I'm programming a piece of math that requires solving an equation of a form $f(x)/x=y$. Now I already have $f^{-1}(z)$ coded (efficiently, and not by me) so I'd prefer using this implementation instead of, say, coding bisection for $f(x)/x$. Is there a way to do that?
If it were $f(x)=y$, I would just use $x=f^{-1}(y)$. But with $f(x)/x=y$ I doubt it's even possible to have a closed-form solution with $f$ and $f^{-1}$ but I can't come up with a reasoning for that.
Not possible in general. For example, if $f(x) = e^x$ then $f^{-1}(x) = \log x$. But the solution $x$ of $$ \frac{e^x}{x} = y $$ is not elementary: $$ x = -W\left(\frac{-1}{y}\right) $$ where W is the Lambert W function. Knowing how to compute $e^x$ and $\log x$ does not tell you how to compute this solution.