Given $F(x) = L(x)G(x)$, with $L$ and $G$ real function strictly greater than zero. Suppose that F and G are decreasing functions (so that $F^{-1}$ and $G^{-1}$ exists). What can we say about the inverse of $F$?
$$F^{-1}(y)$$
Do some representations of this type $$F^{-1} = H G^{-1}$$ exist, for some $H$? (and which would be the link between $L$ and $H$?) Please report more results as you can, even with stronger assumptions on what you want. Thanks. $$$$ And last but not least, numerically speaking, is it a difficult problem? Suppose I can invert numerically $F$ and $G$, so that I can find $H$. Do I have some info about $L$? What can I do?