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Study Guides > MATH 1314: College Algebra

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.

  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)=x+3x29f\left(x\right)=\frac{x+3}{{x}^{2}-9}\\.

Solution

Begin by setting the denominator equal to zero and solving.

{x29=0 x2=9 x=±3\begin{cases} {x}^{2}-9=0 \hfill \\ \text{ }{x}^{2}=9\hfill \\ \text{ }x=\pm 3\hfill \end{cases}\\

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

Analysis of the Solution

A graph of this function confirms that the function is not defined when x=±3x=\pm 3\\.

Graph of f(x)=1/(x-3) with its vertical asymptote at x=3 and its horizontal asymptote at y=0.Figure 8

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

Try It 4

Find the domain of f(x)=4x5(x1)(x5)f\left(x\right)=\frac{4x}{5\left(x - 1\right)\left(x - 5\right)}\\.

Solution

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