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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.