Let $G$ be any group, let $r_x,l_x: G \to G$ defined by $r_x(g)=gx$, and $l_x(g)=xg$. Let $R=\{r_x:x \in G\}$ and $L=\{l_x:x \in G\}$. Show that
$L=\{f \in \mathrm{Sym}(G): \forall r \in R~~~ fr=rf\}$
I have thought about it for hours, but I can't prove the right side belongs to the left side. Please help me.