# power function

A real $f\!:\,\mathbb{R}_{+}\to\mathbb{R}$  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$x\mapsto ax^{a-1}$  and strictly increasing if  $a>0$  and strictly decreasing if  $a<0$  (and 1 if  $a=0$).

The power functions comprise the natural power functions$x\mapsto x^{n}$  with  $n=0,\,1,\,2,\,\ldots$,  the root functions$x\mapsto\sqrt[n]{x}=x^{\frac{1}{n}}$  with  $n=1,\,2,\,3,\,\ldots$  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