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This is a question posed to me by a friend two days ago. It goes as follows:

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that $f\left(f(x)^2 + f(y)\right)= xf(x) + y$ for all $x, y \in \mathbb{R}$.

I tried putting $x=y=0$ but didn't get a satisfactory result. Then I decided to plug in a few values and check, but that didn't help either. So far, I haven't made any progress. Any help (either a full solution or a hint) will be appreciated. Thanks!

Edit 1: $f(x)=x$ seems to be a solution, but the million-dollar question is are there any other solutions...

Edit 2: Functions satisfying: $f(f(x)^2+f(y))=xf(x)+y$

I found a link to my question. However, what I didn't understand is how $f(x)^2=x(f(x))$ implies that $f(x)=x$. If anyone can explain this part, I'll be grateful. Thanks!

insomniaddict
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    Well, one solution should be obvious. Perhaps you could try to prove that this is the only one? Either that or try to find another solution. – lulu Apr 21 '24 at 10:07
  • Yes you are correct!... Thanks – insomniaddict Apr 21 '24 at 10:15
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    Pro-tip for the future: a lot of posts about functional equations like this one can be found by searching through this site called Approach$0$, see for example the results for this query. It's also useful for other kinds of equations but it can get get difficult to get what you want outside of functional equations and contest-type problems, but it's still useful to me from time to time! It mostly provides pages from MSE (this site) and AoPS (Academy of Problem Solving). – Bruno B Apr 21 '24 at 10:58
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    @BrunoB Will keep in mind... thanks! – insomniaddict Apr 21 '24 at 11:30

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