Answer
Domain = $\{(x,y)|-|x|\leq y\leq|x|\}$
(see image)
Work Step by Step
Because of the restrictions for square roots, f is defined when
$x^{2}-y^{2}\geq 0$
$y^{2}\leq x^{2}$
$|y|\leq|x|$
... here we use the property: If $a\geq 0$ and $|y|\leq a$, then $-a\leq y\leq a$
$-|x|\leq y\leq|x|$
that is, for (x,y) from the $ \mathbb{R}^{2}$ plane such that the graph of $y=|x| $ is above the point, and the graph of $y=-|x| $ is below the point.
The graphs of $y=\pm|x|$ are included in the domain because of the sign $\leq$.
In the image, the graph of $y=|x|$ is blue, and the graph of $y=-|x|$ is red.
Shaded are the points that satisfy the inequality (the domain of f).
Domain = $\{(x,y)|-|x|\leq y\leq|x|\}$
(see image)
