Solution
= Solution
The <chain rule> gives
$$
DG(x,y)=\begin{pmatrix}
f_x(x,y)&f_y(x,y)\\
f_x(x,y^2)&2y f_y(x,y^2)
\end{pmatrix},\qquad DG(0,0)=\begin{pmatrix}1&1\\1&0\end{pmatrix}.
$$
Its <determinant> is $-1$, so it is invertible. The map $G$ is <smooth> and $G(0,0)=(0,0)$. The <inverse function theorem> therefore yields an <open neighbourhood> $U$ of the origin for which $G(U)$ is <open>, contains the origin, and
$$
\boxed{G:U\to G(U)\text{ is a diffeomorphism}.}
$$