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# Rational
Functions

A rational function is
a function of the form

_{}

where _{} are polynomial functions. Since division by
zero is undefined, the domain of a rational function is the set of
all real numbers, x, for which the denominator _{}

As an example let’s consider
the rational function

_{.}

Notice that the denominator
is equal to 0 when x = -2, so the domain is

_{}.

The graph is shown in figure
1.

Figure 1

Since the domain consists
of all real numbers _{}, the graph will never cross the vertical line _{}. We can see from the graph that, for values of
x close to –2 on the left, the function increases without bound and,
for values of x close to –2 on the right, the function decreases without
bound. We say that the line x = -2 is a vertical asymptote for the
graph of *f. *Also, we can see that as x increases or decreases
without bound, the values of _{}are getting closer to 1. The horizontal line y =
1 is called a horizontal asymptote for the graph of *f.*
The range is the set _{}

In general, the line _{}is a **vertical asymptote** for the graph of
a function _{}if either

_{}

from either the left side
or the right side.

The line _{} is a **horizontal asymptote** for the graph
of a function _{}if

_{}

A rational function,

_{},

will have a vertical asymptote
at _{}if *a* is a zero of the denominator _{}and not of the numerator_{}. That is, if _{}