130 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.
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].
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.