Defining an Exponential Function

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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.

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

• Exponential growth refers to the original value from the range increases by the same percentage over equal increments found in the domain. Another way to say this that there is a constant multiplier over equal increments 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 [latex]f\left(x\right)=a{b}^{x},[/latex] where [latex]a[/latex] is any nonzero number, and [latex]b[/latex] is a positive real number not equal to 1.

• 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\left(x\right)={2}^{x}[/latex] from our example. We will create a table to determine the corresponding outputs over an interval in the domain from [latex]-3[/latex] to [latex]3.[/latex] See Table 2.

Let us examine the graph of [latex]f[/latex] by plotting the ordered pairs we observe on the table, and then make a few observations. See Figure 1.

Figure 1.

Let’s define the behavior of the graph of the exponential function [latex]f\left(x\right)={2}^{x}[/latex] and highlight some its key characteristics.

• The domain is [latex]\left(-\infty ,\infty \right).[/latex]
• The range is [latex]\left(0,\infty \right).[/latex]
• As x gets larger and larger so does [latex]f\left(x\right).[/latex]
• The graph of [latex]f\left(x\right)[/latex] will never touch the x-axis because base two raised to any exponent never has the result of zero.
• [latex]f\left(x\right)[/latex] is always increasing.
• The vertical intercept is [latex]\left(0,1\right)[/latex].

Defining Exponential Growth and Decay

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 or decay.

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 [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 in 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. 

Notice that the contextual 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, [latex]r,[/latex] and [latex]b=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]

Applying the Compound-Interest Formula

Savings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use compound interest. The term compounding refers to interest earned not only on the original value, but on the accumulated value of the account.

The annual percentage rate (APR) of an account, also called the nominal rate, is the yearly interest rate earned by an investment account. The term nominal is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being greater than the nominal rate! This is a powerful tool for investing.

We can calculate the compound interest using the compound interest formula, which is an exponential function of the variables time [latex]t,[/latex] principal [latex]P,[/latex] APR [latex]r,[/latex] and number of compounding periods in a year [latex]n:[/latex]

For example, observe Table 5, which shows the result of investing $1,000 at 10% for one year. Notice how the value of the account increases as the compounding frequency increases.

Evaluating Functions with Base e

As we saw earlier, the amount earned on an account increases as the compounding frequency increases. Table 6 shows that the increase from annual to semi-annual compounding is larger than the increase from monthly to daily compounding. This might lead us to ask whether this pattern will continue.

Examine the value of $1 invested at 100% interest for 1 year, compounded at various frequencies, listed in Table 6.

These values appear to be approaching a limit as [latex]n[/latex] increases without bound. In fact, as [latex]n[/latex] gets larger and larger, the expression [latex]{\left(1+\frac{1}{n}\right)}^{n}[/latex] approaches a number used so frequently in mathematics that it has its own name: the letter [latex]e.[/latex] This value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its approximation to six decimal places is shown below.

Investigating Continuous Growth

So far we have worked with rational bases for exponential functions. For many real-world phenomena, however, e is used as the base for exponential functions. Exponential models that use [latex]e[/latex] as the base are called continuous growth or decay models. We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.