Intermediate Algebra (11th edition)

This function his called the compositionof functions gand ƒ, written gƒ.

SECTION 5.3 Polynomial Functions, Graphs, and Composition 287

x

X

Y

Z

f

h = g ° f

g

y = f(x) z = g(y) = g(f(x))

z = (g ° f)(x)

FIGURE 1

Read as “ ”.

As a real-life example of how composite functions occur, consider the following

retail situation.

A $40pair of blue jeans is on sale for 25%off. If you purchase the jeans

before noon, the retailer offers an additional 10%off. What is the final

sale price of the blue jeans?

You might be tempted to say that the blue jeans are off and cal-

culate giving a final sale price of

This is not correct.

To find the correct final sale price, we must first find the price after taking 25% off,

and then take an additional 10% off that price.

$30 1 0.10 2 = $3, giving a final sale price of $30- $3=$27.

$40 1 0.25 2 = $10, giving a sale price of $40- $10= $30.

$40-$14= $26.

$40 1 0.35 2 = $14,

25%+10%= 35%

gƒ g of ƒ

Composition of Functions

If and are functions, then the composite function,or composition,of gand

ƒ is defined by

for all xin the domain of ƒ such that ƒ 1 x 2 is in the domain of g.

1 gƒ 2 1 x 2 g 1 ƒ 1 x 22

ƒ g

Take 25% off

original price.

Take additional

10% off.

This is the idea behind composition of functions.

Evaluating a Composite Function

Let and Find

Definition

Use the rule for

Add.

Use the rule for

= 49 Square 7.

= 72 ƒ 1 x 2 ; ƒ 172 = 72.

=ƒ 172

=ƒ 14 + 32 g 1 x 2 ; g 142 = 4 +3.

=ƒ 1 g 1422

1 ƒg 2142

ƒ 1 x 2 = x^2 g 1 x 2 =x+3. 1 ƒg 2142.

EXAMPLE 5

Now evaluate the

“outside” function.

Evaluate the “inside”

function value first.

NOW TRY

NOW TRY
EXERCISE 5
Let

and.

Find. 1 ƒg 2172

g 1 x 2 =x- 2

ƒ 1 x 2 = 3 x+ 7

NOW TRY ANSWER

5. 22