from finding the inverse function evaluating function and op

from finding the inverse function, evaluating function and operation with functions, Does the order of composition matters?

Solution

Short answer: Yes it does.

Here is an example.

f(x) = 3x
g(x) = x2

(f^og)(x)=f(x2)=3*(x2)=3x2
(g^of)(x)=g(3x)=(3x)2=9x2

I haven\'t proven this vigorously, but I conjecture that there are more examples where the order matters than when it doesn\'t.