Step 1 :Determine the domain of the function of two variables $f(x, y)=\sqrt{y+x}$.
Step 2 :The domain is the set of all $(x, y)$ that satisfy the inequality $x + y \geq 0$.
Step 3 :\(\boxed{\text{The domain of the function is }\{(x, y) | x + y \geq 0\}}\)