Chapter 3: Power Functions

We next turn our attention to a large and useful family of functions called power functions. This family includes transformations of several of the basic functions, such as

$$F(d)=\frac{k}{d^2}\text{ and }S(T)=20.06\sqrt{T}$$

The first function gives the gravitational force, $F$, exerted by the sun on an object at a distance, $d$. The second function gives the speed of sound, $S$, in terms of the air temperature, $T$.

By extending our definition of exponent to include negative numbers and fractions, we will be able to express such functions in the form $f(x)=k{x}^n$. Here is an example of a power function with a fractional exponent.

In 1932, Max Kleiber published a remarkable equation for the metabolic rate of an animal as a function of its mass. The table at right shows the mass of various animals in kilograms and their metabolic rates, in kilocalories per day. A plot of the data, resulting in the famous “mouse-to-elephant” curve, is shown in the figure.

Kleiber modeled his data by the power function

$$P(m)=73.3m^{0.74}$$

where $P$ is the metabolic rate and $m$ is the mass of the animal. Kleiber's rule initiated the use of allometric equations, or power functions of mass, in physiology.