How do you find the domain and range of y = (x^2 - 1) / (x+1)?

Apr 4, 2017

Domain: $x \in \left\{\mathbb{R} - \left\{- 1\right\}\right)$
Range: $y \in \left\{\mathbb{R} - \left\{- 2\right\}\right\}$

Explanation:

$\frac{{x}^{2} - 1}{x + 1}$ is defined for all value of $x$ except any value that would make the denominator equal to $0$.
That is $\left(x + 1\right) \ne 0 \textcolor{w h i t e}{\text{XX")rarrcolor(white)("XX}} x \ne - 1$
(This gives us the Domain).

Notice that if $x \ne - 1$
then $\frac{{x}^{2} - 1}{x + 1} = x - 1$
Any real number $r \in \mathbb{R}$ can be generated from $x - 1$ by picking a value of $x = r + 1$ except for the already excluded $\textcolor{b l a c k}{x = - 1}$ That is, we can not generate $r = x - 1$ when $x = - 1$. We can not generate $r = - 2$.
So the Range must exclude $\left\{- 2\right\}$