Following the discussion on bijectivity and given that function inverse is studied on domain/codomain, I would like to ask whether we can find subsets where this function can be bijective.

Let's say the subset of all alphanumeric strings of exactly 5 characters. The cardinal of this subset should be Arangements of 62 characters in a line of 5 characters = 62! / 57! ~ 776 mln . This is definetely lower than 2^256 . Is this enough to say the function is injective on this domain or do I need to calculate every single value to make this conclusion? What happens to surjectivity in this case?

OBS: I might make confusion with continuous functions so do let me know if I have mistaken something.