# Functional equation problem; chain of functions

I have tried more than an hour but couldn't solve it, can somebody please give me a clue?

$$f:\mathbb R\rightarrow\mathbb R$$

$$f(f(f(X)))+f(f(X))+X=3f(X).$$ Find $$f(X)$$

I know that $$f(X)=X$$ is a solution, and I know that the function being (obviously) injective might help, but that's all the useful stuff I could gather.

• Where is this problem from?
– Sil
Jul 27 '19 at 16:53

Probably not a complete solution, I tried $$f(x)=kx$$, $$k$$ constant, $$k\ne0$$, which results in $$(1 - 3 k + k^2 + k^3) x=0.$$ If valid for all $$x$$ then $$1 - 3 k + k^2 + k^3=0 \quad\Leftrightarrow\quad k=\{1,-1-\sqrt{2},-1+\sqrt{2}\}.$$