Does the inverse of a function from $\mathbb{R}^n$ to $\mathbb{R}^m$ exist? Suppose $T:\mathbb{R}^2 \to \mathbb{R}^3$ is defined by $T(x,y) = (x,y,xy).$  What then is the inverse of $T$?  Is it $$T^{-1}: \mathbb{R}^3 \to \mathbb{R}^2; (a,b,c) \mapsto (a,b)?$$  It is clear that, in this case, $T^{-1}(T(x,y)) = (x,y).$ But then $$T(T^{-1}(x,y,z)) = (x, y, xy) \neq (x,y,z).$$  It doesn't seem that this function has an inverse.  This problem seems to extend to any map from $\mathbb{R}^n \to \mathbb{R}^m$ when $n \neq m.$  When are such functions invertible?

Hint: The inverse show be "pulling" things back from $\bf{R}^3$, so the domain can't be $\bf{R}^2$.
Edit by "sea turtles": We can just restrict the domain of the inverse like so: Take $E=\{(x,y,z):z=xy\}$ and take $(T^{-1}|_E)(x,y,xy)$.
