Simple question. Suppose $\displaystyle f(g(x)) = g(f(x))$. Then must one of the following be true?
1. One of the functions is the identity.
2. The two functions are inverses.
f o g:x = g o f:x if one of the following statement holds
i) one of the functions is the identity
ii)the two function are inverses
iii) the two functions are the same
iv) the two functions are of the form f(x) = ax + b = x + b that is a =1
g(x) = cx + d = x + d that is b = 1
we call this functions translations
v) f and g have the same fixed point.
fixed point is the point X0 = (xo,xo) such that f(xo) = xo and g(x0) = xo
fixed point of f = -b/(a-1)