Composition of Mappings/Examples/Compositions of sin x with 2x+1

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Example of Compositions of Mappings

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = \sin x$

Let $g: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map g x = 2 x + 1$


Then the compositions of $f$ with $g$ are:

$f \circ g: \R \to \R$:

$\forall x \in \R: \map {\paren {f \circ g} } x = \map \sin {2 x + 1}$

$g \circ f: \R \to \R$:

$\forall x \in \R: \map {\paren {g \circ f} } x = 2 \sin x + 1$


Note that:

$\map {\paren {f \circ g} } 0 = \map \sin {2 \times 0 + 1} \approx 0 \cdotp 84$
$\map {\paren {g \circ f} } 1 = 2 \times \sin 0 + 1 = 1$

demonstrating that composition of mappings is in general not commutative.


Sources