All wrapped in functions

Question

Find all functions $f:mathbb{R}rightarrowmathbb{R}$ such that $$f(x)fbig(f(x)+ybig)=fbig(x^2big)+f(xy)$$ for all $x,yinmathbb R$

Problem by me

Most elegant solution wins!

Paul Panzer · Accepted Answer

Not sure this rates as elegant but it is certainly short:

Case 1: $f(0)ne 0$

Case 2: $f(0)=0$
Case 2a: there is another zero $f(z)=0,zne 0$

Case 2b: $f(0)=0$ is the only zero

Steve · Answer

I'm not sure about "all functions", but I've found 2:

and

Michael Moschella · Answer

Along with the solutions provided by Steve, I've found that:

Is also a valid solution

msh210 · Answer

Not an answer, but a step toward an answer:

(The examples already found by other answerers —

— all meet this criterion.)

Ankoganit · Answer

First, let's get rid of all constant solutions. If $f(x)=c$ is a solution, then the equation gives

If some $x$ satisfies $f(x)=0$, then plugging that in,

Plugging $y=0$ now gives

Now we claim that

And thus, to prove injectivity,

Phew! Now we are ready for the finish. Indeed, plug in

Thus these are all such $f$:

5 Answers