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mathematical analysis > function > elementary function > algebraic function > power function

mathematical analysis > real analysis > real-valued function > algebraic function > power function

Preferred term

power function

Real functions of the form ${\\displaystyle f(x)=cx^{n}}$ , where ${\\displaystyle c\ eq 0}$ , are sometimes called power functions. When ${\\displaystyle n}$ is an integer and ${\\displaystyle n\\geq 1}$ , two primary families exist: for ${\\displaystyle n}$ even, and for ${\\displaystyle n}$ odd. In general for ${\\displaystyle c>0}$ , when ${\\displaystyle n}$ is even ${\\displaystyle f(x)=cx^{n}}$ will tend towards positive infinity with increasing ${\\displaystyle x}$ , and also towards positive infinity with decreasing ${\\displaystyle x}$ . All graphs from the family of even power functions have the general shape of ${\\displaystyle y=cx^{2}}$ , flattening more in the middle as ${\\displaystyle n}$ increases. Functions with this kind of symmetry ( ${\\displaystyle f(-x)=f(x)}$ ) are called even functions. When ${\\displaystyle n}$ is odd, ${\\displaystyle f(x)}$ 's asymptotic behavior reverses from positive ${\\displaystyle x}$ to negative ${\\displaystyle x}$ . For ${\\displaystyle c>0}$ , ${\\displaystyle f(x)=cx^{n}}$ will also tend towards positive infinity with increasing ${\\displaystyle x}$ , but towards negative infinity with decreasing ${\\displaystyle x}$ . All graphs from the family of odd power functions have the general shape of ${\\displaystyle y=cx^{3}}$ , flattening more in the middle as ${\\displaystyle n}$ increases and losing all flatness there in the straight line for ${\\displaystyle n=1}$ . Functions with this kind of symmetry ( ${\\displaystyle f(-x)=-f(x)}$ ) are called odd functions. For ${\\displaystyle c<0}$ , the opposite asymptotic behavior is true in each case
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Exponentiation#Power_functions)

algebraic function

square function

fonction puissance

French

URI

http://data.loterre.fr/ark:/67375/PSR-KF6VNW4H-5

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