Problem 388 (difficulty: 7/10)

Find a function \(\displaystyle f:[-1,1]\to[-1,1]\) such that \(\displaystyle f(f(x))=-x \forall x\in[-1,1]\).

Solution:

All such \(\displaystyle f\) has the following form: Let \(\displaystyle (0,1]=A\coprod B\) disjoint union, \(\displaystyle \varphi:A\to B\) bijection. Let

\(\displaystyle f(x)= \begin{cases}\phantom{m}\varphi(x) &\text{ if } \ x\in A \\ -\varphi(x) &\text{ if } \ x\in B \\ -\varphi(-x) &\text{ if }-x\in A \\ \varphi(-x) &\text{ if }-x\in B \\ \phantom{m}0 &\text{ if } \ x=0. \end{cases}\)

Easy to check that these will do e.g. \(\displaystyle A=(0,1/2], B=(1/2,1]\). On the other hand if \(\displaystyle f\) has the property required, then choose \(\displaystyle A:=(f>0),\ B:=(f<0)\) .

Give me another random problem!