let R be a set of Real numbers and let f:R→R is defined as f(x)=αx (where α∈R) then show that f is group homomorphism

Question

Let $\mathbb{R}$ be the set of real numbers and let $f:\mathbb{R} \to \mathbb{R}$ be defined as $f(x)=\alpha x$, (where $\alpha \in \mathbb{R}$). Show that f is a group homomorphism.

I know if we define $f:\mathbb{R} \to \mathbb{R}$ and $f(x)=xH$ where $H$ is a subgroup of $\mathbb{R}$, then i know i can find group homomorphism as

Let $x,y \in \mathbb{R}$ then $f(xy)=(xy)H=xH•yH=f(x)f(y)$ So it $f$ group homomorphism

But my question is that ... As i ask in question if $α \in \mathbb {R}$ is arbitrary than how i can find f is group homomorphism?

Answer 1

Let $x, y \in \mathbb{R}$. Then, $f(x + y) = a(x + y) = ax + ay = f(x) + f(y)$