inverse function theorem problem

Question

I am interested in finding a formula for the inverse of the function $$f(x,y) = (x^2 + y^2, xy)$$ which works on the set $S = \{(x,y) \in \mathbb{R}^2 : -x < y < x\}$. I determined that the inverse should be $$f^{-1}(x,y) = \left(\frac{\sqrt{x + 2y} + \sqrt{x - 2y}}{2}, \frac{\sqrt{x + 2y} - \sqrt{x - 2y}}{2}\right),$$ but $S$ is not contained in the domain of $f^{-1}$. I know the function has a local inverse at all points in $S$ since the Jacobian is non-singular on $S$, but how do I find a formula that will work on all of $S$? Thanks!