Assume $f \in C^1(\mathbb{R}^2)$ such that $f(0,0)=0$, $f_x(0,0) \ne −1$, and $f_y(0,0) \ne 0$. Assume $g : \mathbb{R}^2 → \mathbb{R}$ such that $$g(x,y) = f(f(x,y),y).$$ Let $t$ be a function defined on an open interval around the origin s.t. $g(x, t(x)) = 0, \forall x$ in the open interval. Prove such function $t$ exists.
To find such a function, $g(x,t(x)) = f(f(x,y),t(x))=0$, but does this imply that $f(x,y)=0$ and $t(x)=0$ since no other points equal to $0$ around the origin for $f$, because $f_y \ne 0$.
I am stuck and also am not sure if there is a potential open interval theorem I can use to prove it. Or I have thinking perhaps inverse function thm.