Properties of Exponential Functions

First, we will see how to identify an exponential function given an equation, a graph, and a table of values. You will be able to determine whether an exponential function is growing or decaying over time and how to define its domain and range.

Identifying Exponential Functions

When exploring linear growth, we observed a constant rate of change – a constant number by which the output increased for each unit increase in input. For example, in the equation $f(x)=3x+4$ , the slope tells us the output increases by 3 each time the input increases by 1. The scenario in the India population example is different because we have a percent change per unit time (rather than a constant change) in the number of people.

Defining an Exponential Function

A study found that the percent of the population who are vegans in the United States doubled from 2009 to 2011. In 2011, 2.5% of the population was vegan, adhering to a diet that does not include any animal products – no meat, poultry, fish, dairy, or eggs. If this rate continues, vegans will make up 10% of the U.S. population in 2015, 40% in 2019, and 80% in 2021.

What exactly does it mean to grow exponentially? What does the word double have in common with percent increase? People toss these words around errantly. Are these words used correctly? The words certainly appear frequently in the media.

Percent change refers to a change based on a percent of the original amount.
Exponential growth refers to an increase based on a constant multiplicative rate of change over equal increments of time, that is, a percent increase of the original amount over time.
Exponential decay refers to a decrease based on a constant multiplicative rate of change over equal increments of time, that is, a percent decrease of the original amount over time.

For us to gain a clear understanding of exponential growth, let us contrast exponential growth with linear growth. We will construct two functions. The first function is exponential. We will start with an input of 0, and increase each input by 1. We will double the corresponding consecutive outputs. The second function is linear. We will start with an input of 0, and increase each input by 1. We will add 2 to the corresponding consecutive outputs. See Table 1.

$x$	$f(x)=2x$	$g(x)=2x$
0	1	0
1	2	2
2	4	4
3	8	6
4	16	8
5	32	10
6	64	12

Table 1

From Table 1 we can infer that for these two functions, exponential growth dwarfs linear growth.

Exponential growth refers to the original value from the range increases by the same percentage over equal increments found in the domain.
Linear growth refers to the original value from the range increases by the same amount over equal increments found in the domain.

Apparently, the difference between "the same percentage" and "the same amount" is quite significant. For exponential growth, over equal increments, the constant multiplicative rate of change resulted in doubling the output whenever the input increased by one. For linear growth, the constant additive rate of change over equal increments resulted in adding 2 to the output whenever the input was increased by one.

The general form of the exponential function is $f(x)=ab^x$ , where $a$ is any nonzero number, $b$ is a positive real number not equal to 1.

If $b > 1$ , the function grows at a rate proportional to its size.
If $0 < b < 1$ , the function decays at a rate proportional to its size.

Let's look at the function $f(x)=2^x$ from our example. We will create a table (Table 2) to determine the corresponding outputs over an interval in the domain from $−3$ to $3$ .

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$f(x)=2^x$	$2^{−3} = \frac{1}{8}$	$2^{−2} = \frac{1}{4}$	$2^{−1} = \frac{1}{2}$	$2^0 = 1$	$2^1 = 2$	$2^2 = 4$	$2^3 = 8$

Table 2

Let us examine the graph of $f$ by plotting the ordered pairs we observe on the table in Figure 1, and then make a few observations.

Figure 1

Let's define the behavior of the graph of the exponential function $f(x)=2^x$ and highlight some its key characteristics.

the domain is $(-\infty, \infty)$ ,
the range is $(0, \infty)$ ,
as $x \rightarrow \infty, f(x) \rightarrow \infty$ ,
as $x \rightarrow-\infty, f(x) \rightarrow 0$ ,
$f(x)$ is always increasing.
the graph of $f(x)$ will never touch the $x$ -axis because base two raised to any exponent never has the result of zero.
$y=0$ is the horizontal asymptote.
the $y$ -intercept is $1$ .

EXPONENTIAL FUNCTION

For any real number $x$ , an exponential function is a function with the form

$f(x)=ab^x$

where

$a$ is a non-zero real number called the initial value and
$b$ is any positive real number such that $b \neq 1$ .
The domain of $f$ is all real numbers.
The range of $f$ is all positive real numbers if $a > 0$ .
The range of $f$ is all negative real numbers if $a < 0$ .
The $y$ -intercept is $(0,a)$ , and the horizontal asymptote is $y=0$ .

EXAMPLE 1

Identifying Exponential Functions

Which of the following equations are not exponential functions?

Solution

By definition, an exponential function has a constant as a base and an independent variable as an exponent. Thus, $g(x)=x^3$ does not represent an exponential function because the base is an independent variable. In fact, $g(x)=x^3$ is a power function.

Recall that the base $b$ of an exponential function is always a positive constant, and $b \neq 1$ . Thus, $j(x)=(−2)^x$ does not represent an exponential function because the base, $−2$ , is less than $0$ .

TRY IT #1

Which of the following equations represent exponential functions?