# Question #8cbdf

Sep 23, 2016

Domain: $\left[0 , \infty\right)$

Range: $\left[0 , \infty\right)$

#### Explanation:

Assuming we are restricted to $\mathbb{R}$ (the real numbers), the principal square root function $\sqrt{\cdot}$ has a domain $\left[0 , \infty\right)$ and a range $\left[0 , \infty\right)$.

No negative values are in the domain, as the square root of a negative value is an imaginary number.

(specifically, if $a > 0$, then $\sqrt{- a} = \sqrt{a} i$)

No negative values are in the range, as the square of a negative is a positive, and the principal square root of that positive is defined as its positive real root.

We can see the domain and range clearly when examining the graph $y = \sqrt{x}$ by noticing that it follows the restrictions $x \ge 0$ and $y \ge 0$.

graph{sqrt(x) [-10, 10, -5, 5]}