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.

### A General Note: 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.

- Set the denominator equal to zero.
- Solve to find the
*x*-values that cause the denominator to equal zero. - 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 [latex]f\left(x\right)=\frac{x+3}{{x}^{2}-9}[/latex].

### Solution

Begin by setting the denominator equal to zero and solving.

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

### Try It 4

Find the domain of [latex]f\left(x\right)=\frac{4x}{5\left(x - 1\right)\left(x - 5\right)}[/latex].

## Analysis of the Solution

A graph of this function confirms that the function is not defined when [latex]x=\pm 3[/latex].

Figure 8There is a vertical asymptote at [latex]x=3[/latex] and a hole in the graph at [latex]x=-3[/latex]. We will discuss these types of holes in greater detail later in this section.