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