\begin{definition}[Definition:Affine Group of One Dimension]
Let $S$ be the set of mappings $f_{a b}: \R \to \R$ defined as:
:$S := \set {f_{a b}: x \mapsto a x + b: a \in \R_{\ne 0}, b \in \R}$
The algebraic structure $\struct {S, \circ}$, where $\circ$ denotes composition of mappings, is called the '''$1$-dimensional affine group on $\R$''' and can be denoted $\map {\operatorname {Af}_1 } \R$.
\end{definition}