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SectionPower Functions

Before we begin discussing the behavior of power functions, it is important that we remember our laws of exponents. Recall that a positive integer exponent tells us how many times its base occurs as a factor in an expression. For example,

\begin{equation*} 4a^3b^2 \text{ means } 4aaabb. \end{equation*}

Recall too the definition of a negative or zero exponent:

Definition of Negative and Zero Exponents
\begin{equation*} \begin{aligned} a^{-n} \amp = \frac{1}{a^n} \amp\amp (a \ne 0) \\ a^0 \amp = 1 \amp\amp (a \ne 0) \end{aligned} \end{equation*}

As we begin combining both positive and negative exponents, we adhere to the following rules:

Laws of Exponents
  1. \(\displaystyle{a^m\cdot a^n = a^{m+n}}\)

  2. \(\displaystyle{\frac{a^m}{a^n}=a^{m-n}}\)

  3. \(\displaystyle{\left(a^m\right)^n=a^{mn}}\)

  4. \(\displaystyle{\left(ab\right)^n=a^n b^n}\)

  5. \(\displaystyle{\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}}\)

Example298

\begin{align*} \text{a}\amp. ~x^3\cdot x^{-5} = x^{3-5} = x^{-2} \amp\amp \text{Apply the first law: Add exponents.} \\ \text{b}\amp. ~\frac{8x^{-2}}{4x^{-6}}= \frac{8}{4}x^{-2-(-6)} = 2x^4 \amp\amp \text{Apply the second law: Subtract exponents.} \\ \text{c}\amp. ~\left(5x^{-3}\right)^{-2}= 5^{-2}(x^{-3})^{-2}=\frac{x^6}{25} \amp\amp \text{Apply laws IV and III.} \end{align*}

You can check that each of the calculations in Example298 is shorter when we use negative exponents instead of converting the expressions into algebraic fractions.

Example299

Simplify by applying the laws of exponents.

  1. \(\left(2a^{-4}\right) \left(-4a^2\right)\)

  2. \(\dfrac{(r^2)^{-3}}{3r^{-4}}\)

Solution
  1. \((2a^{-4})(-4a^2)=-8a^{-4+2}=-8a^{-2}=\dfrac{-8}{a^2} \)

  2. \(\dfrac{(r^2)^{-3}}{3r^{-4}}=\dfrac{r^{-6}}{3r^{-4}}=\dfrac{1}{3r^2} \)

Caution300

The laws of exponents do not apply to sums or differences of powers. We can add or subtract like terms, that is, powers of the same variable with the same exponent. For example,

\begin{equation*} 6x^{-2} + 3x^{-2} = 9x^{-2} \end{equation*}

but we cannot add or subtract terms with different exponents. Thus, for example, \begin{align*} 4x^2 \amp- 3x^{-2} \amp\amp \text{cannot be simplified, and}\\ x^{-1} \amp + x^{-3}\amp\amp \text{cannot be simplified}. \end{align*}

SubsectionPower Functions

Power Function

A function of the form

\begin{equation*} f(x) = k(x)^p \end{equation*}

where \(k\) is a nonzero constant and \(p\) is any constant, is called a power function. In particular, if \(p=0\text{,}\) then \(f(x)\) is the constant function \(f(x) = k\text{.}\)

Examples of power functions are

\begin{equation*} V(r ) = \frac{4}{3}\pi (r)^3 ~~\text{ and }~~L(T ) = 0.8125(T)^2. \end{equation*}

In addition, the basic functions

\begin{equation*} f (x) = \frac{1}{x} ~~\text{ and }~~ g(x) = \frac{1}{x^2} \end{equation*}

can be written as

\begin{equation*} f (x) = x^{-1} ~~\text{ and }~~ g(x) = x^{-2}. \end{equation*}

Their graphs are shown in Figure301. Note that the domains of power functions with negative exponents do not include zero.

graphs of the two basic reciprocal functions
Figure301
Example302

Which of the following are power functions?

  1. \(f (x) = \dfrac{1}{3}x^4 + 2\)

  2. \(g(x) = \dfrac{1}{3x^4}\)

  3. \(h(x) = \dfrac{x + 6}{x^3}\)

Solution
  1. This is not a power function, because of the addition of the constant term.

  2. We can write \(g(x) = \frac{1}{3}x^{-4}\text{,}\) so \(g\) is a power function.

  3. This is not a power function, but it can be treated as the sum of two power functions, because \(h(x) = x^{-2} + 6x^{-3}\text{.}\)

Example303

Write each function as a power function in the form \(y = kx ^p\text{.}\)

  1. \(f (x) = {12}{x^2}\)

  2. \(g(x) = \dfrac{1}{4x}\)

  3. \(h(x) = \dfrac{2}{5x^6}\)

Solution
  1. \(f (x) = 12x^{-2}\)

  2. \(g(x)=\dfrac{1}{4}x^{-1} \)

  3. \(h(x)=\dfrac{2}{5}x^{-6} \)

SubsectionLong Run Behavior

While power functions do not in general have to have integer exponents, these will be the types of power functions we are most interrested in. In particular, we will look at the graphs and long run behavior of power functions with integer exponents. By long run behavior of a function we mean the value the function approaches as the input \(x \) goes to infinity or negative infinity.

Example304

Recall that the graph of \(f(x)=\frac{1}{x}\) and \(g(x)=\frac{1}{x^2}\) are given by:

graphs of the two basic reciprocal functions
Figure305
Identify the long run behavior of each of these functions.

Solution

Let's begin with the long run behavior of \(f(x) \text{.}\) Looking at the graph, we see that as \(x \) goes to infinity (i.e. as we move farther to the right on the \(x \)-axis) the output values of \(f(x)\) are getting close to 0. Similarly, as \(x \) goes to negative infinity (as we move farther to the left on the \(x\)-axis) the output values of \(f(x)\) are also getting closer to 0. We write this as

\begin{equation*} \text{as}\ \ x\to\pm\infty\ \ f(x)\to 0\text{.} \end{equation*}

By identical reasoning as above, we also have that

\begin{equation*} \text{as}\ \ x\to\pm\infty\ \ g(x)\to 0\text{.} \end{equation*}

In general, we have the following behaviors of power functions with integer exponents:

  1. The graph of

    \begin{equation*} y=x^\text{(odd positive number)} \end{equation*}

    is shown to the right. The long run behavior of such a function is given by \begin{align*} x^\text{(odd positive number)}\to -\infty\ \amp \text{as}\ x\to -\infty\ ,\ \ \text{and}\\ x^\text{(odd positive number)}\to \infty\ \amp \text{as}\ x\to \infty. \end{align*} Examples of such functions are \(x,x^3,x^5,\) etc.

    graphs positive odd powers
    Figure306

  2. The graph of

    \begin{equation*} y=x^\text{even positive number} \end{equation*}

    is shown to the right. The long run behavior of such a function is given by \begin{align*} x^\text{even positive number}\to \infty\ \amp \text{as}\ x\to \pm\infty. \end{align*} Examples of such functions are \(x^2,x^4,x^6,\) etc.

    graphs positive even powers
    Figure307

  3. The graph of

    \begin{equation*} y=x^\text{odd negative number} \end{equation*}

    is shown to the right. The long run behavior of such a function is given by \begin{align*} x^\text{odd negative number}\to 0\ \amp \text{as}\ x\to \pm\infty. \end{align*} Examples of such functions are \(x^{-1},x^{-3},x^{-5},\) etc.

    graphs negative odd powers
    Figure308

  4. The graph of

    \begin{equation*} y=x^\text{even negative number} \end{equation*}

    is shown to the right. The long run behavior of such a function is given by \begin{align*} x^\text{even negative number}\to 0\ \amp \text{as}\ x\to \pm\infty \end{align*} Examples of such functions are \(x^{-2},x^{-4},x^{-6},\) etc.

    graphs negative even powers
    Figure309

SubsectionExercises