Basic functions: Power functions
Definition
A power function has the form \[f(x)=c\cdot x^p,\] in which \(c\) and \(p\) are nonzero parameters. If the exponent \(p\) in the term \(x^p\) is a nonzero natural number, then it is also called the degree of the power function and we speak of a \(p\)th degree power function.
The domain of a power function depends on the value \(p\): If \(p\) is a natural number, then the domain consists of all real numbers. If \(p=\tfrac{1}{n}\) with a natural number \(n\), then the domain consists of all real numbers for odd \(n\) and the domain is equal to \(\ivco {0}{\infty}\) for even \(n\). In this case we also speak of a square root function. Negative values of \(p\) always exclude the number \(0\) in the domain.
The range and characteristics of a power function also strongly depend on the exponent.
When two variables \(x\) and \(y\) are related via \(y=c\cdot x^p\), then we speak of a power relationship between \(y\) and \(x\). If \(c>0\), then this relationship is reciprocal, i.e., if there is a power relationship between \(x\) and \(y\), then there is also a power relationship between \(y\) and \(x \). After all, in this case \(x=c^{-\frac{1}{p}}\cdot is y^{\frac{1}{p}}\).
Example The area \(A(r)\) and the volume \(V(r)\) of a sphere as functions of the radius \(r\) are 2-and 3-degree power functions: \(\;\displaystyle A(r)=4\pi\,r^2,\;\; V(r)=\frac{4}{3}\pi\,r^3\) .
Coulomb's Law The electric force \(F_\mathrm{el} (r)\) that two point charges \(Q\) and \(q\) exert on each other at a distance \(r\) is, according to Coulomb's Law, a power function with exponent \(-2\), i.e., given by the formula \(\;\displaystyle F_\mathrm{el} (r)=f\cdot \frac{Q\cdot q}{r^2}\), for some constant \(f\).
Graph of a power function with a natural number greater then 1 as exponent The shape of the graph of the function \(f(x)=c\cdot x^n\) with natural number \(n\) as exponent depends on whether \(n\) is even or odd. See the interactive figures below. The illustrate that the characteristics of the power functions also differ.
even exponent
The power function \(f(x)=x^{n}\) with an even number \(n\) (here nonzero) is an even function
with range \(\ivco{0}{\infty}\).
odd exponent
The power function \(f(x)=x^{n}\) with an odd number \(n\) (here not equal to 1) is an odd function with range \(\ivoo{-\infty}{\infty}\).
Calculation rules for power functions Let \(a\) and \(b\) are real numbers and \(x\) and \(y\) are positive real numbers, then the following equalities are true:
- \(\left(x^a\right)^b = x^{a\cdot b}\)
- \(\left(x\cdot y\right)^a = x^a\cdot y^a\)
- \(x^a\cdot x^b = x^{a+b}\)
- \(\frac{x^a}{x^b}=x^{a-b}\)
Matchcentre video
Indices or Powers (32:26)
