Q. 9 - Precalculus Enhanced with Graphing Utilities

Question

Find the domain of each function. 

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

Step-by-Step Solution

Answer

The domain of the function $f\left(x\right)=\frac{x}{x^2+2x-3}$ is $\left\{x\right|x\ne -3,1\}$.

Step 1. Given information

We have been given a function $f\left(x\right)=\frac{x}{x^2+2x-3}$.

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 $x$.

$x^2+2x-3=0$

Factor the trinomial.

$$\begin{gathered} x^2+2x-3 \\ =x^2+3x-x-3 \\ =x(x+3)-1(x+3) \\ =(x+3)(x-1) \end{gathered}$$

$$\begin{gathered} (x+3)(x-1)=0 \\ x+3=0 \mathrm{or} x-1=0 \\ x=-3,1 \end{gathered}$$

Therefore, the domain will be $\left\{x\right|x\ne -3,1\}$.