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SectionGraphs of Exponential Functions

In the previous section we listed some properties of exponential functions without much justification. In this section we will discuss why these properties hold, and investigate the long-run behavior of exponential functions.

SubsectionGraphs of Exponential Functions

The graphs of exponential functions have two characteristic shapes, depending on whether the base, \(b\text{,}\) is greater than \(1\) or less than \(1\text{.}\) As typical examples, consider the graphs of \(f (x) = 2^x\) and \(g(x) =\left(\dfrac{1}{2}\right)^x\) shown in Figure181. Some values for \(f\) and \(g\) are recorded in Tables179 and 180.

\(x\) \(f(x)\)
\(-3\) \(\frac{1}{8}\)
\(-2\) \(\frac{1}{4}\)
\(-1\) \(\frac{1}{2}\)
\(0\) \(1\)
\(1\) \(2\)
\(2\) \(4\)
\(3\) \(8\)
\(x\) \(g(x)\)
\(-3\) \(8\)
\(-2\) \(4\)
\(-1\) \(2\)
\(0\) \(1\)
\(1\) \(\frac{1}{2}\)
\(2\) \(\frac{1}{4}\)
\(3\) \(\frac{1}{8}\)
graphs of exponential growth and decay
Table179
Table180
Figure181

Notice that \(f (x) = 2^x\) is an increasing function and \(g(x) = \left(\dfrac{1}{2}\right)^x\) is a decreasing function. In general, exponential functions have the following properties.

Properties of Exponential Functions, \(f(x) = ab^x\text{,}\) \(a \gt 0\)
  1. Domain: all real numbers; \((-\infty,\infty)\text{.}\)

  2. Range: all positive numbers; \((0,\infty)\text{.}\)

  3. If \(b \gt 1\text{,}\) the function is increasing and said to have exponential growth;

    if \(0 \lt b \lt 1\text{,}\) the function is decreasing and said to have exponential decay.

  4. The \(y\)-intercept is \((0, a)\text{.}\) There is no \(x\)-intercept.

In Table179 you can see that as the \(x\)-values decrease toward negative infinity, the corresponding \(y\)-values decrease toward zero. As a result, the graph of \(f\) decreases toward the \(x\)-axis as we move to the left. Thus, the \(x\)-axis, \(y=0 \text{,}\) is a horizontal asymptote for exponential functions no matter what value \(b\text{,}\) the growth factor, happens to be. We can see this illustrated in Figure181, since the \(y\)-values of each exponential get closer to zero.

For exponential functions with \(0 \lt b \lt 1\text{,}\) this takes place close the positive \(x\)-axis, as illustrated in Figure181b. For exponential functions with \(1 \lt b\text{,}\) this takes place close the negative \(x\)-axis, as illustrated in Figure181a.

In Example182, we compare two increasing exponential functions. The larger the value of the base, \(b\text{,}\) the faster the function grows. In this example, both functions have \(a = 1\text{.}\)

Example182

Compare the graphs of \(f (x) = 3^x\) and \(g(x) = 4^x\text{.}\)

Solution

We evaluate each function for several convenient values, as shown in Table183.

Plot the points for each function and connect them with smooth curves. For positive \(x\)-values, \(g(x)\) is always larger than \(f (x)\text{,}\) and is increasing more rapidly. In Figure184, \(g(x) = 4^x\) climbs more rapidly than \(f (x) = 3^x\text{.}\) Both graphs cross the \(y\)-axis at (0, 1).

\(x\) \(f(x)\) \(g(x)\)
\(-2\) \(\dfrac{1}{9}\) \(\dfrac{1}{16}\)
\(-1\) \(\dfrac{1}{3}\) \(\dfrac{1}{4}\)
\(0\) \(1\) \(1\)
\(1\) \(3\) \(4\)
\(2\) \(9\) \(16\)
two exponential growth functions
Table183
Figure184

For decreasing exponential functions, those with bases between \(0\) and \(1\text{,}\) the smaller the base, the more steeply the graph decreases. For example, compare the graphs of \(p(x) = 0.8^x\) and \(q(x) = 0.5^x\) shown in Figure185.

two exponential decay functions
Figure185
Example186
  1. State the ranges of the functions \(f\) and \(g\) in Figure184 from Example182 on the domain \([-2, 2]\text{.}\)
  2. State the ranges of the functions \(p\) and \(q\) shown in Figure185 on the domain \([-2, 2]\text{.}\) Round your answers to two decimal places.
Solution
  1. On the domain \([-2,2]\text{,}\) the smallest values for both \(f\) and \(g\) occur when \(x=-2\text{,}\) and the largest values occur when \(x=2\) (since both of these functions are increasing). Since
    \begin{equation*} \begin{aligned} f(-2) \amp= 3^{-2}=\frac{1}{9} \\ f(2) \amp= 3^2=9 \\ g(-2) \amp= 4^{-2}=\frac{1}{16} \\ g(2) \amp= 4^2= 16, \end{aligned} \end{equation*}
    the range of \(f\) on the domain \([-2,2]\) is \(\left[\frac{1}{9},9\right]\text{;}\) the range of \(g\) on the domain \([-2,2]\) is \(\left[\frac{1}{16},16\right]\text{.}\)
  2. On the domain \([-2,2]\text{,}\) the smallest values for both \(p\) and \(q\) occurs when \(x=2\text{,}\) and the largest values occurs when \(x=-2\) (since both of these functions are decreasing). Since
    \begin{equation*} \begin{aligned} p(-2) \amp= 0.5^{-2}=4 \\ p(2) \amp= 0.5^2=0.25 \\ q(-2) \amp= 0.8^{-2}\approx1.56 \\ q(2) \amp= 0.8^2= 0.64, \end{aligned} \end{equation*}
    the range of \(p\) on the domain \([-2,2]\) is \(\left[0.25,4\right]\text{;}\) the range of \(q\) on the domain \([-2,2]\) is \(\left[0.64,1.5625\right]\text{.}\)

SubsectionExercises