The only requirement to answer the above question is to know the definition of function- a relation becomes a function if every element in domain is mapped to some element in co-domain and no element is mapped to more than one element.
Now, we have $a,b \subseteq X$. Their intersection can be even empty set. So, lets try out options:
options a and d don't even need a check.
Lets take a case where $a \cap b = \phi$. Now, $f(a \cap b) = \phi$, but $f(a) \cap f(b)$ can be non empty. So, option B can be false.
Option C is always true provided "proper subset" is replaced by "subset". This is because no element in domain of a function can be mapped to more than one element. And the subset needn't be "proper" as for a one-one mapping, we get $f(a \cap b) = f(a) \cap f(b)$.