Composition of Linear Real Functions

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:

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

$\ds \map {\paren {\theta_{c, d} \circ \theta_{a, b} } } x$

$=$

$\ds \map {\theta_{c, d} } {\map {\theta_{a, b} } x}$

$\ds $

$=$

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

$\ds $

$=$

$\ds c \paren {a x + b} + d$

$\ds $

$=$

$\ds \paren {a c} x + \paren {b c + d}$

$\ds $

$=$

$\ds \theta_{a c, b c + d}$

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $3$. Mappings: Exercise $5$