# 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