The method of finding the inverse of a simple function $y = f(x)$ involves the following steps:-
1) Change the subject to $x$ instead of $y$.
2) Interchange $x$ and $y$.
3) The newly formed function ($y = g(x)$, say) is then the required inverse.
We know that the method works but why does it work? My question is how to convince others that the interchange part of the above can do the magic and is logically sound? A proof would be even better.