3.1 Exponential Functions

What we learn here:

Definition and Domain of an Exponential Function
Bahvior and Range of an Exponential Function
Growth Rate of Exponential Functions
The Natural Exponential Function

Table of Contents

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 $R$ .
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 $b \neq 0$ ). 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 $b > 0$ ):
$D o m (f) = R = (- \infty, \infty) and R n g (f) = (0, \infty) .$


(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.

Graphs of

y = b^{x}

and

y = b^{- x} = (1 / b)^{x}

are symmetrical with respect to the

y

-axis.

Figure 2

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 $\approx 2.47 \times 10^{41}$ (approximately $2.5 \times 10^{11}$ times larger)!

Graphs of

y = b^{x}

and

y = b^{- x} = (1 / b)^{x}

are symmetrical with respect to the

y

-axis.

Figure 3

Example

Sketch the graph of each function and determine its domain and range.
(a) $f (x) = 1 - 3^{x}$
(b) $g (x) = 3^{- x} - 2$

Solution

(a) There is no restriction on $x$ because base is positive, so the domain of $f (x)$ is $(- \infty, \infty)$ or $R$ . 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
$0 < 3^{x} < \infty$ Therefore
$- \infty < - 3^{x} < 0$ $\Rightarrow - \infty < 1 - 3^{x} < 1$


(a) $y = 3^{x}$	(b) $y = - 3^{x}$	(c) $y = 1 - 3^{x}$

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 $R$ . 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
$0 < 3^{- x} < \infty .$ Therefore
$- 2 < 3^{- x} - 2 < \infty .$


(a) $y = 3^{x}$	(b) $y = 3^{- x}$	(c) $y = 3^{- x} - 2$

Figure 5

The Natural Exponential Function

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 $π \approx 3.14159265$ which is irrational), and its value approximately is
$e \approx 2.71828182845904523536 .$ 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 \approx 2.718281828

is an irrational number

Figure 6

The natural exponential function $e^{x}$ is often referred to as the exponential function and is also denoted by $\exp (x)$ . Many calculators have a special key for calculating $\exp (x)$ . Obviously, $e = \exp (1)$ .

Later we will learn more about Euler’s number and how the use of it simplifies the calculations.