Domain of functions

Question

The example in my math text book has this explanation:

$g(x) = \sqrt{x+3}$ where $[-3, \infty)$ is the domain of this function.

Why is $-3$ part of the domain?

Answer 1

This is because $\sqrt x$ is defined for $x\ge 0$

and thus $\sqrt {g(x)}$ is defined for $g(x)\ge 0$.

In this case

$$\sqrt {x+3} \implies x+3\ge0 \iff x\ge -3$$

Otherwise

$$h(x) = \sqrt{\frac1{x+3}}$$ is defined for $x>-3$.