If f and g are both invertible prove that gof is invertible and (gof)^-1 = f^-1 o g^-1?

Question

Any help would be SO greatly appreciated. Many thanks in advance. )

Answer 1

Look at g*f *f^-1 * g^-1 = x
We can do this because f^-1 and g^-1 exist.
Now use what what we call "collapsing the middle".
f* f^-1 = e
So we get x = g*g^-1
But g*g^-1 = e
so x = e.
Thus the inverse of g*f = f^-1*g^-1.