Domain and Range of Rational Functions

Finding the Domains of Rational Functions

A vertical asymptote represents a value at which a rational function is undefined, so that value is not in the domain of the function. A reciprocal function cannot have values in its domain that cause the denominator to equal zero. In general, to find the domain of a rational function, we need to determine which inputs would cause division by zero.


The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.


Given a rational function, find the domain.

1. Set the denominator equal to zero.

2. Solve to find the x-values that cause the denominator to equal zero.

3. The domain is all real numbers except those found in Step 2.


Finding the Domain of a Rational Function

Find the domain of f(x)=\frac{x+3}{x^{2}-9}.


Begin by setting the denominator equal to zero and solving.

x^{2}-9 &=0 \\
x^{2} &=9 \\
x &=\pm 3

The denominator is equal to zero when x=\pm 3 . The domain of the function is all real numbers except x=\pm 3.


A graph of this function, as shown in Figure 8, confirms that the function is not defined when x=\pm 3 .

Figure 8

There is a vertical asymptote at x=3 and a hole in the graph at x=−3. We will discuss these types of holes in greater detail later in this section.


Find the domain of f(x)=\frac{4 x}{5(x-1)(x-5)}.

Source: Rice University,
Creative Commons License This work is licensed under a Creative Commons Attribution 4.0 License.