Proving a function is one way hash function I am working on a project which computes the hash of the given input, I have to prove that the function is pre-image resistant that is given an output we cannot get back to the input.

While proving the function to be pre-image resistant. The function that I am using has a property that for every output there are more than one possible input. In other words, I have proved my function is many to one function. Is it sufficient to show that we cannot calculate the inverse of the function because it is a many to one function and not a one-one function.
The inverse cannot be calculated, you're right. However, in can be guessed.

In many scenarios, for example cracking hashed password, it's enough for the attacker to know a set of possible inputs which result in certain output (of course if their number is reliable) instead of  one certain value. 

So if the one-to-many function is weak, given the output $y$ the attacker can calculate $x_1,x_2,...,x_n$ such that for each $x_i$ $H(x_i)=y$ and try them one by one.

Concluding - it's not enough to show that it's one-to-many function. You must also show that finding the set mentioned above is computationaly hard for your function.

PS. You might wonder why $n$ can't be so big that trying all possible inputs resulting in a certain output will be impossible. This would cause another weakness - low collision resistance.