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 $x=c^{-\frac{1}{p}}\cdot is y^{\frac{1}{p}}$.