Show that $F$ is not a one-to-one transformation

Question

Given

$$F(x,y)=(x-y,y^2-x-2)=(u,v),$$

how to show that this transformation is not one-to-one? And at which points $F$ is locally one to one?

While I was drawing this transformation I found that the effect of $F$ on some horizontal lines ($y=-1$ and $y=-2$) are the same. Does this demonstrate that the transformation is not one-to-one?

Also, the effect of $F$ on vertical lines are one-to-one locally, I think.

Answer 1

Note that $F$ is not injective because $F(0,0)=F(1,1)$.

The inverse function theorem implies $F$ is locally invertible at a point $p$ if the Jacobian $J$ of $F$ at $p$ is invertible. Since $$ \det J = \det \begin{bmatrix} 1 & -1 \\ -1 & 2\,y \end{bmatrix} =2\,y-1 $$ we see that this ensures that $F$ is locally invertible at every point $p=(x,y)$ where $y\neq 1/2$.

Can you determine if $F$ is locally invertible when $y=1/2$?