This is a somewhat frivolous question, so I won't mind if it gets closed.  One of the categories of Olympiad-style problems (e.g. at the IMO) is solving various functional equations, such as those given in <a href="http://www.imomath.com/tekstkut/funeqn_mr.pdf">this handout</a>.  While I can see the pedagogical value in doing a few of these problems, I never saw the point in practicing this particular type of problem much, and now that I'm a little older and wiser I still don't see anywhere that problems of this type appear in a major way in modern mathematics. 

(There are a few notable exceptions, such as the functional equation defining modular forms, but the generic functional equation problem has much less structure than a group acting via a cocycle.  I am talking about a contrived problem like finding all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying

$$f(x f(x) + f(y)) = y + f(x)^2.$$

When would this condition ever appear in "real life"?!)

Is this impression accurate, or are there branches of mathematics where these kinds of problems actually appear?  (I would be particularly interested if the condition, like the one above, involves function composition in a nontrivial way.)

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**Edit:**  Thank you everyone for all of your answers.  As darij correctly points out in the comments, I haven't phrased the question specifically enough.  I am aware that there is a lot of interesting mathematics that can be phrased as solving certain nice functional equations; the functional equations I wanted to ask about are specifically the really contrived ones like the one above.  The implicit question being: "relative to other types of Olympiad problems, would it have been worth it to spend a lot of time solving functional equations?"