# How do you find the domain and range for y=sqrt(x+4)?

Jan 4, 2016

Domain: $x \ge \left(- 4\right)$ (could also be written as: $x \in \left[- 4 , + \infty\right)$ )
Range: $y \ge 0$ could also be written as: $y \in \left[0 , + \infty\right)$ )

#### Explanation:

To determine the domain we need to ask:
$\textcolor{w h i t e}{\text{XXX}}$For what values of $x$ is $\sqrt{x + 4}$ valid?

We know that $\sqrt{\text{something}}$ is valid if and only if $\text{something} \ge 0$ (assuming we are restricted to Real numbers).

Therefore $\left(x + 4\right)$ must be $\ge 0$
$\Rightarrow x \ge - 4$

To determine the range we need to ask:
$\textcolor{w h i t e}{\text{XXX}}$What values can be generated using legal values of $x$ by the expression $\sqrt{x + 4}$

When $x = - 4$
$\textcolor{w h i t e}{\text{XXX}} \sqrt{x + 4} = 0$
When $x > - 4$
$\textcolor{w h i t e}{\text{XXX}} \sqrt{x + 4} > 0$ with no upper limit as $x \rightarrow + \infty$