Consider the function
$$ y = f(x,w) $$
I am looking for "proper" notation for the "partial inverse" of $f(\cdot)$ with respect to each variable. Something like:
$$ x = f^{-1}(y,w) $$
$$ w = f^{-1}(x,y) $$
But this does not differentiate them. Perhaps
$$ x = f_x^{-1}(y,w) $$
$$ w = f_w^{-1}(x,y) $$
But this might be confused with derivative of the inverse.
Actually, it seems using "inverse" notation is not proper (see chat for full discussion).
I can always revert to new functions, like
$$ x = h(y,w) $$
$$ w = g(x,y) $$
but this is inferior in two ways: it does not show the direct relation between $f(\cdot)$, $h(\cdot)$ and $g(\cdot)$; and it introduces more notation. I'm looking for economy and efficacy.
So, the question is:
What is the notation used to express the "partial inverse" of a multivariate function, like above?