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

Nov 15, 2017

Domain = $\left\{x \in \mathbb{R} | x \ne \pm \sqrt{2}\right\}$
Range = $\left\{y \in \mathbb{R} | y \ne 0\right\}$

#### Explanation:

This will be undefined where the denominator is 0:

${x}^{2} - 2 = 0$
$\left(x - \sqrt{2}\right) \left(x + \sqrt{2}\right) = 0$
$x = \pm \sqrt{2}$

So the domain will consist of all points where x doesn't equal $\pm \sqrt{2}$

Domain = $\left\{x \in \mathbb{R} | x \ne \pm \sqrt{2}\right\}$

Since the denominator has a higher degree than the numerator, the rational equation has a horizontal asymptote at y = 0.

This means the range will consist of all points where y doesn't equal 0

Range = $\left\{y \in \mathbb{R} | y \ne 0\right\}$