Why is f( f^(-1)(x) ) = x and f^(-1)( f(x) ) = x true?

Question

Why is f( f^(-1)(x) ) = x and f^(-1)( f(x) ) = x true?

Answer 1

Edward, these two identities are inherent to the idea of a function operating on a variable (or variables, or on anything else!). A function is a mapping of the first thing onto a second thing -- that is, each element of the first thing becomes a distinct element of the second thing. The inverse function takes the "items" back from their second thing positions, into the starting positions on the first thing again.

You should probably be aware that the statements are only true where there are monotonically arranged elements in the two function domains & ranges -- thus as a counterexample, if f(x) takes both x=2 and x=4 into y=6, you're going to have an ambiguity going back again from y=6 using f^(-1) to a single value of x. Should it give 2? Should it give 4? That's not a function anymore, it's a multiple-valued relation!

Hope this helps you,

-- S.