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
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