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[parent] power function (Definition)

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

$\displaystyle f(x) = x^a$
where $ a$ is a given real number.
Theorem 1   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 constant 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.



"power function" is owned by pahio.
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See Also: fraction power, cube of a number, power tower sequence, Laplace transform of logarithm

Also defines:  natural power function, root function, fraction power function

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Cross-references: general power, integer, positive, power, fractional number, strictly decreasing, strictly increasing, derivative, differentiable, real
There are 11 references to this entry.

This is version 11 of power function, born on 2004-10-25, modified 2006-11-18.
Object id is 6421, canonical name is PowerFunction.
Accessed 16427 times total.

Classification:
AMS MSC: 26A99 (Real functions :: Functions of one variable :: Miscellaneous)
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