Intermediate Algebra (11th edition)

While ƒ is the most common letter used to represent functions, recall that other

letters, such as gand h, are also used. The capital letter P is often used for polyno-

mial functions.The function defined as

yields the same ordered pairs as the function ƒ in Example 1.

P 1 x 2 = 4 x^3 - x^2 + 5

284 CHAPTER 5 Exponents, Polynomials, and Polynomial Functions

OBJECTIVES OBJECTIVE 1 Recognize and evaluate polynomial functions.In Chapter 3,

we studied linear (first-degree polynomial) functions, defined as

Now we consider more general polynomial functions.

ƒ 1 x 2 =ax+ b.

Polynomial Functions, Graphs, and Composition

5.3

1 Recognize and

evaluate polynomial

functions.

2 Use a polynomial

function to model

data.

3 Add and subtract

polynomial

functions.

4 Find the

composition of

functions.

5 Graph basic

polynomial

functions.

Polynomial Function

A polynomial function of degree nis defined by

for real numbers an, an- 1 ,Á, a 1 ,and a 0 ,where anZ 0 and nis a whole number.

ƒ 1 x 2 an xnan 1 xn^1 Áa 1 xa 0 ,

Another way of describing a polynomial function is to say that it is a function

defined by a polynomial in one variable, consisting of one or more terms. It is usually

written in descending powers of the variable, and its degree is the degree of the poly-

nomial that defines it.

We can evaluate a polynomial function ƒ 1 x 2 at different values of the variable x.

NOW TRY
EXERCISE 1
Let.
Find .ƒ 1 - 32

ƒ 1 x 2 =x^3 - 2 x^2 + 7

Evaluating Polynomial Functions

Let Find each value.

(a)

Given function

Substitute 3 for x.

Apply the exponents.

Multiply.

Subtract, and then add.

Thus, ƒ 132 = 104 and the ordered pair 13 , 1042 belongs to ƒ.

ƒ 132 = 104

ƒ 132 = 108 - 9 + 5

ƒ 132 = 41272 - 9 + 5

ƒ 132 = 41323 - 32 + 5

ƒ 1 x 2 = 4 x^3 - x^2 + 5

ƒ 132

ƒ 1 x 2 = 4 x^3 - x^2 + 5.

EXAMPLE 1

(b)

Let.

Multiply.

Subtract, and then add.

So, ƒ 1 - 42 = - 267 .The ordered pair 1 - 4 , - 2672 belongs to ƒ. NOW TRY

ƒ 1 - 42 = - 267

ƒ 1 - 42 =- 256 - 16 + 5

ƒ 1 - 42 = 4 # 1 - 642 - 16 + 5

ƒ 1 - 42 = 4 # 1 - 423 - 1 - 422 + 5 x=- 4

ƒ 1 x 2 = 4 x^3 - x^2 + 5

ƒ 1 - 42

Read this as “ƒ of 3,”

not “ƒ times 3.”

Use parentheses.

Be careful with signs.

NOW TRY ANSWER

1. 
   
   
   
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