.
2
fg
x

x
     6 x  8  x  2 
 x3  2 x 2  6 x 2  12 x  8 x  16
 x3  8 x 2  20 x  16
f  x
 f 
x  6x  8
f.    x  

g  x
x2
g
 x  2  x  4 

 x  4, x  2
x2
2
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COMPOSITION OF FUNCTIONS
If f and g are two functions, the composition
of function f with function g is written as
f g and is defined by the equation
f
g  x   f  g  x   ,
where the domain of f g consists of those
values x in the domain of g for which g(x) is
in the domain of f.
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COMPOSITION OF FUNCTIONS
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EXAMPLE 2
Evaluating a Composite Function
3
f
x

x
and g x   x  1.
Let  
Find each of the following.
a.  f
g 1
Solution
a.  f
b.  g f 1
c.  f
f  1
g 1  f  g 1 
 f  2
2
3
8
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EXAMPLE 2
Evaluating a Composite Function
Solution continued
f x   x 3 and g x   x  1
b.  g f 1  g  f 1 
 g 1  1  1  2
c.  f
f  1  f  f  1 
 f  1   1  1
3
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EXAMPLE 3
Finding Composite Functions
2
f
x

2x

1
and
g
x

x
 3.

Let  
Find each composite function.
a.  f g  x 
b.  g f  x 
c.  f
Solution
a.  f
f  x 
g  x   f  g  x  
 f  x 2  3
 2  x  3  1
2
 2x  6  1
2
 2x  5
2
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EXAMPLE 3
Finding Composite Functions
Solution continued
f x   2x  1 and g x   x 2  3.
b.  g f  x   g  f  x  
 g  2 x  1
  2 x  1  3  4 x 2  4 x  2
2
c.  f
f  x   f  f  x  
 f  2 x  1
 2  2 x  1  1  4 x  3
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EXAMPLE 4
Finding the Domain of a Composite Function
1
Let f x   x  1 and g x   .
x
a. Find  f g  x  and its domain.
b. Find  g f  x  and its domain.
Solution
1 1
a.  f g  x   f  g  x    f     1
 x x
Domain is (–∞, 0) U (0, ∞).
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EXAMPLE 4
Finding the Domain of a Composite Function
Solution continued
1
f x   x  1 and g x  
x
1
b.  g f  x   g  f  x    g  x  1 
x 1
Domain is (–∞, –1) U (–1, ∞).
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EXAMPLE 5
Decomposing a Function
1
Write H  x  
2x  1
functions f and g.
2
as f g for some
Solution
Step 1 Define g(x) as any expression in the
defining equation for H.
Let g(x) = 2x2 + 1.
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EXAMPLE 5
Decomposing a Function
Solution continued
Step 2 Replace the letter H with f and replace
the expression chosen for g(x) with x.
becomes
Step 3 Now we have:
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EXAMPLE 6
Calculating the Area of an Oil Spill from a
Tanker
Oil is spilled from a tanker into the Pacific
Ocean and the area of the oil spill is a perfect
circle.
The radius of this oil slick increases at the rate
of 2 miles per hour.
a. Express the area of the oil slick as a function
of time.
b. Calculate the area covered by the oil slick in
six hours.
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EXAMPLE 6
Calculating the Area of an Oil Spill from a
Tanker
Solution
The area of the oil slick is a function its radius.
2
A  f r    r
The radius is a function time: increasing 2 mph
r  g  t   2t
a. The area is a composite function
2
2
A  f  g  t    f  2t     2t   4 t .
b. Substitute t = 6.
2
A  4  6   4  36   144 square miles.
The area of the oil slick is 144π square miles.
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EXAMPLE 7
Applying Composition to Sales
A car dealer offers an 8% discount off the
manufacturer’s suggested retail price (MSRP)
of x dollars for any new car on his lot.
At the same time, the manufacturer offers a
$4000 rebate for each purchase of a car.
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EXAMPLE 7
Applying Composition to Sales
a. Write a function r (x) that represents the
price after the rebate.
b. Write a function d(x) that represents the price
after the dealer’s discount.
c. Write the functions  r d  x  and  d r  x  .
What do they represent?
d. Calculate  d r  x    r d  x  .
Interpret this expression.
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EXAMPLE 7
Applying Composition to Sales
Solution
a. The MSRP is x dollars, rebate is $4000, so
r (x) = x – 4000
represents the price of the car after the rebate.
b. The dealer’s discount is 8% of x, or 0.08x, so
d(x) = x – 0.08x = 0.92x
represents the price of the car after the
dealer’s discount.
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EXAMPLE 7
Applying Composition to Sales
Solution continued
c. (i)
r
d  x   r  d  x    r  0.92 x 
 0.92 x  4000
represents the price when the dealer’s discount is
is applied first.
(ii)  d r  x   d  r  x    d  x  4000 
 0.92  x  4000 
 0.92 x  3680
represents the price when the manufacturer’s
rebate is applied first.
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EXAMPLE 7
Applying Composition to Sales
Solution continued
d.
d
r  x    r d  x   d  r  x    r  d  x  
  0.92 x  3680    0.92 x  4000 
 320 dollars
This equation shows that it will cost $320 more
for any car, regardless of its price, if you apply
the rebate first and then the discount.
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