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Sullivan PreCalculus
Section 3.3
Rational Functions I
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
p( x)
R( x) 
q( 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.
x 1
x 1
(a) R ( x )  2

x  8 x  12
 x  6 x  2
All real numbers x except -6 and -2.
x4
x4

(b) R ( x )  2
 x  4 x  4
x  16
All real numbers x except -4 and 4.
5
(c) R ( x )  2
x 9
All Real Numbers
1
Recall that the graph of f ( x)  x
(1,1)
(-1,-1)
is
1
Graph the function f ( x )  x  2  1 using transformations
(1,1)
(3,1)
(2,0)
(1,-1)
(-1,-1)
f ( x) 
1
x
1
f ( x) 
x2
(3,2)
(0,1)
(2,0)
(1,0)
1
f (x) 
1
x2
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)
(0,1)
(2,0)
(1,0)
1
f (x) 
1
x2
In the previous example,
there was a vertical
asymptote at x = 2 and a
horizontal asymptote at
y = 1.
Examples of Horizontal Asymptotes
y
y = R(x)
y=L
x
y
y=L
x
y = R(x)
Examples of Vertical Asymptotes
x=c
y
x=c
y
x
x
If an asymptote is neither horizontal nor vertical
it is called oblique.
y
x
Theorem: Locating Vertical Asymptotes
A 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.
3
3

(a) R ( x )  2
x  1 ( x  1)( x  1)
Vertical asymptotes: x = -1 and x = 1
x 5
(b) R( x )  2
x 1
No vertical asymptotes
1
x3
x3


(c) R ( x )  2
x  x  12 ( x  3)( x  4) x  4
Vertical asymptote: x = -4
Consider the rational function
p ( x ) an x n  an 1 x n 1    a1 x  a0
R( x) 

q ( x ) bm x m  bm 1 x m 1    b1 x  b0
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
3x  4 x  15
(a) R ( x )  3
2
x  4x  7x  1
2
Horizontal asymptote: y = 0
2 x2  4 x  1
(b) R ( x ) 
3x 2  x  5
Horizontal asymptote: y = 2/3
x  4x  1
(c) R ( x ) 
x2
2
x6
x  2 x2  4 x  1
-  x 2  2 x
6x  1
- 6 x  12
13
Oblique asymptote: y = x + 6