Condition for Composition of Linear Real Functions to be Commutative

Theorem
Let $a, b, c, d \in \R$ be real numbers.

Let $\theta_{a, b}: \R \to \R$ be the real function defined as:
 * $\forall x \in \R: \map {\theta_{a, b} } x = a x + b$

Let $\theta_{c, d} \circ \theta_{a, b}$ denote the composition of $\theta_{c, d}$ with $\theta_{a, b}$.

Then:
 * $\theta_{c, d} \circ \theta_{a, b} = \theta_{a, b} \circ \theta_{c, d}$


 * $b c + d = a d + b$
 * $b c + d = a d + b$