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

re: Almost there? by Steve Herman)

Ah, very clever.

I agree that either

(a) f(2) = 2 and f(-1) = -1

or

(b) f(2) = -1 and f(-1) = 2