I was intrigued by this question Function composition and Bell polynomials where given that you have a final power series that resulted from compositing two others, how one could solve for what the original two series were.
The answer however is too brief for me to know exactly what it is saying. Is it saying such a technique is impossible? But the fact that the formula exists means there must be some kind of pattern to it. Perhaps as with integrating functions, there needs to be at least one other value of the original two functions known? I'm not sure. It says di Bruno's formula isn't helpful here. Okay, well, would about some trivial modification of it? How do we know there can't be any possible way at all to deduce the original $b_k$ terms? It doesn't seem the "answer" holistically addressed the problem.
I could see a possible issue with inverse relations over a domain since $f(f^{-1}(x)=f^{-1}(f(x))=f(f^{-1}(f(f^{-1}(x))...$, but seeing as how that situation is specific to only two functions, how can that ambiguity be resolved?