Question

Find the domain of each function.

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

Short answer

The domain of the function$f\left(x\right)=\frac{x}{x^2+2x-3}$is role="math" localid="1645961847696" $\left\{x\right|x\ne -3,1\}$.

Step-by-step solution

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 $\left\{x\right|x\ne -3,1\}$.