Let f:X->Y and g:Y->Z be functions. Suppose A is a subset of Z. I'm wondering whether (g o f)(adsbygoogle = window.adsbygoogle || []).push({}); ^{-1}(A)=f^{-1}(g^{-1}(A)).

I'm lost as to where to go with this problem. I know I need to do something using images, but manipulating the inverses of composite functions is proving to be very confusing; I don't know how to properly translate these functions into terms that match up with my definitions, which are as follows:

1) Let f:X->Y be a function: for a subset A of X, the set f(A)={y in Y: y=f(x) for some x in A}.

2) For a subset C of Y, the set f^{-1}(C) = {x in X: f(x) is in C}.

Can you please point me in the right direction? I'm sure that if I could just grasp how to prove this theorem, I'd understand inverses so much better.

Thank you for your time!

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# Working with the inverses of composite functions

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