Answer
The Domain is given by the set $\{(x,y)|-1\leq x+y\leq 1\}$.
The Range is given by the set (a segment) $[0,\pi]$.
Work Step by Step
1) Domain
The given function is $f(x,y)=\arccos(x+y)$.
Now, the $\arccos$ function is defined on the real segment $[-1,1]$. This means that any argument of $\arccos$ can only take values from that segment:
$$-1\leq x+y\leq 1.$$
We don't have any other constraints so the domain is simply the set of all ordered pairs $(x,y)$ such that $-1\leq x+y \leq 1$ and we will denote that set by $\{(x,y)|-1\leq x+y \leq 1\}$.
2) Range
The values of $\arccos$ function are angles between $0$ and $\pi$. Since we will have all the values from the domain of $\arccos$ (all numbers between $-1$ and $1$ can be expressed as a sum of some real $x$ and $y$) we can say that the range of $f(x,y)=\arccos(x+y)$ is the segment of the real line $[0,\pi]$.
