Power Functions with Odd Exponents

Power functions of the form f of x equals a times x raised to the power n, where n is a positive odd integer, are symmetric about the origin.

Rotating their graphs 180 degrees around the origin leaves them unchanged.

As the odd exponent increases, the graph stays closer to the x-axis near the origin and becomes steeper beyond negative 1 and 1. For example, x to the fifth changes more slowly near zero than x cubed, but changes more rapidly farther from zero. The domain and range of a power function with an odd exponent are all real numbers.

In a real-world application, a model may use only part of this domain. For example, in an idealized model, the available power in wind is proportional to the cube of the wind speed.

So, if the wind speed doubles, the available power increases by a factor of eight, assuming the other factors remain constant.

Because wind speed is nonnegative, the practical domain includes only nonnegative values. The model also gives power greater than or equal to zero, so its practical range is nonnegative.

This shows how a real-world application can restrict the domain of a power function.