Exponential Functions and Growth/Decay, precalculus lesson

A culture of 100 bacteria that doubles every hour has 200 after one hour, 400 after two and 800 after three. Each hour adds a different amount, yet each hour multiplies the count by the same number. That steady multiplying is what makes a function exponential, and it is why such quantities look slow at first and then shoot up.

Key terms

Exponential function

Definition Let $b$ be a real number with $b > 0$ and $b \neq 1$ . The exponential function with base $b$ is

f(x) = b^x,

defined for every real number $x$ . For a rational exponent $x = m/n$ , with $m$ an integer and $n$ a positive integer,

b^{m/n} = \bigl(\sqrt[n]{b}\bigr)^m,

and for an irrational $x$ the value $b^x$ is the number that $b^r$ approaches as rational numbers $r$ approach $x$ . More generally, a function of the form $f(x) = a \cdot b^x$ with a constant $a \neq 0$ is also called exponential. Its initial value is $f(0) = a$ .

In plain words The variable sits in the exponent. That is the difference from a power function such as $x^2$ , where the variable is the base and the exponent is fixed.

Example For $f(x) = 3 \cdot 2^x$ the initial value is $3$ and each step of $1$ in $x$ doubles the output.

$x$	$-1$	$0$	$1$	$2$	$3$
$f(x)$	$1.5$	$3$	$6$	$12$	$24$

Inputs between the integers make sense too, for example $f(1/2) = 3\sqrt{2} \approx 4.243$ .

Positive base other than one

Definition The base $b$ of an exponential function must satisfy $b > 0$ and $b \neq 1$ . If $b = 1$ , then $b^x = 1$ for every $x$ , which is a constant function. If $b = 0$ , then $b^x$ is $0$ for every $x > 0$ and undefined for every $x \le 0$ . If $b < 0$ , then $b^x$ is not a real number for many inputs, for instance whenever $x = 1/n$ with $n$ even, so it is not defined on all real numbers.

In plain words A base of $1$ never changes anything, and a zero or negative base breaks down at fractional or negative inputs. A positive base other than $1$ works for every real input.

Example With base $-4$ , the input $1/2$ would give

(-4)^{1/2} = \sqrt{-4},

which is not a real number. With base $1$ , every input gives $1^x = 1$ , so the graph is the flat line $y = 1$ and nothing grows or shrinks.

Growth factor

Definition In $f(x) = a \cdot b^x$ the base $b$ is the growth factor per unit of input, because

\begin{gathered} f(x + 1) = a \cdot b^{x + 1} = b \cdot f(x) \\[0.4em] \text{for every real } x. \end{gathered}

If $b = 1 + r$ , the number $r$ is the percent rate of change per unit of input, written as a decimal. When $a > 0$ , a positive $r$ means growth and $-1 < r < 0$ means decay. For $0 < b < 1$ the base is often called the decay factor instead.

In plain words The growth factor is the number you multiply by each time the input goes up by $1$ . A 3 percent increase means multiplying by $1.03$ , and a 20 percent decrease means multiplying by $0.8$ .

Example A town of $8000$ people grows by 3 percent per year, so its growth factor is $1.03$ and its population after $t$ years is

P(t) = 8000 \cdot 1.03^t.

After one year $P(1) = 8240$ , and after ten years $P(10) \approx 10{,}751$ .

Equal input increments and equal output ratios

Definition For $f(x) = a \cdot b^x$ and any fixed step $h$ ,

\begin{gathered} \frac{f(x + h)}{f(x)} = \frac{a \cdot b^{x + h}}{a \cdot b^x} = b^h \\[0.8em] \text{for every real } x. \end{gathered}

So equal input increments give equal output ratios. Conversely, suppose data points have equally spaced inputs $x_0, x_0 + h, x_0 + 2h, \ldots$ with $h > 0$ , and outputs $y_0, y_1, y_2, \ldots$ with $y_0 \neq 0$ and a constant ratio $y_{k+1}/y_k = R$ , where $R > 0$ and $R \neq 1$ . Then all the points lie on the graph of the exponential function

f(x) = y_0 \cdot R^{(x - x_0)/h},

whose base is $R^{1/h}$ . A linear function behaves differently, since equal input increments give it equal output differences.

In plain words To spot an exponential pattern in a table with evenly spaced inputs, divide each output by the one before. Constant ratios mean exponential, constant differences mean linear.

Example A bacteria count is measured every 3 hours.

Time $t$ in hours	$0$	$3$	$6$	$9$
Count $N$	$200$	$600$	$1800$	$5400$

The differences $400$ , $1200$ and $3600$ are not equal, but every ratio is $3$ . So the count triples every 3 hours, and

N(t) = 200 \cdot 3^{t/3}.

The hourly growth factor is $3^{1/3} \approx 1.442$ , so after one hour there are about $288$ bacteria.

Exponential growth

Definition The function $f(x) = a \cdot b^x$ with $a > 0$ and $b > 1$ models exponential growth. It is increasing on the whole real line, $f(x) \to \infty$ as $x \to \infty$ , and $f(x) \to 0$ as $x \to -\infty$ . Writing $b = 1 + r$ , the number $r > 0$ is the growth rate per unit of input.

In plain words The quantity is multiplied by the same number bigger than $1$ in each time step, so the amount added gets larger every step.

Example Suppose $1000$ dollars earns 5 percent interest, added once a year. The balance after $t$ full years is

A(t) = 1000 \cdot 1.05^t \text{ dollars.}

Years $t$	$0$	$10$	$20$	$30$
Balance $A(t)$	$1000.00$	$1628.89$	$2653.30$	$4321.94$

Each 10 years multiplies the balance by $1.05^{10} \approx 1.629$ . The first decade adds about $629$ dollars and the third adds almost $1669$ .

Exponential decay

Definition The function $f(x) = a \cdot b^x$ with $a > 0$ and $0 < b < 1$ models exponential decay. It is decreasing on the whole real line, $f(x) \to 0$ as $x \to \infty$ , and $f(x) \to \infty$ as $x \to -\infty$ . Writing $b = 1 + r$ as before, the rate $r$ satisfies $-1 < r < 0$ , and the fraction lost per unit of input is $-r = 1 - b$ . Since $b^x = (1/b)^{-x}$ , the graph of a decay function is the mirror image in the $y$ -axis of a growth function with base $1/b > 1$ .

In plain words The quantity is multiplied by the same number between $0$ and $1$ in each time step. It keeps shrinking but never reaches zero.

Example Suppose the body removes 30 percent of a $200$ milligram dose of a medicine each hour. The amount left after $t$ hours is

M(t) = 200 \cdot 0.7^t \text{ milligrams.}

The values for $t = 0, 1, 2, 3$ are $200$ , $140$ , $98$ and $68.6$ milligrams. Each hour removes less than the hour before, because 30 percent of a smaller amount is a smaller amount.

Domain and range

Definition Let $f(x) = a \cdot b^x + k$ with $a \neq 0$ , $b > 0$ and $b \neq 1$ . The domain of $f$ is the set of all real numbers. The function $b^x$ takes every positive value exactly once and no other value, so the range of $f$ is

\begin{gathered} (k, \infty) \text{ if } a > 0, \\[0.4em] (-\infty, k) \text{ if } a < 0. \end{gathered}

With $k = 0$ this gives the range $(0, \infty)$ for $f(x) = a \cdot b^x$ with $a > 0$ .

In plain words You can put in any real number. A positive base raised to any power is positive, so the outputs all lie on one side of the height $k$ and never reach it.

Example For $g(x) = -2 \cdot 3^x + 5$ the domain is all real numbers. Since $3^x > 0$ , the term $-2 \cdot 3^x$ is always negative, so $g(x) < 5$ . Every height below $5$ is reached, since $g(x) = y$ means $3^x = (5 - y)/2$ , which has a solution whenever $(5 - y)/2 > 0$ . The range is $(-\infty, 5)$ . For instance $g(0) = 3$ and $g(2) = -13$ .

Horizontal asymptote

Definition For $f(x) = a \cdot b^x + k$ with $a \neq 0$ , $b > 0$ and $b \neq 1$ , the line $y = k$ is a horizontal asymptote. It is approached as $x \to -\infty$ when $b > 1$ and as $x \to \infty$ when $0 < b < 1$ . In the other direction $\lvert f(x) \rvert$ grows without bound, so there is no horizontal asymptote on that side. The graph never meets the line $y = k$ , because $a \cdot b^x \neq 0$ .

In plain words On one side the graph flattens out toward the height $k$ , getting closer and closer without touching it. On the other side it climbs or falls steeply.

Example Take $f(x) = 3 \cdot (1/2)^x + 1$ . Here $0 < b < 1$ , so the graph approaches $y = 1$ as $x$ grows.

$x$	$0$	$2$	$4$	$6$	$10$
$f(x)$	$4$	$1.75$	$1.1875$	$1.046875$	$1.0029\ldots$

The values stay above $1$ , because $3 \cdot (1/2)^x > 0$ . To the left the graph rises quickly, for instance $f(-3) = 25$ .

Check yourself

Is the data in this table exponential? If so, find a formula.

$x$	$0$	$1$	$2$	$3$
$y$	$48$	$36$	$27$	$20.25$

Show the answer

The inputs are equally spaced and every ratio of consecutive outputs is $0.75$ , for example $36/48 = 0.75$ . So the data is exponential with initial value $48$ and growth factor $0.75$ , that is $y = 48 \cdot 0.75^x$ . This is decay by 25 percent per step.

A culture of 500 bacteria doubles every 4 hours. Write a formula for the count after $t$ hours and find the count after 12 hours.

Show the answer

Doubling every 4 hours gives $N(t) = 500 \cdot 2^{t/4}$ . After 12 hours the culture has doubled three times, and $N(12) = 500 \cdot 2^3 = 4000$ .

Give the domain, the range and the horizontal asymptote of $f(x) = 6 - 4 \cdot (0.5)^x$ , and say from which side the graph approaches the asymptote.

Show the answer

The domain is all real numbers. Here $a = -4 < 0$ and $k = 6$ , so the range is $(-\infty, 6)$ . The base $0.5$ is between $0$ and $1$ , so the horizontal asymptote $y = 6$ is approached as $x \to \infty$ . The term $-4 \cdot (0.5)^x$ is always negative, so the graph approaches $y = 6$ from below.

Exponential Functions and Growth/Decay

Key terms

Exponential function

Positive base other than one

Growth factor

Equal input increments and equal output ratios

Exponential growth

Exponential decay

Domain and range

Horizontal asymptote

Check yourself

How this lesson connects

Builds on

Used later in

Where this leads in Calculus

Resources for this lesson

Function transformation explorer