# 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.

#### 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 -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

##### Solution

Begin by setting the denominator equal to zero and solving.

The denominator is equal to zero when . The domain of the function is all real numbers except .

##### Analysis

A graph of this function, as shown in Figure 8, confirms that the function is not defined when .

**Figure 8**

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

#### TRY IT #4

Source: Rice University, https://openstax.org/books/college-algebra/pages/5-6-rational-functions

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