I have a function:
$$ f(x)=\int_0^m g(z)p(\frac{z}{x})\,dz $$
where $m > 0$ and $p(x)$ is an unknown probability distribution (where $x > 0$). $g(x)$ could be anything (I've been using $g(x)=\frac{1}{2+2x}$ as a generic not-super-trivial function, in case there are some restrictions on this function). I can query the distribution to get the PDF, CDF, and quantile.
My goal is to solve for $x$ given some output $f(x)=y$.
I've tried some pretty basic things like:
$$ y=g(m)C(\frac{m}{x})-\int_0^mg'(z)C(\frac{z}{x})\,dz $$
However, nothing I can think of seems to get me much closer than that to solving it.
I'm currently able to search over the domain to find an approximate answer, but I'm hoping there's a way to get a definite answer. Basically I don't really know if solving this is even possible or where I should be looking to figure that sort of thing out since it's just something I've come up with while playing with math in programming (with no real math background), so any information about where to look for an answer, or if it's not solvable either advice about what information I would need to get from the probability distribution in order to solve it, or an explanation of how it's impossible would be very helpful!
(This question seems to address something vaguely similar, but that's the only relevant thing I've found on my search.)