Power Functions with Even Exponents

A power function has the form f of x equals a times x raised to the power n, where a is nonzero and n is a positive integer.

When n is even, the graph is symmetric about the y-axis and its domain is all real numbers. If a is positive, its range is all nonnegative real numbers. If a is negative, its range is all nonpositive real numbers.

As the even exponent increases, the graph stays closer to the x-axis near the origin and becomes steeper beyond -1 and 1, as seen by comparing x to the fourth with x squared.

A practical example is water flow through a pipe. Under ideal laminar flow conditions, the flow rate is proportional to the fourth power of the pipe's radius.

For this physical model, the pipe radius is restricted to positive real numbers, and the corresponding flow rates are positive. So, the model’s domain and range are positive real numbers.

Because the flow rate depends on the fourth power of the radius, it increases rapidly as the radius grows. For example, doubling the radius increases the flow rate by a factor of 16. Engineers use this relationship to design efficient piping systems.