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.