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

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


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

demonstrating that composition of mappings is in general not commutative.