A rational function is a function that can
be written as the quotient of two polynomials. Any rational function
*r*(*x*) =
, where
*q*(*x*)
is not the zero polynomial. Because by
definition a rational function may have a variable in its denominator, the
domain and range of rational functions do not usually contain all the
real numbers.

There is special symbolism to describe the behavior of a function in certain
situations, depending on the behavior of the independent variable. In speaking
one might say a function approaches a certain value as
*x*
increases, decreases,
or approaches a certain value. To mathematically say "approaches," an arrow is
used. For example, to say that the function
*f* (*x*)
increases without bound as
*x*
increases without bound, one would write
*f* (*x*)âÜ’âàû
as
*x*âÜ’âàû
. Or to say the function
*f*
decreases without bound as
*x*
approaches
0
, you would write
*f* (*x*)âÜ’ - âàû
as
*x*âÜ’ 0
.

Rational functions often have what are called asymptotes. Asymptotes are
lines that functions approach but never reach. There are three kinds of
asymptotes: vertical, horizontal, and oblique. A vertical asymptote is a line
with the equation
*x* = *h*
if
*f* (*x*)âÜ’±âàû
as
*x*âÜ’*h*
from either direction. A horizontal asymptote is a line with the equation
*y* = *k*
if
*f* (*x*)âÜ’*k*
as
*x*âÜ’±âàû
. Oblique asymptotes are
linear functions.

Study the graph below of the rational function
*f* (*x*) =
.

Figure %: The graph of
*f* (*x*) =
.

A line
*x* = *h*
is a vertical asymptote of a function
*f* (*x*) =
if
*p*(*h*)≠ 0
and
*q*(*h*) = 0
. This is the general form of all vertical
asymptotes of rational functions.

Horizontal asymptotes are a little trickier to understand. Let
*f* (*x*) =
. If the degree of
*p*
is less than that of
*q*
, then
*y* = 0
is a horizontal asymptote of
*f*
. If the degree of
*p*
is greater than
that of
*q*
, then
*f*
does not have a horizontal asymptote. If
*p*
and
*q*
have
the same degree, then the horizontal asymptote occurs at the line
*y* =
, where
*candd*
are the leading coefficients of
*p*
and
*q*
,
respectively.

An oblique asymptote occurs when the degree of the numerator function is one
greater than the degree of the denominator function. If this situation arises,
divide
*p*(*x*)
by
*q*(*x*)
using long division. The result will be
(
*x* + *k*) +
, where
*r*(*x*)
is the remainder. An oblique asymptote
will occur at
*y* =
*x* + *k*
.

One of the most important parts of working with rational functions is making sure that the numerator and denominator are completely factored, and that the common factors are canceled before you try to calculate any asymptotes. And also keep in mind that not all rational functions have asymptotes. We only focused on those that do because with long division, you can calculate which rational functions reduce to simple polynomials, and we already know how to deal with them.