Composition of Linear Real Functions
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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:
- $\theta_{c, d} \circ \theta_{a, b} = \theta_{a c, b c + d}$
Proof
\(\ds \map {\paren {\theta_{c, d} \circ \theta_{a, b} } } x\) | \(=\) | \(\ds \map {\theta_{c, d} } {\map {\theta_{a, b} } x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {\theta_{c, d} } {a x + b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds c \paren {a x + b} + d\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a c} x + \paren {b c + d}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \theta_{a c, b c + d}\) |
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $3$. Mappings: Exercise $5$