# How do you find domain and range of a rational function?

Oct 30, 2014

The domain of a rational function is all real numbers that make the denominator nonzero, which is fairly easy to find; however, the range of a rational function is not as easy to find as the domain. You will have to know the graph of the function to find its range.

Example 1

$f \left(x\right) = \frac{x}{{x}^{2} - 4}$

${x}^{2} - 4 = \left(x + 2\right) \left(x - 2\right) \ne 0 R i g h t a r r o w x \ne \pm 2$,

So, the domain of $f$ is

$\left(- \infty , - 2\right) \cup \left(- 2 , 2\right) \cup \left(2 , \infty\right)$.

The graph of $f \left(x\right)$ looks like: Since the middle piece spans from $- \infty$ to $+ \infty$, the range is $\left(- \infty , \infty\right)$.

Example 2

$g \left(x\right) = \frac{{x}^{2} + x}{{x}^{2} - 2 x - 3}$

${x}^{2} - 2 x - 3 = \left(x + 1\right) \left(x - 3\right) \ne 0 R i g h t a r r o w x \ne - 1 , 3$

So, the domain of $g$ is:

$\left(- \infty , - 1\right) \cup \left(- 1 , 3\right) \cup \left(3 , \infty\right)$.

The graph of $g \left(x\right)$ looks like this: Since $g$ never takes the values $\frac{1}{4}$ or $1$, the range of $g \left(x\right)$ is
$\left(- \infty , \frac{1}{4}\right) \cup \left(\frac{1}{4} , 1\right) \cup \left(1 , \infty\right)$.

I hope that this was helpful.