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

May 13, 2017

Domain: $x \in \mathbb{R} \textcolor{w h i t e}{\text{XXX}}$or, in interval notation, $\textcolor{w h i t e}{\text{XXX}} x \in \left(- \infty , + \infty\right)$
Range: $y \in \left[- 2 , + \infty\right)$

#### Explanation:

I have assumed that the Universe of Discourse is the set of Real numbers ($\mathbb{R}$); the answer will be a bit different if you are working in Complex ($\mathbb{C}$) numbers.

${x}^{2} - 2$ is defined for all Real values of $x$
$\Rightarrow$ the Domain is all Real values.

Since (for Real values of $x$) ${x}^{2} \ge 0$
$\Rightarrow y = {x}^{2} - 2 \ge - 2$
and
since ${x}^{2} - 2 \rightarrow + \infty$ as $x \rightarrow \pm \infty$
$\Rightarrow y$'s upper limit is $+ \infty$
$\Rightarrow$ the Range is $\left[- 2 , + \infty\right)$