Composition of Identical Functions

Question

I have come across a problem which asks to find $f(x)$ such that $f(f(x))=-x$. Nothing I can find has anything pertaining to the composition of two identical functions. Is there a way that I can dissect this in order to help in finding a possible $f(x)$?

Answer 1

Through late night serendipity and later verification, I have found that $f(x)=ix$ works for all real numbers.

Answer 2

Here are some possible answers:

• Multiply by sqrt(-1).

• Rotate by 90 degrees.

• Map an even number x to x+1, and an odd number to 1-x.

I don't think it is possible if f is continuous on real numbers.