Let $A: \mathbb{R^2} \to \mathbb{R^2}$ be a linear map and consider $f:\mathbb{R^2} \to \mathbb{R}$ such that $\nabla f(x,y) \neq (0,0); \forall (x,y) \in \mathbb{R^2}$. Determine sufficient conditions that $A$ must obey such that $\forall (x_0,y_0) \in N_{(0,0)}(f)$, the equation $f \circ A(x,y) =0$ must define, in a neighborhood of $(x_0,y_0)$, $x$ as a function of $y$ or $y$ as a function of $x$.
I have that $J{(f \circ A)}(x,y) = Jf(A(x,y)) JA(x,y)=Jf(A(x,y)) A(x,y)$. Now $$J{f(A(x,y))} = \begin{pmatrix} \frac{\partial f}{\partial x}_{A(x,y)} & \frac{\partial f}{\partial y}_{A(x,y)} \end{pmatrix} $$ and let $$A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} $$
Now $$\frac{\partial(f \circ A)}{\partial x}_{|(x,y)}= \frac{\partial f}{\partial x}_{|A(x,y)}a_{11} + \frac{\partial f}{\partial y}_{|A(x,y)}a_{21}$$ and $$\frac{\partial(f \circ A)}{\partial y}_{|(x,y)}= \frac{\partial f}{\partial x}_{|A(x,y)}a_{12} + \frac{\partial f}{\partial y}_{|A(x,y)}a_{22}$$ To define $x$ as a function of $y$, I could consider that $a_{11};a_{21}>0$ and to define $y$ as function of $x$, I could have that $a_{12},a_{22}>0$. Is this approach valid?