# What is the domain and range of y= -x/(x^2-1)?

Jan 28, 2018

$x \in \mathbb{R} , x \ne \pm 1$
$y \in \mathbb{R} , y \ne 0$

#### Explanation:

The denominator of y cannot be zero as this would make y undefined. Equating the denominator to zero and solving gives the values that x cannot be.

$\text{solve } {x}^{2} - 1 = 0 \Rightarrow \left(x - 1\right) \left(x + 1\right) = 0$

$\Rightarrow x = \pm 1 \leftarrow \textcolor{red}{\text{excluded values}}$

$\text{domain is } x \in \mathbb{R} , x \ne \pm 1$

$\text{divide terms on numerator/denominator by } {x}^{2}$

$y = \frac{\frac{x}{x} ^ 2}{{x}^{2} / {x}^{2} - \frac{1}{x} ^ 2} = \frac{\frac{1}{x}}{1 - \frac{1}{x} ^ 2}$

$\text{as } x \to \pm \infty , y \to \frac{0}{1 - 0}$

$\Rightarrow y = 0 \leftarrow \textcolor{red}{\text{excluded value}}$

$\text{range is } y \in \mathbb{R} , y \ne 0$
graph{-x/(x^2-1) [-10, 10, -5, 5]}