Group Direct Product/Examples/R-0 x R/Isomorphism to Set of Affine Mappings on Real Line under Composition

Theorem
Let $G$ be the Cartesian product of $\R \setminus \set 0$ with $\R$:


 * $G = \set {\tuple {a, b} \in \R^2: a \ne 0}$

Let $\circ$ be the group product on $G$ defined as:
 * $\tuple {a_1, b_1} \circ \tuple {a_2, b_2} = \tuple {a_1 a_2, a_1 b_2 + b_1}$

Let $S$ be the set of all real functions $f: \R \to \R$ of the form:


 * $\forall x \in \R: \map f x = r x + s$

where $r \in \R_{\ne 0}$ and $s \in \R$

Let $\struct {S, \circ}$ be the algebraic structure formed from $S$ and the composition operation $\circ$.

Then $\struct {G, \circ}$ is (group) isomorphic to $\struct {S, \circ}$.