f:X→Y and g:Y→Z
both are one-one and onto functions.
It's mean gof:X→Z defined.
we know that if both functions are ono-one , onto and gof defined,
then (gof)−1 defined and exist too.
f:X→Y is one-one onto ⟹f−1:Y→X exist.
g:Y→Z is one-one onto ⟹g−1:Z→Y exist.
thus (f−1og−1):Z→X exist
so domain and codomain of (gof)−1 and (f−1og−1) are same.
f(x)=y and g(y)=z
........from equation (3)
from eq (1) and the last eq,