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Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in [link] .
Year | $$2009$$ | $$2010$$ | $$2011$$ | $$2012$$ | $$2013$$ |
Bird Population | $$800$$ | $$897$$ | $$992$$ | $$1,083$$ | $$1,169$$ |
The population can be estimated using the function $\text{\hspace{0.17em}}P(t)=-0.3{t}^{3}+97t+800,\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}P(t)\text{\hspace{0.17em}}$ represents the bird population on the island $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.
Before we can understand the bird problem, it will be helpful to understand a different 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.
As an example, consider functions for area or volume. The function for the area of a circle with radius $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ is
and the function for the volume of a sphere with radius $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ is
Both of these are examples of power functions because they consist of a coefficient, $\text{\hspace{0.17em}}\pi \text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\frac{4}{3}\pi ,\text{\hspace{0.17em}}$ multiplied by a variable $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ raised to a power.
A power function is a function that can be represented in the form
where $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ are real numbers, and $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is known as the coefficient .
Is $\text{\hspace{0.17em}}f(x)={2}^{x}\text{\hspace{0.17em}}$ 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.
Which of the following functions are power functions?
$$\begin{array}{cccc}\hfill f(x)& =& 1\hfill & \phantom{\rule{2em}{0ex}}\text{Constantfunction}\hfill \\ \hfill f(x)& =& x\hfill & \phantom{\rule{2em}{0ex}}\text{Identifyfunction}\hfill \\ \hfill f(x)& =& {x}^{2}\hfill & \phantom{\rule{2em}{0ex}}\text{Quadraticfunction}\hfill \\ \hfill f(x)& =& {x}^{3}\hfill & \phantom{\rule{2em}{0ex}}\text{Cubicfunction}\hfill \\ \hfill f(x)& =& \frac{1}{x}\hfill & \phantom{\rule{2em}{0ex}}\text{Reciprocalfunction}\hfill \\ \hfill f(x)& =& \frac{1}{{x}^{2}}\hfill & \phantom{\rule{2em}{0ex}}\text{Reciprocalsquaredfunction}\hfill \\ \hfill f(x)& =& \sqrt{x}\hfill & \phantom{\rule{2em}{0ex}}\text{Squarerootfunction}\hfill \\ \hfill f(x)& =& \sqrt[3]{x}\hfill & \phantom{\rule{2em}{0ex}}\text{Cuberootfunction}\hfill \end{array}$$
All of the listed functions are power functions.
The constant and identity functions are power functions because they can be written as $\text{\hspace{0.17em}}f(x)={x}^{0}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f(x)={x}^{1}\text{\hspace{0.17em}}$ respectively.
The quadratic and cubic functions are power functions with whole number powers $\text{\hspace{0.17em}}f(x)={x}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f(x)={x}^{3}.$
The reciprocal and reciprocal squared functions are power functions with negative whole number powers because they can be written as $\text{\hspace{0.17em}}f(x)={x}^{-1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f(x)={x}^{-2}.$
The square and cube root functions are power functions with fractional powers because they can be written as $\text{\hspace{0.17em}}f(x)={x}^{\frac{1}{2}}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}f(x)={x}^{\frac{1}{3}}.$
Which functions are power functions?
$\begin{array}{ccc}\hfill f(x)& =& 2x\cdot 4{x}^{3}\hfill \\ \hfill g(x)& =& -{x}^{5}+5{x}^{3}\hfill \\ \hfill h(x)& =& \frac{2{x}^{5}-1}{3{x}^{2}+4}\hfill \end{array}$
$f(x)\text{\hspace{0.17em}}$ is a power function because it can be written as $\text{\hspace{0.17em}}f(x)=8{x}^{5}.\text{\hspace{0.17em}}$ The other functions are not power functions.
[link] shows the graphs of $\text{\hspace{0.17em}}f(x)={x}^{2},\text{\hspace{0.17em}}g(x)={x}^{4}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h(x)={x}^{6},\text{\hspace{0.17em}}$ which are all power functions with even, whole-number powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.
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