# 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:

$\theta_{c, d} \circ \theta_{a, b} = \theta_{a, b} \circ \theta_{c, d}$
$b c + d = a d + b$

## Proof

 $\ds \map {\theta_{c, d} \circ \theta_{a, b} } x$ $=$ $\ds \map {\theta_{a, b} \circ \theta_{c, d} } x$ $\ds \leadsto \ \$ $\ds \theta_{a c, b c + d}$ $=$ $\ds \theta_{c a, a d + b}$ $\ds \leadsto \ \$ $\ds b c + d$ $=$ $\ds a d + b$

$\blacksquare$