An example of why $f(f^{-1}(B))\neq B$ Let $f:X\rightarrow Y$ be a function and $B\subseteq Y$ a subset of $Y$. I know (and have proven) that $f(f^{-1}(B))\subseteq B$. I've also found an example where $f(f^{-1}(B))\neq B$ for $B= \mathbb{R}$. I want to find another example, because I find my example very silly. Any help?

Hint: The relation $f(f^{-1}(B)) = B$ holds for all $B \subset Y$ iff $f$ is onto.
