Question

Find the domain of each function.

$f\left(x\right)=\frac{x}{x^2-9}$

Short answer

The domain of the function is $\left\{x\right|x\ne -3,3\}$.

Step-by-step solution

Step 1. Given information

We have been given a function $f\left(x\right)=\frac{x}{x^2-9}$.

We have to find its domain.

Step 2. Find the values where the denominator becomes zero

The given function will not be defined where the denominator becomes zero.

We need to exclude those values of $x$.

$$\begin{gathered} x^2-9=0 \\ {x}^2-9+9=0+9(\mathrm{Add}9\mathrm{on}\mathrm{both}\mathrm{sides}) \\ {x}^2=9 \\ x=\pm \sqrt{9}(\mathrm{Take}\mathrm{square}\mathrm{root}\mathrm{on}\mathrm{both}\mathrm{sides}) \\ x=\pm 3 \end{gathered}$$

Therefore, the domain will be $\left\{x\right|x\ne -3,3\}$.