Equality of functions

**MachinePL1993** · December 4th 2012, 10:41 AM

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.

**Link** · December 4th 2012, 10:47 AM

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

**MachinePL1993** · December 4th 2012, 10:49 AM

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.