Domain and Range of Rational Functions

Domain of a Rational Function

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

HOW TO

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.

Example 4

Finding the Domain of a Rational Function

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

Solution

Begin by setting the denominator equal to zero and solving.

$\begin{aligned} x^{2}-9 &=0 \\ x^{2} &=9 \\ x &=\pm 3 \end{aligned}$

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

Analysis

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.

Try It #4

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