Answer
Domain = $\{(x,y)|\quad 1\leq x^{2}+y^{2}\leq 3\}$
(see image)
Work Step by Step
Arcsin is defined for arguments in $[-1,1]$ so f is defined only when
$-1\leq x^{2}+y^{2}-2\leq 1\qquad/+2$
$1\leq x^{2}+y^{2}\leq 3$
This is the region between two concetric circles
$x^{2}+y^{2}=1$ and $x^{2}+y^{2}=3,$
one with radius 1, and the other with radius $\sqrt{3}$, both centered at the origin.
The circles themseles are included, so we graph both with solid lines.
Domain = $\{(x,y)|\quad 1\leq x^{2}+y^{2}\leq 3\}$
(see image)