# How do you find the domain and range of y=x^2 - 5?

Oct 24, 2017

$- \infty < x < \infty$
$y \ge - 5$

#### Explanation:

The domain is the set of $x$ values a function can take to give a real $y$ value, which in the function $y = {x}^{2} - 5$ is simply any $x$ value. For instance, when $x = - 6$ then $y = 36 - 5 = 31$. Similarly, when $x = 1000$, then $y = 1000000 - 5 = 999995$.

Therefore, $- \infty < x < \infty , x \in \mathbb{R}$.

However, for $x \in \mathbb{R}$, ${x}^{2} \ge 0$. In other words, a square number is always positive (greater than 0), so a square number minus five must be always greater than minus five. So,

${x}^{2} \ge 0$

$\therefore$

${x}^{2} - 5 \ge - 5$

$\therefore$

$y \ge - 5$

This is the range of the function, which is defined as the set of $y$ values that can be taken by the function. You'll never find a (real) solution for anything less than $y = - 5$, for which $x = 0$.