## Domain and Range of Rational Functions

Domain and range are essential for rational functions since some inputs make them undefined. It is crucial to understand how to define the domain and range of rational functions because it allows you to determine asymptotes and the long-run behavior 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.

#### 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