Intermediate Algebra (11th edition)

(Marvins-Underground-K-12) #1

362 CHAPTER 7 Rational Expressions and Functions


OBJECTIVES OBJECTIVE 1 Define rational expressions.In arithmetic, a rational number is


the quotient of two integers, with the denominator not 0. In algebra, a rational expres-


sion,or algebraic fraction,is the quotient of two polynomials, again with the denomi-


nator not 0.


and x^5 aor


x^5


1


b


8 x^2 - 2 x+ 5


4 x^2 + 5 x


,


m+ 4


m- 2


,


- a


4


,


x


y


,


Rational Expressions and Functions; Multiplying and Dividing


7.1


1 Define rational
expressions.
2 Define rational
functions and
describe their
domains.
3 Write rational
expressions in
lowest terms.
4 Multiply rational
expressions.
5 Find reciprocals of
rational expressions.
6 Divide rational
expressions.

Examples of rational
expressions

Rational expressions are elements of the set


OBJECTIVE 2 Define rational functions and describe their domains.A


function that is defined by a quotient of polynomials is called a rational functionand


has the form


where


The domain of a rational function consists of all real numbers except those that make


— that is, the denominator — equal to 0. For example, the domain of


Cannot equal 0

includes all real numbers except 5, because 5 would make the denominator equal to 0.


FIGURE 1shows a graph of the function defined by Notice that the


graph does not exist when (It does not intersect the dashed vertical line whose


equation is ) We discuss graphs of rational functions in Section 7.4.


Finding Domains of Rational Functions

For each rational function, find all numbers that are not in the domain. Then give the


domain, using set-builder notation.


(a)


The only values that cannot be used for xare those that make the denominator 0.


To find these values, set the denominator equal to 0 and solve the resulting equation.


Add 14.
Divide by 7.

The number 2 cannot be used as a replacement for x. The domain of ƒ includes all real


numbers except 2, written using set-builder notation as 5 x|xZ 26.


x= 2


7 x= 14


7 x- 14 = 0


ƒ 1 x 2 =


3


7 x- 14


EXAMPLE 1


x=5.


x= 5.


ƒ 1 x 2 =x^2 - 5.


ƒ 1 x 2 =


2


x- 5


Q 1 x 2


ƒ 1 x 2  Q 1 x 2 Z0.


P 1 x 2


Q 1 x 2


,


e


P


Q


` P and Q are polynomials, with Q 0 f.


⎧⎨⎩


x

y

0
5

2

x = 5

f(x) =x – 5^2

FIGURE 1
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