### Basic functions: Power functions

### Power functions with negative integer exponents

A power function with a negative integer exponent can be written according to the rules of calculation as a quotient of a constant function and a power function with a natural number as exponent. In formula form: \[f(x)=c\cdot x^{-n}=\frac{c}{x^n}\quad\text{for }n=1,2,3,\ldots\] The graph of such a power function looks different from the graph of a power function with a natural number greater than 1 as exponent.

Graph of a power function with a negative integer exponent The shape of the graph of the function \(f(x)=c\cdot x^n\) with a negative integer exponent depends on whether \(n\) is even or odd. See the interactive figures below. They illustrate that the characteristics of the power function also differ because of this.

negative even exponent

The power function \(f(x)=x^{-n}\) with an even natural number \(n\) unequal \(0\) is an even function with range \(\ivoo{0}{\infty}\).

negative odd exponent

The power function \(f(x)=x^{-n}\) with an odd natural number \(n\) unequal \(1\) is an odd function with range \(\mathbb{R}\backslash\{0\}\).