f(x+y)=x+y+4. It would appear that g(x,y)=f(x+y)=x+y+4.
This would seem to be the equation of a plane: x+y-z+4=0 where z=g(x,y).
There are no constraints on x and y so the domain of x and y are all real numbers and the range of z is infinite (z belongs to the set of real numbers). Given any value for one of the variables, there are always many possible values for the remaining two.
(If y=f(x) then f(x+y)=f(x+x+4)=f(2x+4)=2x+8=2(x+4)=2y. But the question doesn't define y in terms of x.)