power functionPowerFunction2004-10-25 16:13:222006-11-18 15:52:46Definitionreal functionnatural power functionroot functionfraction power function% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{thmplain}{Theorem}A real {\em power function} \,$f:\,\mathbb{R}_+\to\mathbb{R}$\, has the form
$$f(x) = x^a$$
where $a$ is a given real number.
\begin{thmplain}
\,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 \PMlinkescapetext{constant} 1 if\,
$a = 0$).
\end{thmplain}
The power functions comprise the {\em natural power functions}\, $x\mapsto x^n$\, with\, $n = 0,\,1,\,2,\,\ldots$,\, the {\em root functions}\, $x\mapsto \sqrt[n]{x} = x^{\frac{1}{n}}$\, with\, $n = 1,\,2,\,3,\,\ldots$\, and other {\em fraction power functions}\, $x\mapsto x^a$\, with $a$ any fractional number.
\textbf{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.