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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
\(\displaystyle{a^m\cdot a^n = a^{m+n}}\)
\(\displaystyle{\frac{a^m}{a^n}=a^{m-n}}\)
\(\displaystyle{\left(a^m\right)^n=a^{mn}}\)
\(\displaystyle{\left(ab\right)^n=a^n b^n}\)
\(\displaystyle{\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}}\)
Example308
\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 Example308 is shorter when we use negative exponents instead of converting the expressions into algebraic fractions.
Example309
Simplify by applying the laws of exponents.
\(\left(2a^{-4}\right) \left(-4a^2\right)\)
\(\dfrac{(r^2)^{-3}}{3r^{-4}}\)
Solution
\((2a^{-4})(-4a^2)=-8a^{-4+2}=-8a^{-2}=\dfrac{-8}{a^2} \)
\(\dfrac{(r^2)^{-3}}{3r^{-4}}=\dfrac{r^{-6}}{3r^{-4}}=\dfrac{1}{3r^2} \)
Caution310
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 Figure311. Note that the domains of power functions with negative exponents do not include zero.
Example312
Which of the following are power functions?
\(f (x) = \dfrac{1}{3}x^4 + 2\)
\(g(x) = \dfrac{1}{3x^4}\)
\(h(x) = \dfrac{x + 6}{x^3}\)
Solution
This is not a power function, because of the addition of the constant term.
We can write \(g(x) = \frac{1}{3}x^{-4}\text{,}\) so \(g\) is a power function.
This is not a power function, but it can be treated as the sum of two power functions, because \(h(x) = x^{-2} + 6x^{-3}\text{.}\)
Example313
Write each function as a power function in the form \(y = kx ^p\text{.}\)
\(f (x) = {12}{x^2}\)
\(g(x) = \dfrac{1}{4x}\)
\(h(x) = \dfrac{2}{5x^6}\)
Solution
\(f (x) = 12x^{-2}\)
\(g(x)=\dfrac{1}{4}x^{-1} \)
\(h(x)=\dfrac{2}{5}x^{-6} \)
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 interested 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.
Example314
Recall that the graph of \(f(x)=\frac{1}{x}\) and \(g(x)=\frac{1}{x^2}\) are given by:
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:
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.
Figure316 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.
Figure317 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.
Figure318 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.
Figure319