How to find the inverse of this multivariable function?

Question

I have a function $f(1+x,1-x)=(e^y+xe^{-y},e^y-xe^{-y})$ where $x,y\in\mathbb{R}$. I can show that the function is smooth, but I want to show that the inverse is also smooth. How do I construct the inverse?

Thanks in advance.