Intermediate Algebra (11th edition)

If we interchange the order of the functions in Example 5,the composition of g

and ƒ is defined by. To find , we let

Definition

Use the rule for

Square 4.

Use the rule for

Add.

Here we see that because. In general,

1 ƒg 21 x 2  1 gƒ 21 x 2.

1 ƒg 2142 Z 1 gƒ 2142 49 Z 19

= 19

= 16 + 3 g 1 x 2 ; g 1162 = 16 +3.

=g 1162

=g 1422 ƒ 1 x 2 ; ƒ 142 = 42.

=g 1 ƒ 1422

1 gƒ 2142

g 1 ƒ 1 x 22 1 gƒ 2142 x=4.

288 CHAPTER 5 Exponents, Polynomials, and Polynomial Functions

Finding Composite Functions

Let and Find the following.

(a)

Work inside the parentheses.

Multiply, and then subtract.

(b)

Use as the input for the function ƒ.

Use the rule for

Distributive property

Combine like terms.

(c) Find again, this time using the rule obtained in part (b).

From part (b)

Let

Square 2.

Multiply.

Same result as in part (a) = 35 Add. NOW TRY

= 16 + 19

= 4142 + 19

1 ƒg 2122 = 41222 + 19 x=2.

1 ƒg 21 x 2 = 4 x^2 + 19

1 ƒg 2122

= 4 x^2 + 19

= 4 x^2 + 20 - 1

= 41 x^2 + 52 - 1 g 1 x 2 =x^2 + 5

= 41 g 1 x 22 - 1 ƒ 1 x 2 ; ƒ 1 x 2 = 4 x-1.

=ƒ 1 g 1 x 22 g 1 x 2

1 ƒg 21 x 2

= 35

= 4192 - 1 ƒ 1 x 2 = 4 x- 1

=ƒ 192

=ƒ 122 + 52 g 1 x 2 =x^2 + 5

=ƒ 1 g 1222

1 ƒg 2122

ƒ 1 x 2 = 4 x- 1 g 1 x 2 =x^2 + 5.

EXAMPLE 6

OBJECTIVE 5 Graph basic polynomial functions. Recall from Section 3.5

that each input (or x-value) of a function results in one output (or y-value). The set of

input values (for x) defines the domain of the function, and the set of output values

(for y) defines the range.

The simplest polynomial function is the identity function,defined by

and graphed in FIGURE 2on the next page. This function pairs each real number with

itself.

ƒ 1 x 2 x

NOW TRY
EXERCISE 6
Let

and.

Find the following.

(a)

(b) 1 ƒg 21 x 2

1 gƒ 21 - 12

g 1 x 2 =-x^2 + 2

ƒ 1 x 2 =x- 5

NOW TRY ANSWERS

6. (a)- 34 (b) -x^2 - 3