TITLE: Functional puzzle: find $f(2)$
QUESTION [5 upvotes]: Here's a little puzzle I came up with, for anyone who wants to try it:

A function $f$ satisfies
  $$f(x)=xf(x^2-3)-x$$
  for all $x$. What is the value of $f(2)$?

REPLY [4 votes]: $$\begin{aligned}
f(2) &= 2f(1)-2 \\
f(1) &= f(-2)-1 \\
f(-2) &= -2f(1)+2\\
\end{aligned}$$
The second and third equation can be solved simultaneously to get
$$f(1)=\frac{1}{3}, f(-2)=\frac{4}{3}.$$
Therefore, $\displaystyle f(2)=-\frac{4}{3}$.