Pointwise Operation/Examples/Cube and Sine Functions

Examples of Pointwise Operations

Let $f$ and $g$ be the real functions defined as

$\ds \forall x \in \R: \, $

$\ds \map f x$

$=$

$\ds x^3$

$\ds \forall x \in \R: \, $

$\ds \map g x$

$=$

$\ds \sin x$

Then for all $x \in \R$:

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

$=$

$\ds x^3 + \sin x$

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

$=$

$\ds x^3 - \sin x$

$\ds \map {\paren {f \times g} } x = \map {\paren {g \times f} } x$

$=$

$\ds x^3 \sin x$

$\ds \map {\paren {f \times f} } x$

$=$

$\ds x^6$

Contrast this with:

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

$=$

$\ds \paren {\sin x}^3$

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

$=$

$\ds \map \sin {x^3}$

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

$=$

$\ds x^9$

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces