Showing some transformation is group isomorphism (topological group).

Question

Let's $Mg$ be a set of all real valued functions defined on topological group $G$. Assume that $f\in Mg$. Let's $a \in G$, then define $f_a(x):=f(ax)$ for all $x \in G$. Now define $h_a(f):=f_a$ we know that $h_a:Mg \to Mg$. Define $\varphi(a):=h_a$. We know, that this is isometry [For some fixed metric on $Mg$. AR], then denote group of all isometries of space $Mg$ into itself by $\operatorname{Is}(Mg)$. I should show that $\varphi$ is a homomorphism of groups $G$ and $\operatorname{Is}(Mg)$. But probably throught great number of signs i have had only bad ideas. Please help.