Intermediate Algebra (11th edition)

5.3 Polynomial Functions, Graphs, and Composition

NOW TRY
EXERCISE 2
Use the function in Example 2
to approximate the number of
public school students in
2002.

OBJECTIVE 2 Use a polynomial function to model data.

Using a Polynomial Model to Approximate Data

The number of students enrolled in public schools (grades pre-K–12) in the United

States during the years 1990 through 2006 can be modeled by the polynomial func-

tion defined by

,

where corresponds to the year 1990, corresponds to 1991, and so on,

and is in millions. Use this function to approximate the number of public school

students in 2006. (Source:Department of Education.)

Since corresponds to 2006, we must find

Let

Evaluate.

There were about 49.3 million public school students in 2006. NOW TRY

P 1162 L49.3

P 1162 =-0.01774 11622 + 0.7871 1162 +41.26 x=16.

P 1 x 2 =-0.01774x^2 + 0.7871x+41.26

x= 16 P 1162.

P 1 x 2

x= 0 x= 1

P 1 x 2 =-0.01774x^2 + 0.7871x+41.26

EXAMPLE 2

OBJECTIVE 3 Add and subtract polynomial functions.The operations of

addition, subtraction, multiplication, and division are also defined for functions. For

example, businesses use the equation “profit equals revenue minus cost,” which can

be written in function notation.

xis the number of items

produced and sold.

Profit Revenue Cost

function function function

The profit function is found by subtracting the cost function from the revenue function.

We define the following operations on functions.

P 1 x 2 = R 1 x 2 - C 1 x 2

Adding and Subtracting Functions

If and define functions, then

Sum function

and Difference function

In each case, the domain of the new function is the intersection of the domains of

ƒ 1 x 2 and g 1 x 2.

1 ƒg 21 x 2 ƒ 1 x 2 g 1 x 2.

1 ƒg 21 x 2 ƒ 1 x 2 g 1 x 2

ƒ 1 x 2 g 1 x 2

Adding and Subtracting Functions

Find each of the following for the polynomial functions defined by

and

(a)

Use the definition.

Substitute.

=- 2 x^2 - 10 x+ 14 Add the polynomials.

= 1 x^2 - 3 x+ 72 + 1 - 3 x^2 - 7 x+ 72

=ƒ 1 x 2 + g 1 x 2

1 ƒ+ g 21 x 2

ƒ 1 x 2 = x^2 - 3 x+ 7 g 1 x 2 =- 3 x^2 - 7 x+ 7.

EXAMPLE 3

This notation does

notindicate the

distributive property.

NOW TRY ANSWER

2. about 48.2 million