Functional puzzle: find $f(2)$

Question

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 $f(2)$?

Answer 1

We remark that $f$ is odd, as we plainly have $f(-x)=-f(x)$.

Letting $x=2$ yields $$f(2)=2f(1)-2$$

Letting $x=1$ yields $$f(1)=f(-2)-1=-f(2)-1$$

Combining yields $$f(2)=-2f(2)-2-2\implies 3f(2)=-4\implies \boxed {f(2)=-\frac 43}$$

Answer 2

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