I have a function $\ f:\mathbb R\to\mathbb R$ such that $\ f(x,y)=(xe^y,xe^{-y}) $ Let $\ a=(1,0), b=(1,1) $ and let $\ g$ be the continuous inverse of $\ f$ such that $\ g(b)=a$. Compute $\ g$ explicitly and give an explicit neighborhood of b in the $\ uv$-plane in which $\ g=f^{-1}$
I have computed the Jacobian and it is nonzero so I can use the inverse function theorem. I'm just really stumped on how to compute $\ g$ explicitly.
$g(u,v) = \left(\sqrt{uv}, \frac{1}{2}\ln({\frac{u}{v}})\right)$ should work in a neighborhood of $b = (1,1)$