Identity of Affine Group of One Dimension

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

Then $\map {\mathrm {Af}_1} \R$ has $f_{1, 0}$ as an identity element.

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

Then:

Thus $f_{1, 0}$ is the identity element of $\map {\mathrm {Af}_1} \R$.