power function

A real power function  $f:{ℝ}_{+}\to ℝ$  has the form

$$f(x)=x^a$$

where $a$ is a given real number.

Theorem.

The power function  $x\mapsto x^a$  is differentiable with the derivative  $x\mapsto a{x}^{a-1}$  and strictly increasing if  $a>0$  and strictly decreasing if    (and 1 if $a=0$).

The power functions comprise the natural power functions  $x\mapsto x^n$  with  $n=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots }$,  the root functions  $x\mapsto \sqrt[n]{x}=x^{\frac{1}{n}}$  with  $n=1,\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3},\mathrm{\dots }$  and other fraction power functions  $x\mapsto x^a$  with $a$ any fractional number.

Note.  The power $x^a$ may of course be meaningful also for other than positive values of $x$, if $a$ is an integer.  On the other hand, e.g. ${(-1)}^{\sqrt{2}}$ has no real values — see the general power.

Title	power function
Canonical name	PowerFunction
Date of creation	2013-03-22 14:46:32
Last modified on	2013-03-22 14:46:32
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	17
Author	pahio (2872)
Entry type	Definition
Classification	msc 26A99
Related topic	PropertiesOfTheExponential
Related topic	FractionPower
Related topic	CubeOfANumber
Related topic	Polytrope
Related topic	PowerTowerSequence
Related topic	LaplaceTransformOfLogarithm
Defines	natural power function
Defines	root function
Defines	fraction power function