Existence of a great function

Suppose such a functiom exists. Let $a = \frac{\sqrt{5}-1}{2}$ and $b = \frac{-\sqrt{5}-1}{2}$ . Then clearly, $f(f(a))=b$ and $f(f(b))=a$ . Let $f(a)= p$ . Then $f(b) = f(f(f(a))) = f(f(p))= p^2-2$ .

So we have a cycle of length four, given by

$a \rightarrow p \rightarrow b \rightarrow p^{2}-2 \rightarrow a$ . Notice that $a,b, p^{2} -2$ are all distinct from $p$ :

If $p=a$ , then $b= f(p)= f(a)=p=a$ , which is false.

If $p=b$ , then $f(a)= f(f(a))$ , so $a=f(f(f(f(a))))= f(f(f(f(f(a)))))= f(a)=p$ , which cannot be.

If $p=p^{2}-2$ , then $f(p)= f(p^{2}-2)$ , or $b=a$ , which is false.

Now, $f(f(p^{2}-2))= f(f(f(f(f(a)))))= f(a)=p$ , so

$(p^{2}-2)^{2} -2 = p$ , or $p^{4} - 4p^{2} -p + 2 = 0$ , which factorises as

$(p-2)(p+1)(p^{2} + p -1) = 0$ , which can be written as $(p-(p^{2}-2))(p-a)(p-b)=0$ . But we saw that none of $p=a, p= b$ or $p=p^{2}-2$ is possible. So we have arrived at a contradiction. Hence our assumption of the existence of such a function is incorrect.

Therefore no such function exists.