We know that substituting a power of x into a power of x gives a power of
x, so we try to see if f(x)=xn gives a solution:

If so then try

f(x) = xn

then 

f(f(x)) = f(xn) = (xn)n = xn² = x2

The one and only way that can be true for all x is if the exponents of x are
equal

So n²=2

or



So one solution is



Another solution is



Can there be other solutions???

Edwin