Inverse in Affine Group of One Dimension

Theorem
Let $\operatorname{Af}_1 \left({\R}\right)$ be the $1$-dimensional affine group on $\R$.

Let $f_{a b} \in \operatorname{Af}_1 \left({\R}\right)$.

Let $c = \dfrac 1 a$ and $d = \dfrac {- b} a$.

Then $f_{c d} \in \operatorname{Af}_1 \left({\R}\right)$ is the inverse of $f_{a b}$

Proof
As $a \in \R_{\ne 0}$ by definition of $\operatorname{Af}_1 \left({\R}\right)$ it follows that $\dfrac 1 a \in \R_{\ne 0}$ and $\dfrac {- b} a \in \R$.

So let $c = \dfrac 1 a$ and $d = \dfrac {- b} a$.

Then:

Similarly:

Hence the result.