Suppose x'_i = h_i(x_1, \ldots, x_n) for $i = 1,...,n$. To apply the inverse function theorem, let $f: R^{n} \rightarrow R^n $ with $f^i(x_1,...,x_n) = h_i(x_1,...,x_n)$. Assume that the determinant of f is nonzero for all $x_i$, then there will exist a function g such that g_i(x'_1,...,x'_n) = x_i for x' close to (x'_1,...,x'_n).
But this only provides a localized inverse, right, there's no way for it to give the inverse everywhere? My classical mechanics textbook (corben) talks about this but it does not mention that the inverse is local and makes it sound like there will be a inverse that you can write down on paper. I don't see how the inverse function theorem is useful in coordinate transforms as the existence of local inverses seems pretty useless.