Composition of Mappings is not Commutative/Examples

Examples of Use of Composition of Mappings is not Commutative

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

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

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

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

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

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

$\ds \map {\paren {f \circ g} } x$

$=$

$\ds \paren {x + 1}^2 + 1$

$\ds = x^2 + 2 x + 2$

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

$\ds \map {\paren {g \circ f} } x$

$=$

$\ds \paren {x^2 + 1} + 1$

$\ds = x^2 + 2$

and it is immediately seen that:

$g \circ f \ne f \circ g$

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

$\forall \tuple {x, y} \in \R \times \R: \map f {x, y} = x^2 + y^2$

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

$\forall \tuple {x, y} \in \R \times \R: \map g {x, y} = x + y$

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

$\forall x \in \R: \map g x = x^2$

Then we have that:

$\map h {\map g {x, y} } = \paren {x + y}^2$

while:

$\map g {\map h x, \map h y} = x^2 + y^2 = \map f {x, y}$

Hence the diagram:

$\quad\quad \begin{xy} \xymatrix@L+2mu@+1em{ \R \times \R \ar[r]^*{g} \ar@{-->}[rd]_*{f} & \R \ar[d]^*{h} \\ & \R }\end{xy}$