When given a set of ordered pairs and asked for the domain and range you should first write down the ordered pairs for each point that is given in the relation. Once you write down all of the ordered pairs you can use them to identify your domain range. Remember the domain is all of your X values and the ranges all of your Y values. In this case your ordered pairs would be (-4, 2)(-3, 0), (-1, 3), (0, 2) and (4, -2). From there you can write your domain which is domain:{-4, -3, -1, 0, 4} and the range:{-2, 0, 2, 3}. Be sure to list any repeated value only once.

There are two things you want to avoid in the domain of $f$.

1) $x^2+y^2+4<0$

2) $0=v = y-x$

But $x^2+y^2 +4 \geq 4 >0$ so this shouldn't be a problem. However if $x=y$ then $v=0$ so the domain is $D=\{(x,y)\mid x \neq y\}$.

Edit Here is a "picture" of $D$ as asked in the comment. $D$ is basically everything except the points that are on the red line. Note that this picture only represents a little part of $D$ since it is an unbounded domain (e.g. $(10^{2137612},\pi)$ belongs to $D$ but is not on the picture).