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I know the basics of function. Like the relation between function and sets, how one set of data can image another, concepts of domain, range , co-domain etc, finding inverse functions and so on. But I know the absolute basics and can solve pretty straight forward questions. But whenever asked about any olympiad level question I just get stuck. Like one is:

$$f(f(x))=x^2 +3$$ then find the value of $f(3)$.

I saw one queston of IMO was something like this:

find all the function of $x$ and $y$ such that $f(f(x)f(y)) + f(x+y) = f(xy)$

I get stuck in this type of questions. I dont even know what they are saying and what I should do. I am not really worried about IMO type questions. But I want to get a feel of functions and relate it with real world situations. I want a book which can really give me a practical idea of function and how they works and what they actually mean and at the same time prepare me for the divisional or national level mathematics olympiad. Can anybody help me out.

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    $\begingroup$ The key term is "functional equations". $\endgroup$ – lhf Oct 3 '17 at 12:48
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Try this book:

Introduction to Functional Equations: Theory and problem-solving strategies for mathematical competitions by Costas Efthimiou (AMS+MSRI, 2011)

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