Can injective function has an element that maps to nothing? Can injective function has an element that maps to nothing? I don't think this violate the definition of injective function. If that is the case, is it possible for a function to be bijective but its inverse not bijective since there might be an element that maps to nothing.
 A: Recall that the definition of a function from $A$ to $B$ is that the domain of the function is $A$.
But you can think about functions as just sets of ordered pairs with a particular property; in which case you can then define the domain of a function to be the maximal set for which the function is defined. But even then, an element "mapped to nothing" is simply not in the domain. So we are back to the original issue. 
There is a notion, however, of a partial function. Which is exactly a function whose domain is a subset of a particular set. This seems to be more in sync with what you are asking.
Note that it will not be fortuitous to allow function to be partial functions. Since in that case, every two sets have injections between them, rendering many theorems, such as the Cantor-Bernstein theorem, false. And thus requiring us to add the property that the domains are maximal anyway. So it's much more useful to work with functions as we do today, and when the context permits it, to work with partial functions.
