Exponential Functions

India is the second most populous country in the world with a population of about $1.39$ billion people in 2021. The population is growing at a rate of about $1.2\%$ each year. If this rate continues, the population of India will exceed China’s population by the year $2027$. 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.

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 the table below.

From the table we can infer that for these two functions, exponential growth dwarfs linear growth.

• Exponential growth refers to the original value from the range increasing by the same percentage over equal increments found in the domain.
• Linear growth refers to the original value from the range increasing 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 to determine the corresponding outputs over an interval in the domain from $-3$ to $3$.

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

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 $x\to\infty$, $f(x)\to\infty$,
• as $x\to-\infty$, $f(x)\to0$,
• $f(x)$ 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.

Example. Which of the following equations are not exponential functions?

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

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 $b\ne1$. Thus, $j(x)=(-2)^x$ does not represent an exponential function because the base, $-2$, is less than $0$.

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 $x=\tfrac{1}{2}$. Then $f(x)=f\!\left(\tfrac{1}{2}\right)=(-9)^{\tfrac{1}{2}}=\sqrt{-9}$, 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 $f(3)$?

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 $f(3)$?

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

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

Solution. Follow the order of operations. Be sure to pay attention to the parentheses.

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.

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 the table below.

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

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. 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 model for $t=18$, because 2031 is 18 years after 2013. Rounding to the nearest thousandth,

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

Finding equations of exponential functions

In the previous examples, we were given an exponential function, which we then evaluated for a given input. Sometimes we are given information about an exponential function without knowing the function explicitly. We must use the information to first write the form of the function, then determine the constants $a$ and $b$, and evaluate the function.

Example. In 2006, 80 deer were introduced into a wildlife refuge. By 2012, the population had grown to 180 deer. The population was growing exponentially. Write an exponential function $N(t)$ representing the population $(N)$ of deer over time $t$.

Solution. We let our independent variable $t$ be the number of years after 2006. Thus, the information given in the problem can be written as input-output pairs: $(0,80)$ and $(6,180)$. Notice that by choosing our input variable to be measured as years after 2006, we have given ourselves the initial value for the function, $a=80$. We can now substitute the second point into the equation $N(t)=80b^t$ to find $b$:

Note. Unless otherwise stated, do not round any intermediate calculations. Then round the final answer to four places for the remainder of this section.

The exponential model for the population of deer is $N(t)=80(1.1447)^t$. (Note that this exponential function models short-term growth. As the inputs gets large, the output will get increasingly larger, so much so that the model may not be useful in the long term.)

We can graph our model to observe the population growth of deer in the refuge over time. Notice that the graph below passes through the initial points given in the problem, $(0,80)$ and $(6,180)$. We can also see that the domain for the function is $[0,\infty)$, and the range for the function is $[80,\infty)$.

Example. Find an exponential function that passes through the points $(-2,6)$ and $(2,1)$.

Solution. Because we don’t have the initial value, we substitute both points into an equation of the form $f(x)=ab^x$, and then solve the system for $a$ and $b$.

• Substituting $(-2,6)$ gives $6=ab^{-2}$
• Substituting $(2,1)$ gives $1=ab^2$

Use the first equation to solve for $a$ in terms of $b$:

Substitute $a$ in the second equation, and solve for $b$:

Use the value of $b$ in the first equation to solve for the value of $a$:

Thus, the equation is $f(x)=2.4492(0.6389)^x$.

We can graph our model to check our work. Notice that the graph below passes through the initial points given in the problem, $(-2,6)$ and $(2,1)$. The graph is an example of an exponential decay function.

Example. Find an equation for the exponential function graphed below.

Solution. We can choose the $y$-intercept of the graph, $(0,3)$, as our first point. This gives us the initial value, $a=3$. Next, choose a point on the curve some distance away from $(0,3)$ that has integer coordinates. One such point is $(2,12)$.

Because we restrict ourselves to positive values of $b$, we will use $b=2$. Substitute $a$ and $b$ into the standard form to yield the equation $f(x)=3(2)^x$.

Example. Use a graphing calculator to find the exponential equation that includes the points $(2,24.8)$ and $(5,198.4)$.

Solution. Follow the guidelines above. First press [STAT], [EDIT], [1: Edit…], and clear the lists L1 and L2. Next, in the L1 column, enter the $x$-coordinates, 2 and 5. Do the same in the L2 column for the $y$-coordinates, 24.8 and 198.4.

Now press [STAT], [CALC], [0: ExpReg] and press [ENTER]. The values $a=6.2$ and $b=2$ will be displayed. The exponential equation is $y=6.2\cdot2^x$.

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 $t$, principal $P$, APR $r$, and number of compounding periods in a year $n$:

For example, observe the table below, 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.

Example. If we invest $3{,}000 in an investment account paying $3\%$ interest compounded quarterly, how much will the account be worth in 10 years?

Solution. Because we are starting with $3{,}000, $P=3{,}000$. Our interest rate is $3\%$, so $r=0.03$. Because we are compounding quarterly, we are compounding 4 times per year, so $n=4$. We want to know the value of the account in 10 years, so we are looking for $A(10)$, the value when $t=10$.

The account will be worth about $4{,}045.05 in 10 years.

Example. A 529 Plan is a college-savings plan that allows relatives to invest money to pay for a child’s future college tuition; the account grows tax-free. Lily wants to set up a 529 account for her new granddaughter and wants the account to grow to $40{,}000 over 18 years. She believes the account will earn $6\%$ compounded semi-annually (twice a year). To the nearest dollar, how much will Lily need to invest in the account now?

Solution. The nominal interest rate is $6\%$, so $r=0.06$. Interest is compounded twice a year, so $n=2$.

We want to find the initial investment, $P$, needed so that the value of the account will be worth $40{,}000 in 18 years. Substitute the given values into the compound interest formula, and solve for $P$.

Lily will need to invest $13{,}801 to have $40{,}000 in 18 years.

Evaluating functions with base $e$

As we saw earlier, the amount earned on an account increases as the compounding frequency increases. The table below 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 below.

These values appear to be approaching a limit as $n$ increases without bound. In fact, as $n$ gets larger and larger, the expression $\left(1+\tfrac{1}{n}\right)^n$ approaches a number used so frequently in mathematics that it has its own name: the letter $e$. 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.

Example. Calculate $e^{3.14}$. Round to five decimal places.

Solution. On a calculator, press the button labeled $\left[e^x\right]$. The window shows $\left[e^{\wedge}(\ \right]$. Type $3.14$ and then close parenthesis, $\left[)\right]$. Press [ENTER]. Rounding to 5 decimal places, $e^{3.14}\approx23.10387$. Caution: Many scientific calculators have an “Exp” button, which is used to enter numbers in scientific notation. It is not used to find powers of $e$.

Investigating continuous growth

So far we have worked with rational bases for exponential functions. For most real-world phenomena, however, $e$ is used as the base for exponential functions. Exponential models that use $e$ 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.

Example. A person invested $1{,}000 in an account earning a nominal $10\%$ per year compounded continuously. How much was in the account at the end of one year?

Solution. Since the account is growing in value, this is a continuous compounding problem with growth rate $r=0.10$. The initial investment was $1{,}000, so $P=1000$. We use the continuous compounding formula to find the value after $t=1$ year:

The account is worth $1{,}105.17 after one year.

Example. Radon-222 decays at a continuous rate of $17.3\%$ per day. How much will 100 mg of Radon-222 decay to in 3 days?

Solution. Since the substance is decaying, the rate, $17.3\%$, is negative. So, $r=-0.173$. The initial amount of radon-222 was 100 mg, so $a=100$. We use the continuous decay formula to find the value after $t=3$ days:

So 59.5115 mg of radon-222 will remain.

Key equations

Key concepts

• An exponential function is defined as a function with a positive constant other than $1$ raised to a variable exponent.
• A function is evaluated by solving at a specific value.
• An exponential model can be found when the growth rate and initial value are known.
• An exponential model can be found when the two data points from the model are known.
• An exponential model can be found using two data points from the graph of the model.
• An exponential model can be found using two data points from the graph and a calculator.
• The value of an account at any time $t$ can be calculated using the compound interest formula when the principal, annual interest rate, and compounding periods are known.
• The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and life span of the account are known.
• The number $e$ is a mathematical constant often used as the base of real world exponential growth and decay models. Its decimal approximation is $e\approx2.718282$.
• Scientific and graphing calculators have the key $\left[e^x\right]$ or $\left[\exp(x)\right]$ for calculating powers of $e$.
• Continuous growth or decay models are exponential models that use $e$ as the base. Continuous growth and decay models can be found when the initial value and growth or decay rate are known.

Practice

Evaluate exponential functions

Find the equation of an exponential function

Use compound interest formulas

Evaluate exponential functions with base $e$

This section is adapted from Precalculus 2e, Section 4.1: Exponential Functions by Jay Abramson and OpenStax, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated every graph as an accessible inline SVG generated from its exact equation — the labeled-points curve $f(x)=2^x$ with its $y=0$ asymptote; the Company A/B comparison of $A(x)=100+50x$ against $B(x)=100(1.5)^x$; the deer-population curve $N(t)=80(1.1447)^t$; the decay curve $f(x)=2.4492(0.6389)^x$ of the “initial value not known” example; the graph-reading example’s $f(x)=3(2)^x$; and the Try It graph-reading curve $f(x)=\sqrt{2}(\sqrt{2})^x$; presented every value-versus-frequency comparison (the $1{,}000-at-10% table, the $1-at-100% table, and the store-count and interest tables) as Markdown tables; omitted the two decorative chapter-opener/definition photographs (the linear/quadratic/exponential-function illustration and the E. coli micrograph), which carry no mathematics; omitted the section’s two Media links to external graphing-calculator resources; excluded a “coreq-skills” block present in the pinned CNXML module (a corequisite-course skills review covering evaluating and graphing exponential functions with its own short exercise set) that does not appear in the printed Precalculus 2e text — pages 397–426 of the source PDF confirm the printed section runs directly from the chapter introduction into “Identifying Exponential Functions,” with no corequisite-skills material between them; converted the “Try It” practice problems into interactive exercises with instant feedback — two multiple-choice questions (replacing one four-option “which of these represent exponential functions” prompt with two correct answers, since a single-answer multiple-choice component can only test one designated answer at a time) and a fill-in for the definition Try It, a fill-in for the order-of-operations evaluation, two fill-ins for the India/China population comparison, two fill-ins for the wolf-population model (renaming the population function from the source’s $N$ to $w$, since $N$ is reserved by this project’s compute engine for numeric evaluation), a fill-in each for the two-point and known-graph equation-writing Try Its, a fill-in asking for just the growth factor $b$ (rather than the full calculator-derived equation) for the graphing-calculator Try It, a fill-in each for the two compound-interest Try Its, a fill-in for the power-of-$e$ Try It, a fill-in each for the two continuous growth/decay Try Its (the radon-222 one declaring scientific-notation, since its value is far too small to write as a plain decimal), and adapted eight selected end-of-section exercises — two growth/decay-versus-linear identifications, an exponential evaluation, three find-the-equation problems, two compound-interest applications, a continuous-growth identification, and two base-$e$ evaluations — into eleven interactive components in a closing Practice block, one group per objective. The second “Finding Equations of Exponential Functions” example is reproduced exactly as the source prints it, including its $a=6b^2\approx6(0.6389)^2\approx2.4492$ step, which substitutes the already-rounded $b$ and so sits in tension with the section’s own “do not round any intermediate calculations” note; the unrounded value would be $a=6b^2=\sqrt6\approx2.4495$. That tension is the source’s, and this page does not silently resolve it.