Inverse in Affine Group of One Dimension

Theorem
Let $\map {\operatorname {Af}_1} \R$ denote the $1$-dimensional affine group on $\R$.

Let $f_{a b} \in \map {\operatorname {Af}_1} \R$.

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

Then $f_{c d} \in \map {\operatorname {Af}_1} \R$ is the inverse of $f_{a b}$.

Proof
As $a \in \R_{\ne 0}$ by definition of $\map {\operatorname {Af}_1} \R$ 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.