Section 5.2Properties of Rational Functions

Objectives• Find the Domain of a Rational Function

• Determine the Vertical Asymptotes of a Rational Function

• Determine the Horizontal or Oblique Asymptotes of a Rational Function

A rational function is a function of the form

R xp xq x

( )( )( )

where p and q are polynomial functions and q is not the zero polynomial. The domain consists of all real numbers except those for which the denominator q is 0.

Find the domain of the following rational functions.

128

1)( (a)

2

xx

xxR 26

1

xx

x

All real numbers x except -6 and -2.

16

4)( (b)

2

x

xxR 44

4

xx

x

All real numbers x except -4 and 4.

9

5)( (c)

2

xxR

All Real Numbers

Recall that the graph of isxxf

1)(

(1,1)

(-1,-1)

Graph the function using transformations

12

1)(

xxf

(1,1)

(-1,-1)x

xf1

)(

(3,1)

(1,-1)(2,0)

2

1)(

xxf

(3,2)

(1,0)(2,0)

(0,1)

12

1)(

xxf

If, as x or as x - , the values of R(x) approach some fixed number L, then the line y = L is a horizontal asymptote of the graph of R.

If, as x approaches some number c, the values |R(x)| , then the line x = c is a vertical asymptote of the graph of R.

(3,2)

(1,0)(2,0)

(0,1)

12

1)(

xxf

In the previous example, there was a vertical asymptote at x = 2 and a horizontal asymptote at y = 1.

Examples of Horizontal Asymptotes

y = L

y = R(x)y

x

y = L

y = R(x)

y

x

Examples of Vertical Asymptotes:

x = cy

x

x = c

y

x

If an asymptote is neither horizontal nor vertical it is called oblique.

y

x

Theorem: Locating Vertical AsymptotesA rational function R(x) = p(x) / q(x), in lowest terms, will have a vertical asymptote x = r, if x - r is a factor of the denominator q.

Example: Find the vertical asymptotes, if any, of the graph of each rational function.

1

3)( (a)

2

xxR

)1)(1(

3

xx

Vertical asymptotes: x = -1 and x = 1

1

5)( (b)

2

x

xxR

No vertical asymptotes

12

3)( (c)

2

xx

xxR

)4)(3(

3

xx

x4

1

x

Vertical asymptote: x = -4

01

1

1

01

1

1

)(

)()(

bxbxbxb

axaxaxa

xq

xpxR

m

m

m

m

n

n

n

n

Consider the rational function

in which the degree of the numerator is n and the degree of the denominator is m.

1. If n < m, then y = 0 is a horizontal asymptote of the graph of R.2. If n = m, then y = an / bm is a horizontal asymptote of the graph of R.

3. If n = m + 1, then y = ax + b is an oblique asymptote of the graph of R. Found using long division.

4. If n > m + 1, the graph of R has neither a horizontal nor oblique asymptote. End behavior found using long division.

Example: Find the horizontal or oblique asymptotes, if any, of the graph of

174

1543)( (a)

23

2

xxx

xxxR

Horizontal asymptote: y = 0

53

142)( (b)

2

2

xx

xxxR

Horizontal asymptote: y = 2/3

2

14)( (c)

2

x

xxxR

13

126-

16

2-

6 142

2

2

x

x

xx

xxxx

Oblique asymptote: y = x + 6