# How do you find the domain and range of sqrt{-x - 2}?

Jan 3, 2018

Domain: $\left\{x | x \le - 2\right\}$
Range: $y \in {\mathbb{R}}_{\ge} 0$

#### Explanation:

Finding the domain
The domain is wherever the function is defined with real numbers. We see that this function is not defined when the bit inside the square root is negative, so let's solve for when that happens:

$- x - 2 < 0$

Multiply both sides by $- 1$, and remember that the inequality flips when multiplying by a negative number:
$x + 2 > 0$

$x + \cancel{2 - 2} > 0 - 2$

$x > - 2$

So, our function is not defined when $x > - 2$. This means that our domain must be:
$\left\{x | x \le - 2\right\}$

Finding the Range
The square root function has values ranging from $0$ to $\infty$, and is continuous on everything in between, which means that the square root function has a range of the positive real numbers and $0$.

Our function's range is the same as the square root function, since the bit inside the square root is still continuous, so the domain of our function is also the positive real numbers and $0$, ${\mathbb{R}}_{\ge} 0$