Find 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.

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.

[latex]\begin{cases} {x}^{2}-9=0 \hfill \\ \text{ }{x}^{2}=9\hfill \\ \text{ }x=\pm 3\hfill \end{cases}\\[/latex]

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].

Analysis of the Solution

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

Figure 8

There 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.

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