/sci/ - Science & Math

>>14823211
f(a) = b
f(f(a)) = f(b) = -a
f(f(f(a))) = f(f(b)) = f(-a) = -b
f(f(f(f(a)))) = f(-b) = a

So you get these cycles: b(-a)(-b)ab...

Since we get a sign flip once for every two applications, we can deduce:

1. f(x) has the opposite sign from x for some, but not all, x
2. if f(x) flips the sign, f(f(x)) doesn't
3. if f(x) doesn't flip the sign, f(f(x)) does

Therefore 'a' values and 'b' values have to come from two distinct subsets of the positive reals; otherwise f would have to both flip and not flip the sign for some arguments. Pic rel: small diagram that sums up the special requirements for OP's functions. Unsurprisingly, it's similar to rotations with complex numbers in most aspects. Constructing them in a piecewise fashion according to these requirements should be straightforward, but you could exploit the regular alternations of the sign-flipping and the way the values alternate between the two sets.