2.1 Power Functions

Introduction

Part of the enjoyment of art comes from its patterns of sound and color and motion.

Patterns are also an essential part of mathematics. The British mathematician G. H. Hardy once said that "A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas".

In this section, we will find an underlying structure that unifies the six basic functions from Chapter 1 and opens up a new category of functions for our use.

Power Functions

When working with a group of functions, the first pattern we look for is any similarity in the form, or equation, of the functions. Once we have identified a pattern, we can create new functions by changing the parameters of that basic equation.

For instance, these three functions

\begin{align} \text{identity function} && f(x) &= x = x^1 \newline \text{square function} && f(x) &= x^{2} \newline \text{cube function} && f(x) &= x^3 \newline \end{align}

are all written in the form $f(x)=x^{p}$, where $p$ is the power of the variable $x$. New functions can be made by taking that basic format and changing the power $p$. Since the power is the important parameter of the equation, we will call these power functions.

In general, a power function is any function that can be written as $f(x)=x^{p}$, where $p$ is a constant. Notice that the variable must be raised to a fixed power. That power can be positive or negative, a whole number, a fraction or a decimal, but it cannot be a variable. As an example, $g(x)=3^{x}$ has a variable exponent, so it cannot be a power function even though it looks somewhat similar.

Use Rules of Exponents to Identify Power Functions

We now know that the identity, square and cube are all power functions, but what about the other basic functions from Chapter 1? At first glance, these functions

$$ \begin{align} \text{reciprocal function} && f(x) &= \frac{1}{x} \newline \text{square root function} && f(x) &= \sqrt{x} \newline \text{cube root function} && f(x) &= \sqrt[3]{x} \end{align} $$

do not appear to be power functions since they are not written in the form $f(x)=x^{p}$.

However, any function that can be written as $f(x)=x^{ p}$ is a power function, even if it is often written in a different format. A brief review of a few rules of exponents will show that these three functions are indeed power functions.

Negative Powers

You might recall the rule $\frac{1}{x}=x^{-1}$. This tells us that the reciprocal function $f(x) = \frac{1}{x}$ is the same as the power function $f(x) = x^{-1}$. The general rule of exponents at work here is

\begin{equation} \frac{1}{x^{n}}=x^{-n} \end{equation}

This rule can be used to convert any reciprocal-type function into a power function. For instance, $f(x)=\frac{1}{x^{5}}$ can be rewritten in the standard power function format as $f(x)=x^{-5}$.

Roots

Another rule of exponents allows us to rewrite roots as fractional powers. Specifically,

$$ \sqrt[m]{x} = x^{1/m} $$

From this we see that the square root function $f(x)=\sqrt{x}$ and the cube root function $f(x)=\sqrt[3]{x}$ can be written as power functions where the powers are $1/2$ and $1/3$, respectively.

\begin{equation} f(x)=\sqrt{{x}} = x^{1/2} \end{equation}

and

$$ f(x) = \sqrt[3]{x} = x^{1/3} $$

Other roots can be rewritten as power functions in the same way. For instance, $f(x)=\sqrt[8]{x} = x^{1/8}$.

Fractional Powers

The family of power functions also includes roots of powers, such as $f(x)=\sqrt[m]{x^{n}}$ , or powers of roots like $f(x)=(\sqrt[m]{x})^{n}$, which happen to be the same thing. These can be rewritten as power functions where the power is a rational number, or faction, of the form $p=n/m$, such as $-3/8$ or $2/5$.

$$ \sqrt[m]{ x^n } = \left(\sqrt[m]{x}\right)^{n} = x^{n/m} $$

Since all three forms are equivalent, you may be asked to convert one into the others. The power function $f(x) = x^{2/5 }$, for instance, can also be written as $f(x) = \sqrt[5]{x^{2}}$ or as $f(x) = (\sqrt[5]{x})^2$.

Evaluating Power functions

When using a calculator to evaluate and graph power functions we need to be careful to wrap $x$ and/or $p$ in parenthesis anytime either one is anything other than a whole number. For example, if $f(x)=x^{2/5}$ then to evaluate $f(-3/8)$ parenthesis should be put around both $-3/8$ and around the power $2/5$. Without parenthesis, the normal order of operations would give a very different answer.

Correct evaluation of $(-3/8)^{2/5}$	Incorrect evaluation of $(-3/8)^{2/5}$
(-3/8)^(2/5) = 0.675480019	-3/8^2/5 = -0.009375

We should also keep in mind that many power functions are not defined when $x<0$, and that if the power is negative then $0$ is not in the domain. Expressions involving either of these will produce an error message on your calculator.

Graphs of Power Functions

While technology can be used to graph power functions, you can often create a "good enough" sketch within seconds just by examining the equation. The shape of the graph of a power function $f(x)=x^{ p}$ is controlled by the value of the power $p$. In this interactive figure you can use the blue slider to change $p$.

Since many powers are undefined when $x< 0$, we will focus on the first quadrant for now. As you change $p$, look for patterns or similarities between the different graphs. In particular, try to group the graphs into three distinct shapes.

As you may have noticed, all power functions pass through the point $(1,\ 1)$, though they do so in three different ways depending on the power $p$. The three behavior patterns are separated from one another by $p=0$ (when we get the horizontal line $y=1$) and $p=1$ (when we get the identity function $y=x$).

In Chapter 1 you learned how to identify several characteristics of a function by reading its graph. Since understanding the properties of these three shapes will impact your ability to use power functions effectively, our next goal is to analyze each of these shapes individually.

Case 1: $\color{red}{p<0}$

Let's start with the case where the power is negative, that is $p<0$. Notice that this graph resembles the right side of the reciprocal function.

Looking at the graph we see that it drops sharply along the y-axis and levels off toward the positive x-axis, as you move from left to right. In mathematical terms, the graph is both decreasing and concave up on the interval $(0, \infty)$, with a vertical asymptote at $x=0$ (the y-axis) and a horizontal asymptote at $y=0$ (the x-axis).

Because of the asymptotes, the graph never reaches a minimum nor a maximum value. Based on the shape of the graph, it would be reasonable to use this type of power function to represent quantities that decay as $x$ increases.

Case 2: $\color{green}{0<p<1}$

If the power is small, that is $0<p<1$, then the graph of the power function $f(x)=x^{p}$ is increasing and concave down over the interval $(0, \infty)$. Note that this shape resembles the square root function.

The graph has a minimum value at $(0, 0)$, but does not have a maximum or any asymptotes. Since the graph is concave down, the rate of increase is decreasing, suggesting that this type of power function might model quantities whose growth is slowing down.

Case 3: $\color{blue}{p>1}$

When $p>1$, the power function $f(x)=x^{p}$ increases and is concave up on the interval $(0, \infty)$. This graph resembles the right side of the square function.

There is a minimum value at the origin $(0, 0)$, but no maximums or asymptotes. These functions will be useful when modeling quantities that grow faster and faster. Lastly, note that this graph, along with the two previous ones, is one-to-one and must have an inverse.

Symmetry of Power Functions

So far, we have restricted our attention to power functions with $x \geq 0$ because many powers are not defined when $x<0$.

In particular, if the power is a rational number, or fraction, of the form $p =\color{blue}{n}/\color{green}{m}$ (and the power has been reduced) then the function is not defined when $x<0$. The reason for this is that fractional powers with even denominators are even roots, and even roots of negatives are not real numbers.

However, if the denominator $\color{green}{m}$ is odd then the function is defined for negative values of $x$. You can explore this in the interactive figure below.

Notice for an odd denominator $\color{green}{m}$, the function exhibits even symmetry when the numerator $\color{blue}{n}$ is even and odd symmetry when the numerator $\color{blue}{n}$ is odd.

So a quick look at the power helps us know where the function is defined and what type of symmetry it has.

Use Symmetry to Graph Power Functions

Knowing how the values of $\color{blue}{n}$ and $\color{green}{m}$ impact the symmetry of $f(x)=x^{p}$ with $p = \color{blue}{n} / \color{green}{m}$ also allows us to quickly sketch the graph of a power function.

Consider, for instance, $f(x) = x^{\color{blue}{2}/\color{green}{3}}$. The power $p =\color{blue}{2}/\color{green}{3}$ is between $0$ and $1$, so the graph of this function looks similar to the basic shape for $0<p<1$, which resembles the square root function. The denominator $\color{green}{3}$ is an odd number, which tells us that this function has a left side. And since the numerator $\color{blue}{2}$ is even, we should reflect the right side across the y-axis to create a graph with even symmetry. Thus, the graph of $f(x) = x^{2/3}$ looks something like this:

If the numerator had been odd, such as $f(x) = x^{5/3}$ then we would have reflected around the origin instead, ending up with this rough sketch:

Notice that the shape of the right side of $f(x) = x^{5/3}$ is similar to the right side of the square function, since $5/3>1$.

In summary, to sketch the graph of a power function with a rational exponent, first choose the basic shape that matches the power $p$.

Then use the symmetry rules to draw the other half of the graph, if it exists.

• If the denominator $\color{green}{m}$ is odd, then the function also exists when $x<0$ so it has a left side.
• When $\color{green}{m}$ is odd, the function has even symmetry if $\color{blue}{n}$ is even and odd symmetry if $\color{blue}{n}$ is odd.

Inverses of Power Functions

Finding the inverse of a power function is a very straightforward process. As an illustration, consider the cube function $f(x) = x^{3}$ and its inverse, the cube root function $f^{-1}(x) = \sqrt[3]{x}$.

Since the cube root can be written as the one-third power $x^{1/3}$, the inverse of $f(x)=x^{3}$ is the power function $f^{-1}(x)=x^{1/3}$.

Notice that the two powers are reciprocals of each other, that is the key.

In general, the inverse of any power function is the power function whose power is the reciprocal of the other.

That is to say, given some power function $f(x)=x^{p}$, then its inverse is always $f^{-1}(x)=x^{1/p}$, assuming $p\neq 0$.

Of course, if the function is not one-to-one (like in the case of $f(x)=x^2$), then we should restrict ourselves to a portion of the domain (usually $x \geq 0$) where the function is one-to-one.

How Power Functions Change

Our last task for this section is to see if we can find any patterns in the way power functions change.

You will recall that the difference quotient $D(x)$ is a function that lets us calculate the average rate of change on a generic interval $[x, x+h]$, where $h$ is any positive number. The difference quotient of a function is given by

\begin{equation} D(x)=\frac{f(x+h)-f(x)}{h} \end{equation}

Below are the difference quotients for the power functions $f(x)=x^2$ and $f(x)=x^3$.

Function	Difference quotient $D(x)$	Power function similar to $D(x)$
$x^{2}$	$2\color{black}{\boldsymbol{x}}+h$ (see steps)	$\color{black}{\boldsymbol{x^1}}$
$x^3$	$3\color{black}{\boldsymbol{x^2}}+3xh+h^2$ (see steps)	$\color{black}{\boldsymbol{x^{2}}}$

Click the link under each difference quotient to see the steps required to arrive at the equation.

Notice that the difference quotient of each power function is similar to the next smaller power function. In general, the difference quotient of $f(x)=x^p$ will involve the power function $f(x)=x^{p-1}$ for any $p\neq0$. This is a fact that is verified in the first term of calculus.

Looking Ahead

In this section we discovered that the basic library functions could all be written in an identical algebraic form. We then used that common form to create a new category of functions, whose properties we explored. This is a process that will be repeated in later chapters as well. We'll start with functions we understand and then create new functions that have slightly different algebraic forms and investigate their properties.

An understanding of the properties and behaviors of functions is essential when working with applications, which is where we will head next.