A function is Invertible iff it is Bijection because if suppose f:A->B that means "every" element in A has an image in B ie there is no element in A which doesnot have any mapping.(multiple elements of A can have same image in B.)
Now suppose g:B->A that means every element in B has a image in A(multiple elements of B can have same image in A.)
If we say g is inverse of f than
number of elements in A and B must be same and has unique image ie it must be Bijection.
Hope it helps..!
For a function to be invertible it has to be both one to one and onto i.e. Bijection.