* Let $f:A\rightarrow B$ be a function, and $X\subseteq A$. Prove or disprove that $f(f^{-1}(f(X)))=f(X)$.

Let $A=\mathbb{N}$, $B=\mathbb{R}$ and $X=\mathbb{N\setminus\left\{0\right\}}.$ Hence, $f$ is not invertible so we don't have $f(f^{-1}(f(X)))=f(X)$ every time. 

**Can you check my solution?**