Answer
Domain of the given function is
$$
\left\{(x, y) | -1 \leq (x+y) \leq 1 \right\}.
$$
Work Step by Step
$$
f(x,y)=arcsin (x+y)
$$
This function is defined only when:
$$
-1 \leq (x+y) \leq 1
$$
Thus the domain of $f $ is
$$
\left\{(x, y) | -1 \leq (x+y) \leq 1 \right\}.
$$
