I just had a thought…for ease of reference, we should probably use post titles like "Exercise 0.24(b)" instead of "HW 3 24(b)." This way no one has to wonder what chapter HW 3 is from and we remove any ambiguity about which 24(b) we are talking about.

OK, now for a hint. You are trying to prove that given a function $f:X\to Y$ with $A_1, A_2 \subseteq X$ that $f(A_1\cap A_2) \subseteq f(A_1) \cap f(A_2)$. First, you should draw a picture using "bubble diagrams" to convince yourself that this is even true. It will also help you later to show that $f(A_1)\cap f(A_2)\nsubseteq f(A_1\cap A_2)$$, in general. To prove any set containment, you should choose an arbitrary element in the smaller set and argue that it lies in the larger set. In this case, your proof should start with something like: "Let $y\in f(A_1\cap A_2)$." Now, "unpack" what this means. To do so, you will probably need to introduce a new variable, say $x$ that lives in $A_1\cap A_2$. After doing some mathematical yoga, the last line of your proof should be something like: "Therefore, $y\in f(A_1)\cap f(A_2)$." Of course, the yoga is the hard part.

To show that the reverse containment fails, you should construct a single counterexample. To do this, you should specify exactly what $X, Y, A_1, A_2$ are, as well as exactly what $f$ does to the elements in $A_1$ and $A_2$.

Does this help?