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.
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.
| \(x\) | \(f(x)\) |
| \(-3\) | \(\frac{1}{8}\) |
| \(-2\) | \(\frac{1}{4}\) |
| \(-1\) | \(\frac{1}{2}\) |
| \(0\) | \(1\) |
| \(1\) | \(2\) |
| \(2\) | \(4\) |
| \(3\) | \(8\) |
| \(x\) | \(g(x)\) |
| \(-3\) | \(8\) |
| \(-2\) | \(4\) |
| \(-1\) | \(2\) |
| \(0\) | \(1\) |
| \(1\) | \(\frac{1}{2}\) |
| \(2\) | \(\frac{1}{4}\) |
| \(3\) | \(\frac{1}{8}\) |
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.
Domain: all real numbers; \((-\infty,\infty)\text{.}\)
Range: all positive numbers; \((0,\infty)\text{.}\)
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.
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 more info on horizontal asymptotes, see AppendixB.)
For exponential functions with \(0 \lt b \lt 1\text{,}\) this takes place close to the positive \(x\)-axis, as illustrated in Figure181b. For exponential functions with \(1 \lt b\text{,}\) this takes place close to 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 \(a = 1\text{.}\)
Compare the graphs of \(f (x) = 3^x\) and \(g(x) = 4^x\text{.}\)
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).
| \(x\) | \(f(x)\) | \(g(x)\) |
| \(-2\) | \(\dfrac{1}{9}\) | \(\dfrac{1}{16}\) |
| \(-1\) | \(\dfrac{1}{3}\) | \(\dfrac{1}{4}\) |
| \(0\) | \(1\) | \(1\) |
| \(1\) | \(3\) | \(4\) |
| \(2\) | \(9\) | \(16\) |
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.
