How to go about finding a function that satisfies $f(x)=-f(x-1)^{2}+2^{2^{x-3}}f(x-1)+2^{2^{x-2}}$ (Or determining if such a function exists)

I don't necessarily need an answer to this particular case, but in general I have no idea how to solve this kind of problem or even how to Google for such a method. If it helps for this particular problem, the context in which this expression came up only positive integer values of x $\mathit{really}$ makes sense, but if possible I would delight in finding a function that works for any real number.

(Apologies for the lack of tags, I have no idea what field this would fall under)

Solution 1:

This is a functional equation. Such equations are ubiquitous in recreational math and math competitions, but also in “serious” math research – but then they are more likely to carry further restrictions such as being continuous or differentiable.

AoPS has a nice introduction to the topic. If you really need to solve this particular equation, you’ll want to be very specific about the domain and the codomain (natural numbers or real numbers, with or without 0), because they makes a whole world of difference.