Finding the inverse function of $g(x,y)$ for fixed $x$, in which the least eigenvalue of $\nabla_y g(x,y)$ is always greater than $1$.

Let $g(x,y)\in C^1(\mathbb{R}^n\times\mathbb{R}^n,\mathbb{R}^n)$ such that the least eigenvalue of $\nabla_y g(x,y)$ is always greater than $1$, $\nabla_y g(x,y)$ is always invertible and $\nabla_y g(x,y)$ is bounded. Can we find an inverse of $g(x,\cdot)$ for a fixed $x\in \mathbb{R}^n$? That is, some $g_x^{-1}(z)$ such that $g(x,g_x^{-1}(z))=z$? Moreover, is this $g_x^{-1}(z)$ defined global for $z \in \mathbb{R}^n$?

It seems to me that the inverse function theorem applies, but I do not know if $g(x,y)$ can take value on whole $\mathbb{R}^n$, and that inverse is global.