Copyright 2013, 2009, 2005, 2001, Pearson, Education, Inc.

7.1

Rational Functions and Simplifying

Rational Expressions

Rational Expressions

Q

PRational expressions can be written in the form where P and Q are both polynomials and Q 0.

Examples of Rational Expressions

54

423 2

x

xx22 432

34

yxyx

yx

4

3 2x

A rational expression is undefined if the denominator is 0.

Example:

The rational expression is undefined when x = 3.

Domains of Rational Functions

2 2

3

x

x

Are there any values for x for which each rational expression is undefined?

4515

49 3

x

xx

The expression is undefined when 15x + 45 = 0.

So the expression is undefined when x = 3.

Example

Find the domain of each rational function.

a.

b.

c.

Example

3 28 7 20

3

x x

25 3

1

x

x

2

7 2

2 15

x

x x

{ | is a real number}x x

{ | is a real number and 1}x x x

{ | is a real number and 5, 3}x x x x

Simplifying a rational expression means writing it in lowest terms or simplest form.

Simplifying a Rational Expression

1) Completely factor the numerator and denominator.

2) Divide out factors common to the numerator and denominator. (This is the same as “removing a factor of 1.”)

Simplifying Rational Expressions

Simplify:

xx

x

5

3572

)5(

)5(7

xx

x

x

7

Example

2

7 35

5

x

x x

Simplify:

20

432

2

xx

xx

)4)(5(

)1)(4(

xx

xx

5

1

x

x

Example

2

2

3 4

20

x x

x x

Simplify:

7

7

y

y 7

)7(1

y

y1

Example

7

7

y

y

Simplify:

Example2

2

4

3 5 2

x

x x

2

2

4 (2 )(2 )

3 5 2 ( 2)(3 1)

x x x

x x x x

1( 2)(2 )

( 2)(3 1)

x x

x x

1( 2)x

(2 )

( 2)

x

x

(3 1)x

1(2 ) 2 or

(3 1) 3 1

x x

x x

For the ICL Production Company, the rational function describes the company’s cost per disc of pressing x compact discs. Find the cost per disc for pressing

a. 100 compact discs b. 1000 compact discs

2.6 10,000( )

xC x

x

Example

2.6 100 10,000 10,260(100) 102.6

100 100C

a.

The cost is $102.60.

b. 2.6 1000 10,000 12,600(1000) 12.6

1000 1000C

The cost is

$12.60.