Properties of Exponential Functions

Learning Objectives

In this section, you will:

• Evaluate exponential functions.
• Find the equation of an exponential function.
• Use compound interest formulas.
• Evaluate exponential functions with base $e$

Source: Rice University, https://openstax.org/books/college-algebra/pages/6-1-exponential-functions
This work is licensed under a Creative Commons Attribution 4.0 License.

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.

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 $3$.

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?

• $f(x)=4^{3(x−2)}$
• $g(x)=x^3$
• $h(x)=\left( \frac{1}{3} \right)^x$
• $j(x)=(−2)^x$

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?

• $f(x)=2x^2−3x+1$
• $g(x)=0.875^x$
• $h(x)=1.75x+2$
• $j(x)=1095.6^{−2x}$

Recall that the base of an exponential function must be a positive real number other than $1$. Why do we limit the base $b$ to positive values? To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:

• Let $b=−9$ and $x=12$. Then $f(x)=f \left( \frac{1}{2} \right)=(−9)^{\frac{1}{2}}=\sqrt{−9}$, which is not a real number.

Why do we limit the base to positive values other than $1$? Because base $1$ results in the constant function. Observe what happens if the base is $1$:

• Let $b=1$. Then $f(x)=1^x=1$ for any value of $x$.

To evaluate an exponential function with the form $f(x)=b^x$, we simply substitute $x$ with the given value, and calculate the resulting power. For example:

Let $f(x)=2^x$. What is $f(3)$?

$\begin{aligned} f(x) &=2^{x} & & \\ f(3) &=2^{3} & & \text { Substitute } x=3 \\ &=8 & & \text { Evaluate the power. } \end{aligned}$

To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. For example:

Let $f(x)=30(2)^x$. What is $f(3)$?

$\begin{aligned} f(x) &=30(2)^{x} & & \\ f(3) &=30(2)^{3} & & \text { Substitute } x=3 \\ &=30(8) & & \text { Simplify the power first. } \\ &=240 & & \text { Multiply. } \end{aligned}$

Note that if the order of operations were not followed, the result would be incorrect:

$f(3)=30(2)^3 \neq 60^3=216,000$

EXAMPLE 2

Evaluating Exponential Functions

Let $f(x)=5(3)^{x+1}$. Evaluate $f(2)$ without using a calculator.

Solution

$\begin{aligned} f(x) &=5(3)^{x+1} & & \\ f(2) &=5(3)^{2+1} & & \text { Substitute } x=2 . \\ &=5(3)^{3} & & \text { Add the exponents. } \\ &=5(27) & & \text { Simplify the power. } \\ &=135 & & \text { Multiply. } \end{aligned}$

TRY IT #2

Let $f(x)=8(1.2)^{x−5}$. Evaluate $f(3)$ using a calculator. Round to four decimal places.

Defining Exponential Growth

Because the output of exponential functions increases very rapidly, the term "exponential growth" is often used in everyday language to describe anything that grows or increases rapidly. However, exponential growth can be defined more precisely in a mathematical sense. If the growth rate is proportional to the amount present, the function models exponential growth.

EXPONENTIAL GROWTH

A function that models exponential growth grows by a rate proportional to the amount present. For any real number $x$ and any positive real numbers $a$  and $b$ such that $b \neq 1$, an exponential growth function has the form

$f(x)=ab^x$

where

• $a$ is the initial or starting value of the function.
• $b$ is the growth factor or growth multiplier per unit $x$.

In more general terms, we have an exponential function, in which a constant base is raised to a variable exponent. To differentiate between linear and exponential functions, let's consider two companies, A and B. Company A has 100 stores and expands by opening 50 new stores a year, so its growth can be represented by the function $A(x)=100+50x$. Company B has 100 stores and expands by increasing the number of stores by 50% each year, so its growth can be represented by the function $B(x)=100(1+0.5)^x$.

A few years of growth for these companies are illustrated in Table 3.

Table 3

The graphs comparing the number of stores for each company over a five-year period are shown in Figure 2. We can see that, with exponential growth, the number of stores increases much more rapidly than with linear growth.

Figure 2 The graph shows the numbers of stores Companies A and B opened over a five-year period.

Notice that the domain for both functions is $[0, \infty)$, and the range for both functions is $[100,\infty)$. After year 1, Company B always has more stores than Company A.

Now we will turn our attention to the function representing the number of stores for Company B, $B(x)=100(1+0.5)^x$. In this exponential function, 100 represents the initial number of stores, 0.50 represents the growth rate, and $1+0.5=1.5$ represents the growth factor. Generalizing further, we can write this function as $B(x)=100(1.5)^x$, where 100 is the initial value, $1.5$ is called the base, and $x$ is called the exponent.

EXAMPLE 3

Evaluating a Real-World Exponential Model

At the beginning of this section, we learned that the population of India was about $1.25$ billion in the year 2013, with an annual growth rate of about $1.2 \%$. This situation is represented by the growth function $P(t)=1.25(1.012)^t$, where t is the number of years since 2013. To the nearest thousandth, what will the population of India be in 2031?

Solution

To estimate the population in 2031, we evaluate the models for $t=18$, because 2031 is 18 years after 2013. Rounding to the nearest thousandth,

$P(18)=1.25(1.012)^18 \approx 1.549$

There will be about 1.549 billion people in India in the year 2031.

TRY IT #3

The population of China was about 1.39 billion in the year 2013, with an annual growth rate of about $0.6 \%$. This situation is represented by the growth function $P(t)=1.39(1.006)^t$, where $t$ is the number of years since 2013. To the nearest thousandth, what will the population of China be for the year 2031? How does this compare to the population prediction we made for India in Example 3?