Identify power functions
In order to better understand the bird problem, we need to understand a specific type of function. A power function is a function with a single term that is the product of a real number, a coefficient, and a variable raised to a fixed real number. (A number that multiplies a variable raised to an exponent is known as a coefficient.)
As an example, consider functions for area or volume. The function for the area of a circle with radius r is
and the function for the volume of a sphere with radius r is
Both of these are examples of power functions because they consist of a coefficient, or , multiplied by a variable r raised to a power.
A General Note: Power Function
A power function is a function that can be represented in the form
where k and p are real numbers, and k is known as the coefficient.
Q & A
Is a power function?
No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.
Example 1: Identifying Power Functions
Which of the following functions are power functions?
Solution
All of the listed functions are power functions.
The constant and identity functions are power functions because they can be written as and respectively.
The quadratic and cubic functions are power functions with whole number powers and .
The reciprocal and reciprocal squared functions are power functions with negative whole number powers because they can be written as and .
The square and cube root functions are power functions with fractional powers because they can be written as or .
Try It 1
Which functions are power functions?
Solution