
Rational Functions (Definition)
Definition
of a Rational Function
A rational function is a quotient of polynomials
P(x)/Q(x) 
Examples

(x^{2} + x  1)/(3x^{3 }+ 1)

(x  1)/(x^{2} +1)

x^{2}/(x + 1)
Example
Find the domain of
(x^{2} +
1)/(x^{2} 1)
The domain of this rational function is the set
of all real numbers that do not make the denominator zero. We find
x^{2}
1 = 0
solving
x = 1,
or x = 1
So that the domain is
{x  x is
not 1 or 1}

Vertical Asymptotes
Definition of a Vertical Asymptote
A Vertical Asymptote of a rational function occurs where the denominator
is 0. 
Example: Graph the vertical asymptotes of
y = f(x) =
(x^{2} + 1)/(x^{2} 1)
Solution
From the last example, we see that there are vertical asymptotes at 1 and 1.
Since f(x) is positive a little to the left of 1,
we say that
as
x > 1^{}
("x goes to 1 from the left")
f(x)
>
Similarly since f(x) is negative a little to the
right of 1, we say that
as
x> 1^{+}
("x goes to 1 from the right")
f(x) > 
Since f(x) is negative a little to the left of
1,
as
x > 1^{}
f(x)
>

Similarly since f(x) is positive a little to the
right of 1,
as
x> 1^{+}
f(x) >

Four Types of Vertical Asymptotes
Below are the four types of vertical asymptotes:

Horizontal Asymptotes
Example:
Consider the rational function
f(x) = (3x^{2} + x  1)/(x^{2}  x  2)
For the numerator, the term 3x^{2} dominates when
x is large, while
for the denominator, the term x^{2} dominates when x is large.
Hence
as
x >
3 is called the horizontal asymptote and we have the the left and right behavior
of the graph is a horizontal line y = 3.

Oblique Asymptotes
Consider the function
f(x) = (x^{2}  3x  4)/(x + 3)
f(x) does not have a horizontal asymptote, since
x^{2 }/
x = x
is not
a constant, but we see (on the calculator) that the left and right behavior
of the curve is like a line. Our goal is to find the equation of this
line.
We use synthetic division to see that
(x^{2}  3x  4)/(x + 3) = x  6 + 14/(x + 3)
For very large x,
14/x + 3
is very small, hence
f(x) is approximately equal to
x  6
on the far left and far right of the
graph.
We call this line an Oblique Asymptote
To graph by have we see that there is a vertical asymptote at
x = 3
with
behavior:
left down and right
up
The graph has xintercepts at 4 and 1, and
a y intercept at 4/3.
Exercise:
Graph
(x^{3} + 8)/(x^{2}  3x  4)

Rational Functions With Common Factors
Consider the graph of
y = (x  1)/(x  1)
What
is wrong with the picture?
When
f(x) = g(x)(x  r)/h(x)(x  r)
with neither g(r) nor h(r) zero,
the graph will have a hole at x =
r. We call this hole a removable
discontinuity.
Example:
Graph
f(x) = (x^{2}  2)/(x^{2}  x  2) = (x  2)(x + 2)/(x
 2)(x + 1)
This graph will have a vertical asymptote at x 1 and a
hole at (2,2)
We end our discussion with a list of steps for graphing rational functions.
Steps in graphing rational functions:
Step 1 Plug in x = 0 to find the yintercept
Step 2 Factor the numerator and denominator. Cancel any
common factors remember to put in the appropriate holes if necessary.
Step 3 Set the numerator = 0 to find the xintercepts
Step 4 Set the denominator = 0 to find the vertical asymptotes.
Then plug in nearby values to fine the left and right behavior of the
vertical asymptotes.
Step 5 If the degree of the numerator = degree of the
denominator then the graph has a horizontal asymptote. To determine
the value of the horizontal asymptote, divide the term highest power of the
numerator by the term of highest power of the denominator.
If the degree of the numerator = degree of the denominator + 1 then use
polynomial or synthetic division to determine the equation of the oblique
asymptote.
Step 6 Graph it!