Define $f:\Bbb R^2\to \Bbb R^2$ by $$\begin{align}f(x,y)=(e^{2x+y},4x^2+4xy+y^2+6x+4y)\ . \end{align}$$ Define $U:=f(\Bbb R^2)$. Prove that the inverse function of $f$ (say $f^{-1}:U\to \Bbb R^2$) exists.
I want to apply the inverse function theorem. I know that $f$ is $C^1$ and the Jacorbian determinant $J_f(x,y)=2e^{2x+y}>0$ for all $(x,y)\in \Bbb R^2$. But I have a question. If I know that for all $(x,y)\in \Bbb R^2$, I can find a local inverse, does this imply $f$ has a inverse from its range $U$ onto $\Bbb R^2$? Or I need to find some ways to prove that $f$ is injective from $\Bbb R^2$ to $U$?
Local invertibility without injectivity cannot give you global invertibility. But here injectivity is easy to prove, as well as an explicit formula for the inverse. If $$(e^{2x+y}, 4x^2+4xy+y^2+6x+4y) = (a, b)$$ then $$y = \ln(a) - 2x$$ and you can check that $$b = 4x^2+4xy+y^2+6x+4y = - 2x + 4 \ln(a) + \ln(a)^2.$$ You can easily conclude with this.