Learning Objectives
In this section, you will:
- Evaluate exponential functions.
- Find the equation of an exponential function.
Defining Exponential Functions
Let’s take a look at the following situations:
India is the second most populous country in the world with a population of about 1.25 billion people in 2013. The population is growing at a rate of about 1.2% each year1. If this rate continues, the population of India will exceed China’s population by the year 2031.
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.
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 [latex]f(x)=3x+4[/latex], 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.
When populations grow rapidly, we often say that the growth is “exponential,” meaning that something is growing very rapidly. To a mathematician, however, the term exponential growth has a very specific meaning. In this section, we will take a look at exponential functions, which model this kind of rapid growth.
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.
In order to gain a clear understanding of exponential growth, let us contrast exponential growth with linear growth.
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 [latex]A\left(x\right)=100+50x[/latex]. 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 [latex]B\left(x\right)=100{\left(1+0.5\right)}^{x}[/latex].
A few years of growth for these companies are illustrated below.
| Year, x | Stores, Company A | Stores, Company B |
|---|---|---|
| 0 | 100 + 50(0) = 100 | 100(1 + 0.5)0 = 100 |
| 1 | 100 + 50(1) = 150 | 100(1 + 0.5)1 = 150 |
| 2 | 100 + 50(2) = 200 | 100(1 + 0.5)2 = 225 |
| 3 | 100 + 50(3) = 250 | 100(1 + 0.5)3 = 337.5 |
| x | A(x) = 100 + 50x | B(x) = 100(1 + 0.5)x |
The graphs comparing the number of stores for each company over a five-year period are shown in below. 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 [latex]\left[0,\infty \right)[/latex], and the range for both functions is [latex]\left[100,\infty \right)[/latex]. 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, [latex]B\left(x\right)=100{\left(1+0.5\right)}^{x}[/latex]. In this exponential function, 100 represents the initial number of stores, 0.50 represents the growth rate, and [latex]1+0.5=1.5[/latex] represents the growth factor. Generalizing further, we can write this function as [latex]B\left(x\right)=100{\left(1.5\right)}^{x}[/latex], where 100 is the initial value, 1.5 is called the base, and x is called the exponent.
- 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.
Clearly, 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 multiplying the output by 1.5 whenever the input increased by one. For linear growth, the constant additive rate of change over equal increments resulted in adding 50 to the output whenever the input was increased by one.
The video below shows another example comparing linear and exponential growth.
The general form of the exponential function is [latex]f(x)=a{b}^{x} [/latex], where [latex]a \neq 0, \; b>0, \; b \neq 1[/latex].
- If [latex]b>1[/latex], the function grows at a rate proportional to its size.
- If [latex]0<b<1[/latex], the function decays at a rate proportional to its size.
Let’s look at the function [latex]f(x)={2}^{x} [/latex]. We will create a table (below) to determine the corresponding outputs over an interval in the domain from -3 to 3.
| [latex]x[/latex] | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| [latex]f(x)={2}^{x}[/latex] | [latex]{2}^{-3}=\frac{1}{8}[/latex] | [latex]{2}^{-2}=\frac{1}{4}[/latex] | [latex]{2}^{-1}=f\rac{1}{2}[/latex] | [latex]{2}^{0}=1[/latex] | [latex]{2}^{1}=2[/latex] | [latex]{2}^{2}=4[/latex] | [latex]{2}^{3}=8[/latex] |
Let us examine the graph of f(x) by plotting the ordered pairs from the table, and then make a few observations.
Let’s analyze the behavior of the graph of the exponential function[latex]f(x)={2}^{x} [/latex] and highlight some of its key characteristics.
- the domain is [latex](-\infty ,\infty )[/latex],
- the range is [latex](0 ,\infty )[/latex],
- as [latex] x \rightarrow \infty , f(x)\rightarrow \infty [/latex],
- as [latex] x \rightarrow -\infty , f(x)\rightarrow 0 [/latex],
- [latex] f(x) [/latex] is always increasing,
- the graph of [latex] f(x) [/latex] will never touch the x-axis because two raised to any exponent never has the result of zero,
- [latex] y=0 [/latex] is the horizontal asymptote,
- the y-intercept is 1.
A General Note: EXPONENTIAL FUNCTION
For any real number x an exponential function is a function of the form
[latex]f(x)=a{b}^{x}[/latex],
where
- [latex]a [/latex] is a non-zero real number called the initial or starting value of a function and
- [latex]b [/latex] is any positive real number such that [latex]b \neq 1[/latex], called the growth factor or growth multiplier per unit x.
- The domain of [latex] f(x)[/latex] is all real numbers.
- The range of [latex] f(x)[/latex] is all positive real numbers if [latex] a>0[/latex].
- The range of [latex] f(x)[/latex] is all negative real numbers if [latex] a<0[/latex].
- The y-intercept is [latex] (0,1) [/latex] and the horizontal asymptote is [latex]y=0 [/latex].
Example: identifying exponential functions
Which of the following equations are not exponential functions?
- [latex]f(x)={4}^{3(x-2)}[/latex]
- [latex]g(x)={x}^{3}[/latex]
- [latex]h(x)=(\frac{1}{3})^{x}[/latex]
- [latex]j(x)=(-2)^{x}[/latex]
try it
Which of the following equations represent exponential functions?
- [latex]f(x)=2{x}^{2}-3x+1[/latex]
- [latex]g(x)={0.875}^{x}[/latex]
- [latex]h(x)=1.75x+2[/latex]
- [latex]j(x)={1095.6}^{-2x}[/latex]
Evaluating Exponential Functions
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 [latex]x=\frac{1}{2}[/latex]. Then [latex]f\left(x\right)=f\left(\frac{1}{2}\right)={\left(-9\right)}^{\frac{1}{2}}=\sqrt{-9}[/latex], 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 [latex]f\left(x\right)={1}^{x}=1[/latex] for any value of x.
To evaluate an exponential function with the form [latex]f\left(x\right)={b}^{x}[/latex], we simply substitute x with the given value, and calculate the resulting power. For example:
Let [latex]f\left(x\right)={2}^{x}[/latex]. What is [latex]f\left(3\right)[/latex]?
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 [latex]f\left(x\right)=30{\left(2\right)}^{x}[/latex]. What is [latex]f\left(3\right)[/latex]?
Note that if the order of operations were not followed, the result would be incorrect:
Example: Evaluating Exponential Functions
Let [latex]f\left(x\right)=5{\left(3\right)}^{x+1}[/latex]. Evaluate [latex]f\left(2\right)[/latex] without using a calculator.
Try It
Let [latex]f\left(x\right)=8{\left(1.2\right)}^{x - 5}[/latex]. Evaluate [latex]f\left(3\right)[/latex] using a calculator. Round to four decimal places.
Example: 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 [latex]P\left(t\right)=1.25{\left(1.012\right)}^{t}[/latex], where t is the number of years since 2013. To the nearest thousandth, what will the population of India be in 2031?
Try It
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 [latex]P\left(t\right)=1.39{\left(1.006\right)}^{t}[/latex], 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 our example?