Question

for proving a function invertible function is sufficient to prove it bijective

Answer 1

Dear Student,
Every function is not invertible(i.e all functions do not have an inverse). For a function to be invertible, each element in the co-domain(say $y\in Y$) must correspond to not more than one element in the domain(say $x\in X$) . This means that there is one-to-one relation between the elements of the domain and co-domain, i.e they must be one-to-one functions. Another condition that needs to be satisfied is that every element in the co-domain must correspond to some element in the domain, i.e no element must be left out in the co-domain particularly. Such a function is called a surjective function(or onto function).
Thus, as observed, a function can be called as an invertible only if it is one-to-one and onto. This combination of a one-to-one function and a surjective function is called as a bijective function.
Thus for proving a function invertible, it is sufficient to prove that it is bijective is true.
Regards.