Power function

Graphs of some power functions

As power functions are called elementary mathematical functions of the form

${\displaystyle f\colon x\mapsto ax^{r}\qquad a,r\in \mathbb {R} .}$

If you only consider natural or integer exponents , you usually write for the exponent : ${\displaystyle n}$

${\displaystyle f\colon x\mapsto ax^{n}\qquad n\in \mathbb {Z} .}$

If the exponent is a natural number, the functional term is a monomial . ${\displaystyle n}$ ${\displaystyle ax^{n}}$

Special cases

• constant function : (for ) ${\displaystyle f\colon x\mapsto a}$ ${\displaystyle r=0}$
• (homogeneous) linear function / proportionality : (for ) ${\displaystyle f\colon x\mapsto ax}$ ${\displaystyle r=1}$
• Square function and its multiple: (for ) ${\displaystyle f\colon x\mapsto ax^{2}}$ ${\displaystyle r=2}$
• From the power functions with natural exponents are quite rational functions composed from which with integer exponents to rational functions . ${\displaystyle r}$
• For with there are root functions . ${\displaystyle r=1/n}$ ${\displaystyle n\in \mathbb {N} }$

Set of definitions and values

The maximum possible definition set depends on the exponent. If roots from negative numbers are not allowed, then they can be given with the following table:

	r > 0	r <0
${\displaystyle r\in \mathbb {Z} }$	${\displaystyle \mathbb {R} }$	${\displaystyle \mathbb {R} \setminus \{0\}}$
${\displaystyle r\notin \mathbb {Z} }$	${\displaystyle \mathbb {R} _{0}^{+}}$	${\displaystyle \mathbb {R} ^{+}}$

In the case of the sets of values, you also have to consider the sign of ; if is, it also depends on whether it is an even or odd number: ${\displaystyle a}$ ${\displaystyle r\in \mathbb {Z} }$ ${\displaystyle r}$

	r > 0		r <0
	r straight or ${\displaystyle \notin \mathbb {Z} }$	r odd	r straight or ${\displaystyle \notin \mathbb {Z} }$	r odd
a > 0	${\displaystyle \mathbb {R} _{0}^{+}}$	${\displaystyle \mathbb {R} }$	${\displaystyle \mathbb {R} ^{+}}$	${\displaystyle \mathbb {R} \setminus \{0\}}$
a <0	${\displaystyle \mathbb {R} _{0}^{-}}$	${\displaystyle \mathbb {R} }$	${\displaystyle \mathbb {R} ^{-}}$	${\displaystyle \mathbb {R} \setminus \{0\}}$

Graph

The graphs of power functions with natural hot parables th order that with integer negative hyperbole th order. The parameter expresses a stretching of the graph with respect to the -axis by the factor and also reflection on the -axis, if is. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle a}$ ${\displaystyle y}$ ${\displaystyle |a|}$ ${\displaystyle x}$ ${\displaystyle a<0}$

If a power function has the definition set , then its graph consists of two branches, otherwise there is only one branch. ${\displaystyle \mathbb {R} \setminus \{0\}}$

symmetry

Only the graphs of power functions with are symmetric; more precisely: they are even for even and odd for odd . In the first case your graph is axially symmetric to the -axis, in the second it is point symmetric to the origin. ${\displaystyle r\in \mathbb {Z} }$ ${\displaystyle r}$ ${\displaystyle r}$ ${\displaystyle y}$

Behavior for x → ± ∞ and x → 0

All power functions with positive exponents have a zero at , rise (but always slower than the exponential function ) and go against for . For the behavior for results from the symmetry. ${\displaystyle x^{r}}$ ${\displaystyle x=0}$ ${\displaystyle e^{x}}$ ${\displaystyle +\infty }$ ${\displaystyle x\to +\infty }$ ${\displaystyle r\in \mathbb {Z} }$ ${\displaystyle x\to -\infty }$

All power functions with negative exponents go against for . You fall and go against for . ${\displaystyle x^{r}}$ ${\displaystyle +\infty }$ ${\displaystyle x\to 0\;(x>0)}$ ${\displaystyle 0}$ ${\displaystyle x\to +\infty }$

Steadiness, derivation and integration

Every power function is continuous on its definition set. ${\displaystyle f\colon x\mapsto ax^{r}}$

The corresponding derivative function is (see power rule )

${\displaystyle f'\colon x\mapsto arx^{r-1}.}$

This formula applies to everyone and everyone , if only is defined at the point . It also applies at the point when is. For the function is continuous, but not differentiable at the point . ${\displaystyle x\neq 0}$ ${\displaystyle r}$ ${\displaystyle x^{r}}$ ${\displaystyle x}$ ${\displaystyle x=0}$ ${\displaystyle r\geq 1}$ ${\displaystyle 0<r<1}$ ${\displaystyle x\mapsto a\,x^{r}}$ ${\displaystyle x=0}$

For example, is valid in whole (or even in whole , if one allows odd roots of negative numbers - see below). ${\displaystyle (x{\sqrt[{3}]{x^{2}}})'={\tfrac {5}{3}}{\sqrt[{3}]{x^{2}}}}$ ${\displaystyle \mathbb {R} _{0}^{+}}$ ${\displaystyle \mathbb {R} }$

For any non-negative rational number , the formula is ${\displaystyle r\neq -1}$

${\displaystyle \int x^{r}\;dx={\frac {x^{r+1}}{r+1}}+C}$

valid for all intervals that are subsets of the definition set. for true ${\displaystyle r=-1}$

${\displaystyle \int x^{-1}\;dx=\int {\frac {1}{x}}\;dx=\ln(|x|)+C}$

For example:

${\displaystyle \int x{\sqrt[{3}]{x^{2}}}\;dx=\int x^{5/3}\;dx=3/8~x^{8/3}+C=3/8~x^{2}{\sqrt[{3}]{x^{2}}}+C}$.

Power functions with roots from negative numbers

This section only looks at power functions with rational exponents, where the denominator of the abbreviated exponent is odd, and explains how to extend their definition set to negative numbers. The following then explains which of the above-mentioned properties of the functions are changed by this.

Odd roots from negative numbers

(→ See also potency )

In the previous sections, the convention common in many textbooks was used that roots are only defined for non-negative radicands. However, one can also allow odd roots from negative numbers. For odd and arbitrary one defines, analogously to the known definition for positive radicals: ${\displaystyle n}$ ${\displaystyle x\in \mathbb {R} }$

${\displaystyle {\sqrt[{n}]{x}}}$is the (unique) real number for which applies. ${\displaystyle y}$ ${\displaystyle y^{n}=x}$

For example, according to this definition, the solution of the equation would be given by (whereas according to the usual definition one would have to write negative numbers without roots ). ${\displaystyle x^{3}=-8}$ ${\displaystyle x={\sqrt[{3}]{-8}}=-2}$ ${\displaystyle x=-{\sqrt[{3}]{8}}=-2}$

Set of definitions and values

In power functions with the above mentioned characteristics can now be the domain of negative expand: Be with , , this odd, and be and prime, then: ${\displaystyle x}$ ${\displaystyle r=n/m}$ ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle m\in \mathbb {N} }$ ${\displaystyle m}$ ${\displaystyle m}$ ${\displaystyle n}$

${\displaystyle x^{r}={\sqrt[{m}]{x^{n}}}}$ ${\displaystyle {\qquad }}$(or, which is equivalent, ). ${\displaystyle {\quad }}$ ${\displaystyle x^{r}=({\sqrt[{m}]{x}})^{n}}$

(Note: If , then this is again a power function with an integer exponent.) ${\displaystyle m=1}$

For the definition set of this function is then the same , for it is the same . ${\displaystyle n>0}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle n<0}$ ${\displaystyle \mathbb {R} \setminus \{0\}}$

For the set of values ​​you have to pay attention to the sign of . It also depends on whether one of the numbers is or is even (i.e. the product is even) or whether these two numbers are odd (i.e. the product is odd): ${\displaystyle a}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle mn}$ ${\displaystyle mn}$

	n > 0		n <0
	${\displaystyle mn}$ straight	${\displaystyle mn}$ odd	${\displaystyle mn}$ straight	${\displaystyle mn}$ odd
a > 0	${\displaystyle \mathbb {R} _{0}^{+}}$	${\displaystyle \mathbb {R} }$	${\displaystyle \mathbb {R} ^{+}}$	${\displaystyle \mathbb {R} \setminus \{0\}}$
a <0	${\displaystyle \mathbb {R} _{0}^{-}}$	${\displaystyle \mathbb {R} }$	${\displaystyle \mathbb {R} ^{-}}$	${\displaystyle \mathbb {R} \setminus \{0\}}$

Symmetry and behavior for x → ± ∞ and x → 0

The same applies to symmetry as to integer exponents: the function is even for even and odd for odd . Their behavior for and for is then defined by their symmetry properties and by their behavior on the right semi-axis. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle x\to 0\;(x<0)}$ ${\displaystyle x\to -\infty }$

Applications

Power functions have a variety of applications in business, nature and technology:

• Proportionalities appear in many contexts: ${\displaystyle (r=1)}$
  • Costs and quantity of goods (without volume discount)
  • Conversion between currencies
  • Circumference and radius
  • Mass and volume (at constant density )
  • Elapsed time and distance covered (at constant speed)
  • Distance traveled and amount of fuel consumed (with constant consumption)
  • Force and acceleration (at constant mass)
  • Stretching of a body and attacking force (within certain limits, see Hooke's law )
• Reciprocal proportionalities (also called indirect or anti-proportionality) occur practically just as often : ${\displaystyle (r=-1)}$
  • Number of workers and working hours
  • Time required for a distance and (constant) speed
  • required force and length of a lever ( lever law )
  • Mass and force required for a given acceleration
• Many quantities in geometry and physics depend on one another as a square : ${\displaystyle (r=2)}$
  • Area of a square and its side length
  • Area of a circle and its radius
  • Tension energy and elongation of a body
  • Kinetic energy and speed
  • Distance covered and time with steady acceleration
  • electrical power and resistance
  • Drag force and speed in turbulent flow
• For example, the third power is common in geometry : ${\displaystyle (r=3)}$
  • The radius and volume of a sphere
  • Side length and volume of a cube
• Some physical quantities are related to the fourth power : ${\displaystyle (r=4)}$
  • Radiation power of a black body and its absolute temperature ( Stefan-Boltzmann law )
  • Scattering cross-section for light scattering and light frequency (the Rayleigh scattering responsible for the blue color of the sky, among other things )
  • Volume flow through a thin pipe and pipe radius ( Hagen-Poiseuille's law )
• Non-integer powers also occur in many contexts:
  • Relationship between pressure , volume and absolute temperature with adiabatic changes of state (see also adiabatic exponent )
  • Relationship between the major semi-axis and the period of revolution of planets or moons ( 3rd Kepler's law ) ${\displaystyle a}$ ${\displaystyle T}$
  • Laws of scale , for example in phase transitions , but also in biology
  • In the geometry applies to the relationship between surface area and volume of a cube : ; a similar formula results for a sphere . ${\displaystyle V=(O/6)^{3/2}}$
  • In a universe which is filled with a homogeneous substance comprising a state equation of the form satisfied, the results for the time dependence of the scale factor from the Friedmann equations : . ${\displaystyle p=w\rho }$ ${\displaystyle a(t)\sim t^{2/3(1+w)}}$

literature

• Karl-Heinz Pfeffer: Analysis for technical colleges . Vieweg + teubner 2005, ISBN 3-528-54006-0 , p. 104 ( limited online copy in the Google book search)
• Wolfgang Brauch, Hans-Joachim Dreyer, Wolfhart Haacke: Mathematics for engineers. Vieweg + Teubner 2006, ISBN 3-8351-0073-4 , p. 104 ( limited online copy in the Google book search)
• Horst Stöcker: Pocket book of mathematical formulas and modern procedures . Harri Deutsch Verlag 2009, ISBN 978-3-8171-1812-0 , p. 146 ( limited online copy in the Google book search)

Web links

• Power functions with integer exponents (pdf; 373 kB)
• Power functions with natural exponents (pdf; 105 kB) - ZUM materials for the power function
• All cases of power functions with integer exponents in the form of interactive Java applets