1. ## Equality of functions

Hello,

I have a following problem to solve:

For some function $f:X \rightarrow Y$ and $C \subseteq Y$ I have to find something, that would equal $f(f^{-1}(C)) = ?$.

I managed to show that $f(f^{-1}(C)) \subseteq C$ and I have a counterexample that disproves $C \subseteq f(f^{-1}(C))$. Yet I have no idea what $f(f^{-1}(C))$ could equal to.

2. ## Re: Equality of functions

it's always included in C, that's all we can say. Exemple f R-R x|-> x², and C is [-4,4], we have f(f^-1(C))=[0,4] =/= C

3. ## Re: Equality of functions

Could I just write $f(f^{-1}(C))=\{y| y \in C \wedge \exists_{x \in X}(f(x)=y)\}$? Because $f(f^{-1}(C))$ equals the elements of C that can be reached from X.

4. ## Re: Equality of functions

Originally Posted by MachinePL1993
For some function $f:X \rightarrow Y$ and $C \subseteq Y$ I have to find something, that would equal $f(f^{-1}(C)) = ?$.

This is much like your other post.

$f(f^{-1}(C)) = C$ if and only if $f$ is surjective.

5. ## Re: Equality of functions

Yes you can write that.