What we learn here:
Defintion and Domain
A functions of the form $f(x)=b^{x}$ where $b>0$ is a constant and $x$ is the variable is called an “exponential” function with base $b$ because the variable $x$ appears in the “exponent.”
- When $b<0$, the expression $b^{x}$ is defined only for rational numbers $x=p/q$ where $q$ is an odd integer. But if $b>0$, the expression $b^{x}$ is defined for all $x$. Therefore, in calculus, the exponential functions are studied only on the assumption that the base is positive.
- The expression $b^{x}$ is defined for all $x$. The domain of $f(x)=b^{x}$ is hence $(-\infty,\infty)$ or $\mathbb{R}$.
- We note that $b^{x}>0$ for any $x$. Therefore, the graph of $f(x)=b^{x}$ always lies above the $x$-axis.
- Recall that $b^{0}=1$ (for $b\neq0$). Therefore, the graph of $f(x)=b^{x}$ cuts the $y$-axis at $y=1$. In other words, it passes through $(0,1)$.
Behavior and Range
The behavior of $y=b^{x}$ depends on whether $b>1$ or $0<b<1$. The case $y=1^{x}$ is uninteresting because it reduces to a constant $y=1$.
When $b>1$ (Figure 1(a)), if $x$ is a large positive number, $y$ is also large. The larger numerically the negative value of $x$ becomes, the closer $y$ becomes to zero and the curve approaches asymptotically the negative $x$-axis. Note that no matter how the curve $y=b^{x}$ looks like in a graphing calculator, if you zoom in, you will realize that the curve $y=b^{x}$ never touches the negative $x$-axis. The range of $y=b^{x}$ is thus $(0,\infty)$.
Conversely, when $0<b<1$ (Figure 1(b)), the curve $y=b^{x}$ indefinitely approaches $y=0$ on indefinite increase in $x$, and rises as $x$ gets numerically large but negative. The range of $y=b^{x}$ ($0<b<1$) is also $(0,\infty)$.
If $f(x)=b^{x}$ (with $b>0$):
\[
Dom(f)=\mathbb{R}=(-\infty,\infty)\qquad\text{and}\qquad Rng(f)=(0,\infty).
\]
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| (a) $y=1.5^{x},y=2^{x}$, and $y=5^{x}$. | (b) $y=0.2^{x},y=0.5^{x}$, and $y=0.7^{x}$ |
Figure 1
The graphs of the exponential functions with $b=10,3,2,1/10,1/3,1/2$ are illustrated in Figure 2. It is obvious that the graph of $y=(1/b)^{x}$ is the reflection of the graph of $y=b^{x}$ in the $y$-axis. But why is that? Note that we can write $(1/b)^{x}$ as $b^{-x}$, and obviously $y=b^{-x}$ takes the same values for positive $x$ as the function $y=b^{x}$ takes for negative $x$ with the same absolute value and vice versa. This implies that the graph of $(1/b)^{x}$ and $b^{x}$ are reflections of one another in the $y$-axis.
Because $1/b>1$ if $0<b<1$ and $0<1/b<1$ if $b>1$, there corresponds to every graph of an exponential function with base $<1$, the graph of an exponential with base $>1$, and these graphs are reflections of one another in the $y$-axis.
Figure 2
Graphs of $y=b^{x}$ and $y=b^{-x}=(1/b)^{x}$ are symmetrical with respect to the $y$-axis.
Growth Rate
Figure 3 shows the graphs of $y=2^{x}$ and $y=x^{2}$. The graphs intersect three times, but for $x>4$ the graph $y=2^{x}$ always stays above the graph of $y=x^{2}$ and the difference between them steeply increases at $x$ gets larger.
When $b>1$, the exponential function grows very rapidly for large values of $x$. When $x$ is large even a modest increase in $x$ results in a comparatively large increase in $y$. No matter what the base of the exponential function $y=b^{x}$ is (when $b>1$), its growth finally outstrips the growth of the power function $y=x^{n}$ even when $n$ is very large. For example let $b=1.1$ and $n=10$. At $x=1000$, $y=x^{10}$ is $10^{30}$ but $y=1.1^{x}$ is $\approx2.47\times10^{41}$ (approximately $2.5\times10^{11}$ times larger)!
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| Graphs of $y=b^{x}$ and $y=b^{-x}=(1/b)^{x}$ are symmetrical with respect to the $y$-axis. |
Figure 3
(a) There is no restriction on $x$ because base is positive, so the domain of $f(x)$ is $(-\infty,\infty)$ or $\mathbb{R}$. To graph $f(x)$, we start with the graph of $y=3^{x}$ that we know how it looks like (Figure 4(a)). We reflect it in the $x$-axis to obtain the graph of $y=-3^{x}$ in Figure 4(b) and then shift the result upward one unit to get the graph of $y=1-3^{x}$ in Figure 4(c). As we can see the range of $f$ is $(-\infty,1)$. We can double check the range of $f$ by recalling that Figure 4 (b) Again there is no restriction on $x$ appearing in exponential functions (as long as the base is positive) and thus the domain of $g$ is $\mathbb{R}$. To graph $g$, we start again with the graph of $y=3^{x}$ (Figure 5(a)), and reflect it in the $y$-axis to obtain the graph of $y=3^{-x}$ in Figure 5(b). Finally we shift the result downward 2 units to get the graph of $y=3^{-x}-2$ in Figure 5(c). As we can see the range of $g$ is $(-2,\infty)$. We can double check the range of $g$ by recalling that $0<3^{x}<\infty$. Because reflection in the $y$-axis does not change the range, we have Figure 5 A special case of the exponential functions that gives simpler formulas is the natural exponential function whose base is the special number $e$. The number $e$, which is often called Euler’s number or sometimes Napier’s constant, is an irrational number ( like $\pi\approx3.14159265$ which is irrational), and its value approximately is Figure 6 Later we will learn more about Euler’s number and how the use of it simplifies the calculations.
\[
0<3^{x}<\infty
\]
Therefore
\[
-\infty<-3^{x}<0
\]
\[
\Rightarrow-\infty<1-3^{x}<1
\]
(a) $y=3^{x}$
(b) $y=-3^{x}$
(c) $y=1-3^{x}$
\[
0<3^{-x}<\infty.
\]
Therefore
\[
-2<3^{-x}-2<\infty.
\]
(a) $y=3^{x}$
(b) $y=3^{-x}$
(c) $y=3^{-x}-2$
The Natural Exponential Function
\[
e\approx2.71828182845904523536.
\]
Thus the graph of $y=e^{x}$ is thus between those of $y=2^{x}$ and $y=3^{x}$ (Figure 6).
The natural exponential function $y=e^{x}$ where $e\approx2.718281828$ is an irrational number