R denotes the set of real numbers. Does there exist functions F: R → R such that:

F(F(x)) = x^{2} - 2?

If so, find all such functions.

If not, prove that no such function can exist.

Source: American Mathematical Monthly

(In reply to

Almost there? by Jer)

Jer:

I am not sure that I agree with any of this.

1) Just because f(f(2)) = 2, I don't think it follows that f(2) = 2.

For instance, if g(g(x)) = x

then g(x) could be 1/x if x <> 0 and 0 if x = 0.

2) Even if f(f(f(0))) = 2 (which I don't think is necessarily true), I don't see that it follows that f(0) = 2.

Steve