I Proof about pre-images of functions

[PhysicsRock]

The problem reads: $f:M \rightarrow N$, and $L \subseteq M$ and $P \subseteq N$. Then prove that $L \subseteq f^{-1}(f(L))$ and $f(f^{-1}(P)) \subseteq P$.
My co-students and I can't find a way to prove this. I hope, someone here will be able to help us out. It would be very appreciated.

Thank you in advance and have a great day everyone.

 

[FactChecker]

If this is a textbook homework type of problem, then there is a section and a format for that and we are only allowed to give hints and guidance.
Hint: pick a point in the smaller subset side and track it through the operations.

 

[PhysicsRock]

If this is a textbook homework type of problem, then there is a section and a format for that and we are only allowed to give hints and guidance.

I guess I figured it out anyway, at least I tried. Thank you for the advice. I'll ask for a specific hint etc. next time.

 

[PeroK]

The problem reads: $f:M \rightarrow N$, and $L \subseteq M$ and $P \subseteq N$. Then prove that $L \subseteq f^{-1}(f(L))$ and $f(f^{-1}(P)) \subseteq P$.
My co-students and I can't find a way to prove this. I hope, someone here will be able to help us out. It would be very appreciated.

Thank you in advance and have a great day everyone.

Let $x \in L$. Then $y = f(x) \in f(L)$. Now, what is, by definition, $f^{-1}(f(L))$? And why is $x \in f^{-1}(f(L))$?

Hint: it might help conceptually (be less confusing) to let $X = f(L)$ so that $y = f(x) \in X$ and show that $x \in f^{-1}(X)$.

PS the trick with these proofs is to get all the logical steps in the right order.