When I was doing my exercise about functions, I came across a question that asks to prove that composite function f^-1 g^-1 (x) is equivalent to the composite function (gf)^-1 (x) given random one-one functions for f(x) and g(x).
I answered the question to show they are equivalent by using the example given, and after double checking by using another set of functions for f(x) and g(x), I realized that the rule f^-1 g^-1 (x) = (gf)^-1 (x) applies to all one-one functions.
But I don't know why this particular arrangement (gf)^-1 (x) equals to f^-1 g^-1 (x) rather than (fg)^-1 (x) being equal to it. I just want to ask if there is a certain algebraical method to determine which compound function is equivalent to the other compound function?