Slide 2.7- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slide 2.7- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

Combining Functions; Composite Functions

Learn basic operations on functions.

Learn to form composite functions.

Learn to find the domain of a composite function.

Learn to decompose a function

Learn to apply composition to a practical problem.

SECTION 2.7

1

2

3

4

5


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

quotient are functions whose domains

consist of those values of x that are common to

the domains of f and g. These functions are

defined as follows:

f

g


SUM, DIFFERENCE, PRODUCT, AND QUOTIENT OF FUNCTIONS

f

g

x f x g x , g x 0.(iv) Quotient

(i) Sum f g x f x g x

(ii) Difference f g x f x g x

(iii) Product fg x f x gg x


EXAMPLE 1 Combining Functions

Let f x x2 6x 8, and g x x 2.

Find each of the following functions.

a. f g x b. f g x

c. fg x d. f

g

x Solution

a. f g x f x g x x2 6x 8 x 2 x2 5x 6



Solution continued

f x x2 6x 8 and g x x 2

b. f g x f x g x x2 6x 8 x 2 x2 7x 10

c. fg x x2 6x 8 x 2 x3 2x2 6x2 12x 8x 16

x3 8x2 20x 16



Solution continued

f x x2 6x 8 and g x x 2

d. f

g

x f x g x , g x 0

x2 6x 8

x 2, x 2 0

x 2 x 4

x 2, x 2



Solution continued

Since f and g are polynomials, the domain of f and g is the set of all real numbers, or, in interval notation (–∞, ∞).

The domain forf

gmust exclude x = 2.

Its domain is (–∞, 2) U (2, ∞).

The domain for f +g, f – g, and fg is (–∞, ∞).


COMPOSITION OF FUNCTIONS

If f and g are two functions, the composition of function f with function g is written asf og and is defined by the equation

f og x f g x ,

where the domain of values x in the domain of g for which g(x) is in the domain of f.

consists of thosef og


COMPOSITION OF FUNCTIONS


EXAMPLE 2 Evaluating a Composite Function

LetFind each of the following.

f x x3 and g x x 1.

a. f og 1 b. go f 1 c. f o f 1 d. gog 1

Solution

a. f og 1 f g 1 f 2 23

8


EXAMPLE 2 Evaluating a Composite Function

Solution continued

b. go f 1 g f 1 g 1 11 2

f x x3 and g x x 1

c. f o f 1 f f 1 f 1 1 3 1

d. gog 1 g g 1 g 0 0 1 1


EXAMPLE 3 Finding Composite Functions

LetFind each composite function.

f x 2x 1 and g x x2 3.

a. f og x b. go f x c. f o f x Solution

a. f og x f g x f x2 3 2 x2 3 1

2x2 6 1

2x2 5


EXAMPLE 3 Finding Composite Functions

Solution continued

b. go f x g f x g 2x 1 2x 1 2 3 4x2 4x 2

c. f o f x f f x f 2x 1 2 2x 1 1 4x 3

f x 2x 1 and g x x2 3.


EXAMPLE 4 Finding the Domain of a Composite Function

Let f x x 1 and g x 1

x.

c. Find f og x and its domain.

d. Find go f x and its domain.

b. Find go f 1 .a. Find f og 1 .

Solution

a. f og 1 f g 1 f 1 11 0


EXAMPLE 4 Finding the Domain of a Composite Function

f x x 1 and g x 1

x

c. f og x f g x f

1

x

1

x1

d. go f x g f x g x 1 1

x 1

b. go f 1 g f 1 g 0 not defined

Solution continued

Domain is (–∞, 0) U (0, ∞).

Domain is (–∞, –1) U (–1, ∞).


EXAMPLE 5 Decomposing a Function

Show that each of theLet H x 1

2x2 1.

following provides a decomposition of H(x).

a. Express H x as f g x , where f x 1

x and g x 2x2 1.

b. Express H x as f g x , where f x 1

x and g x 2x2 1.


EXAMPLE 5 Decomposing a Function

Solutiona. f g x f 2x2 1

1

2x2 1

H x b. f g x f 2x2 1

1

2x2 1

H x


EXAMPLE 6Calculating the Area of an Oil Spill from a Tanker

Oil is spilled from a tanker into the Pacific Ocean. Suppose the area of the oil spill is a perfect circle. (In practice, this does not happen, because of the winds and tides and the location of the coastline.) Suppose that the radius of the oil slick is increasing (because oil continues to spill) 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 6 hours.


EXAMPLE 6Calculating the Area of an Oil Spill from a Tanker

Solution

The area of the oil slick is a function its radius. A f r r2 .

The radius is a function time: increasing 2 mph r g t 2t.

a. The area is a composite function A f g t f 2t 2t 2 4t 2 .

b. Substitute t = 6.

A 4 6 2 4 36 144 square miles.

The area of the oil slick is 144π square miles.


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.

a. Write a function f (x) that represents the price after the rebate.

b. Write a function g(x) that represents the price after the dealer’s discount.

c. Write the function f og x and go f x .What do they represent?



d. Calculate go f x f og x .Interpret this expression.

Solution

a. The MSRP is x dollars, rebate is $4000, sof (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 g(x) = x – 0.08x = 0.92x represents the price of the car after the dealer’s discount.



Solution continued

represents the price when the dealer’s discount is is applied first.

represents the price when the manufacturer’s rebate is applied first.

c. (i) f og x f g x f 0.92x 0.92x 4000

(ii) go f x g f x g x 4000 0.92 x 4000 0.92x 3680



Solution continued

This equation shows that it will cost 320 dollars more for any car, regardless of its price, if you apply the rebate first and then the discount.

d. go f x f og x g f x f g x 0.92x 3680 0.92x 4000 320 dollars

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Slide 2.7- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley