socrates wrote: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $$\displaystyle{f(f(x)) = \frac{x^2 - x}{2}\cdot f(x) + 2-x,}$$for all $x \in \mathbb{R}.$ Find all possible values of $f(2).$ Let $P(x)$ be the assertion $f(f(x))=\frac{x^2-x}2f(x)+2-x$ $P(2)$ implies $f(f(2))=f(2)$ But if $f(u)=u$ for some $u$, then $P(u)$ $\implies$ $u=\frac{u^2-u}2u+2-u$ and so $f(2)\in\{-2,1,2\}$ $P(0)$ $\implies$ $f(f(0))=2$ and then $P(f(0))$ $\implies$ $f(2)=f(f(f(0)))=f(0)^2-2f(0)+2$ So $f(2)\ge 1$ and so $f(2)\in\{1,2\}$ If $f(2)=1$ : $P(2)$ implies $f(1)=1$ $f(2)=f(0)^2-2f(0)+2$ implies $f(0)=1$ and so $f(f(0))=f(1)=1$, in contradiction with $f(f(0))=2$ So $\boxed{f(2)=2}$ and this value may indeed be reached for example with $f(x)=\frac 4x$ $\forall x\ne 0$ and $f(0)=2$