 Solution

The inverse function theorem states that a $C^1$ map on an open subset of ${\mathbb{R}}^2$ has a $C^1$ local inverse near a point where its derivative is invertible; the inverse derivative at the image point is the inverse matrix of the original derivative.

Put $s=1+x+y$. Quotient differentiation givesThe entries are smooth and the determinant is nonzero everywhere on the stated domain, so the theorem supplies the required local inverse at every point. The point $(1/3,1/3)$ is the image of $(1,1)$, whereConsequentlyDirectly solving the two rational equations also gives $f^{-1}(u,v)=(u/(1-u-v),v/(1-u-v))$ on $u+v\ne 1$, consistent with the derivative calculation.