Domain and Range 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.

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 $f(x)=\frac{4 x}{5(x-1)(x-5)}$.

Source: Rice University, https://openstax.org/books/college-algebra/pages/5-6-rational-functions
This work is licensed under a Creative Commons Attribution 4.0 License.