# Equality of functions

• Dec 4th 2012, 10:41 AM
MachinePL1993
Equality of functions
Hello,

I have a following problem to solve:

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

I managed to show that $\displaystyle f(f^{-1}(C)) \subseteq C$ and I have a counterexample that disproves $\displaystyle C \subseteq f(f^{-1}(C))$. Yet I have no idea what $\displaystyle f(f^{-1}(C))$ could equal to.
• Dec 4th 2012, 10:47 AM
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
• Dec 4th 2012, 10:49 AM
MachinePL1993
Re: Equality of functions
Could I just write $\displaystyle f(f^{-1}(C))=\{y| y \in C \wedge \exists_{x \in X}(f(x)=y)\}$? Because $\displaystyle f(f^{-1}(C))$ equals the elements of C that can be reached from X.
• Dec 4th 2012, 10:56 AM
Plato
Re: Equality of functions
Quote:

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

This is much like your other post.

$\displaystyle f(f^{-1}(C)) = C$ if and only if $\displaystyle f$ is surjective.
• Dec 4th 2012, 10:57 AM