Inverse of a 2 variable function

Question

I have shown that $f(x,y)=(x^2-y^2,2xy)$ is invertible for all $x\neq 0, y \neq 0$, but I need to find the inverse function. The Jacobi Matrix of the inverse is $$\begin{bmatrix} \frac{x}{2(x^2+y^2)} & \frac{y}{2(x^2+y^2)}\\ \frac{-y}{2(x^2+y^2)} & \frac{x}{2(x^2+y^2)} \\ \end{bmatrix}$$

I tried to integrate the first and second component in order to find the x component and y component of my inverse function, but it didn't work out, can someone help me? Thanks

Answer 1

The function is not invertible, since it is not injective. Note that $f(x,y)=f(-x,-y)$.