Consider a parametrization of $x$ and $y$ by $t$ (all real variables), say $y=f(t)$, $x=g(t)$. Given a function $f$ and a function $h$, we would like to find the function $g$ such that $y=h(x)$. Perhaps this question is trivial, but I am too tired to think at the moment: find a nessecary and sufficient condition concerning $f,h$ such that such a $g$ exists.
For example, given $f=t^2,h=x$ we choose $g=t^2$.
$$ g(t) = x = h^{-1}(y) = h^{-1}(f(t)), $$ so $g = h^{-1}\circ f$.
As @abiessu says, this requires $h$ to be invertible.