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

Example of Compositions of Mappings

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: \, $

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

$=$

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

$\ds = x^2 + 2$