4.1. RATIONAL FUNCTIONS AND
ASYMPTOTES
1
What You Should Learn
• Find the domains of rational functions.
• Find the vertical and horizontal asymptotes of
graphs of rational functions.
• Use rational functions to model and solve
real-life problems.
2
Introduction
A rational function is a quotient of polynomial functions. It
can be written in the form
where N(x) and D(x) are polynomials and D(x) is not the
zero polynomial.
In general, the domain of a rational function of x includes all
real numbers except x-values that make the denominator
zero.
Much of the discussion of rational functions will focus on
their graphical behavior near the x-values excluded from
the domain.
3
Example
Finding the Domain of a Rational Function
Find the domain of
and discuss the behavior of f
near any excluded x-values.
4
Solution
Because the denominator is zero when x = 0, the domain
of f is all real numbers except x = 0.
To determine the behavior of f near this excluded value,
evaluate f (x) to the left and right of x = 0, as indicated in the
following tables.
5
Solution
Note that as x approaches 0 from the left, f (x) decreases
without bound.
In contrast, as x approaches 0 from the right, f (x) increases
without bound.
The graph of f is shown in Figure 4.1.
Figure 4.1
6
Vertical and Horizontal Asymptotes
In the previous Example, the behavior of f near x = 0 is
denoted as follows.
The line x = 0 is a vertical asymptote
of the graph of f, as shown in Figure 4.2.
Figure 4.2
7
Vertical and Horizontal Asymptotes
From this figure, you can see that the graph of f also has a
horizontal asymptote—the line y = 0.
This means that the values of f (x) = 1/x approach zero
as x increases or decreases without bound.
8
Vertical and Horizontal Asymptotes
Eventually (as x 
or x 
), the distance between
the horizontal asymptote and the points on the graph must
approach zero.
9
Vertical and Horizontal Asymptotes
Figure 4.3 shows the vertical and horizontal asymptotes of
the graphs of three rational functions.
(a)
(b)
(c)
Figure 4.3
10
Vertical and Horizontal Asymptotes
The graphs of f (x) = 1/x in Figure 4.2 and
f (x) = (2x + 1)/(x + 1) in Figure 4.3(a) are hyperbolas.
Figure 4.2
11
Vertical and Horizontal Asymptotes
12
Example
Finding Vertical and Horizontal Asymptotes
Find all vertical and horizontal asymptotes of the graph of
each rational function.
a.
b.
13
Solution
a. For this rational function, the degree of the numerator is
less than the degree of the denominator, so the graph
has the line y = 0 as a horizontal asymptote.
To find any vertical asymptotes, set the denominator
equal to zero and solve the resulting equation for x.
Because the equation 3x2 + 1 = 0
has no real solutions,
you can conclude that the graph
has no vertical asymptote.
14
Solution
b. For this rational function, the degree of the numerator is
equal to the degree of the denominator.
The leading coefficient of both the numerator is 2 and
the leading coefficient of the denominator is 1, so the
graph has the line y = 2 as a horizontal asymptote.
To find any vertical asymptotes, set the denominator
equal to zero and solve the resulting equation for x.
x2 – 1 = 0
Set denominator equal to zero.
(x + 1)(x – 1) = 0
x+1=0
Factor.
x = –1
Set 1st factor equal to 0.
15
Solution
x–1=0
x=1
Set 2nd factor equal to 0.
This equation has two real solutions, x = –1 and x = 1, so
the graph has the lines x = –1 and x = 1 as vertical
asymptotes.
16