Does the order of composition matters?

Short answer: Yes it does.

Here is an example.

f(x) = 3x

g(x) = x

^{2}(f°g)(x)=f(x

^{2})=3*(x^{2})=3x^{2}(g°f)(x)=g(3x)=(3x)

^{2}=9x^{2}I haven't proven this vigorously, but I conjecture that there are more examples where the order matters than when it doesn't.