Question

The function F : R^2 → R defined by F(x,y)= (y + 2 , x + y) is locally invertible at any (x, y)∈R^2.

Answer 1

The function is locally invertible if the Jacobian is not zero.

$\begin{vmatrix} (y+20)_x & (x+y)_x \\ (y+2)_y & (x+y)_y \end{vmatrix} = \begin{vmatrix} 0 & 1 \\ 1 & 1 \end{vmatrix} =0-1=-1\ne 0 \forall x,y\in R$ Thus, the function is locally invertible.