If (f:A\to B) is a function and (a_1,a_2\in A) with (a_1\ne a_2), which statement is always true?

Correct answer: Both (f(a_1)) and (f(a_2)) lie in (B)

The direct answer is C. If f:A to B is a function, the definition says that every input in A is assigned exactly one output in B. Therefore, since a1 and a2 belong to A, both f(a1) and f(a2) must belong to B. The two inputs being different does not force their outputs to be different; a function may send several inputs to one common value. Option C is therefore always true. Option A is not always true: for example, a constant function can have f(a1)=f(a2). Option B is also not always true, because the outputs may or may not be equal. Option D is wrong because a1 and a2 are elements of A, not necessarily elements of B; the sets may even be different types of objects. The important lesson is that “function” guarantees one output in the codomain for each input, but it does not guarantee distinct outputs. Distinct outputs require an extra property called one-one or injective.