I am trying to solve the following functional equation: $f(x)$ is a continuous function, satisfying (1) $f(f(x))=x$
(2) $ 3f(-3x)-f(x)= 3x^2 $ for $x>0$.
From (1) & (2), I found that $f(x)$ should be decreasing, and $f(x) = f^{-1} (x)$.
But, I cannot figure out how to use (2).
How can I use (2) to solve this?
Thanks!
There are no solutions.
(The question already notes $f$ is decreasing; I include a proof for completeness).
By (1), $f$ is bijective.
Since $f$ is assumed continuous, this implies $f$ is strictly monotone by this result. That is, $f$ is either strictly increasing or strictly decreasing.
Taking the limit as $x\to0^+$ in (2) gives $f(0)=0$. Setting $x=1$ gives $$ 3f(-3)-f(1)=3.\tag{3} $$ If $f$ is increasing, then $f(-3)<0$ and $f(1)>0$, contradicting (3). Thus $f$ is decreasing. In particular $f(-3)>0$, so (3) gives $$ f(1)=3f(-3)-3>-3. $$ Thus $1=f(f(1))<f(-3)$. Now (3) gives $$ f(1)=3f(-3)-3>0=f(0), $$ a contradiction.