$\def\R{\mathbb R}$Let $ G = \mathrm{Aff}(\R) := \{ f\colon\R\to\R\mid f \colon x\mapsto ax+b, a \in \R^*, b \in \R \} $ Then $(G,\circ)$ is a group where $\circ$ denotes function composition $(f \circ g)(x) = f(g(x))$. Now let $N := \{ f\in G\mid f\colon x\mapsto x+d, d \in \R \}$ and $ H := \{ g\in G \mid x \mapsto cx, c \in \R^*\} $ I have shown that $G$ is a group and that $N$ and $H$ are subgroups of $G$. Now I have to show that $N$ is isomorphic to $\R$ and that $H$ is isomorphic to $\R^*$. After that I have to show that $G$ is the semidirect product of $N$ by $H$.
I am stuck at showing that they are isomorphic. Help is appreciated. Thanks