Suppose we have a functional question, such as the one here. Often a key step in these problems is to try to equate or chain results together (see my hint). From looking at these questions, it seems that there is often only a finite number of ways that we can attempt to join them together. Can this fact be examined more formally?
Note: I am not asking you to turn my answer from that question into something algebraic. It is that way purposely as I was only trying to give a hint, not a complete solution. What I am asking, is whether there are any results that show that the functions can only be "combined" or "equated" in a certain number of ways on some reasonably general class of problems.