Why Can We Swap Variables When Finding Inverse Functions?

[hyper]

We have y=f(x), and get the inverse by uing the first function and solving it for x and get x=g(y). (F and g are different functions.) Then we swap the name of x and y and we get y=g(x).

Buw why can we do this when we want to find the inverse functions? If we got y=f(x) and want to find the inverse we take g(f(x))=x. But how is this related to the first thing I did?

 

[HallsofIvy]

Since y= f(x), g(f(x))= x is the same as g(y)= x.