Section 4.2 Exponential Functions
ΒΆExponential Function.
Note 4.14.
- We do not allow b to be negative, because if b<0, then bx is not a real number for some values of x. For example, if b=β4 and f(x)=(β4)x, then f(1/2)=(β4)1/2 is an imaginary number.
- We also exclude b=1 as a base because 1x=1 for all values of x; hence the function f(x)=1x is actually the constant function f(x)=1.
Subsection Graphs of Exponential Functions
The graphs of exponential functions have two characteristic shapes, depending on whether the base, b, is greater than 1 or less than 1. As typical examples, consider the graphs of f(x)=2x and g(x)=(12)x shown below. Some values for f and g are recorded in the tables.x | f(x) |
β3 | 18 |
β2 | 14 |
β1 | 12 |
0 | 1 |
1 | 2 |
2 | 4 |
3 | 8 |
x | g(x) |
β3 | 8 |
β2 | 4 |
β1 | 2 |
0 | 1 |
1 | 12 |
2 | 14 |
3 | 18 |
Properties of Exponential Functions, f(x)=abx, a>0.
Domain: all real numbers.
Range: all positive numbers.
-
If b>1, the function is increasing and concave up;
if 0<b<1, the function is decreasing and concave up.
The y-intercept is (0,a). There is no x-intercept.
Example 4.15.
Compare the graphs of \(f (x) = 3^x\) and \(g(x) = 4^x\text{.}\)
We evaluate each function for several convenient values, as shown in the table.
Then we plot the points for each function and connect them with smooth curves. For positive \(x\)-values, \(g(x)\) is always larger than \(f(x)\text{,}\) and is increasing more rapidly. In the figure, we can see that \(g(x) = 4^x\) climbs more rapidly than \(f(x) = 3^x\text{.}\) Both graphs cross the \(y\)-axis at (0, 1).
\(x\) | \(f(x)\) | \(g(x)\) |
\(-2\) | \(\dfrac{1}{9}\) | \(\dfrac{1}{16}\) |
\(-1\) | \(\dfrac{1}{3}\) | \(\dfrac{1}{4}\) |
\(0\) | \(1\) | \(1\) |
\(1\) | \(3\) | \(4\) |
\(2\) | \(9\) | \(16\) |
Note 4.16.
For decreasing exponential functions, those with bases between 0 and 1, the smaller the base, the more steeply the graph decreases. For example, compare the graphs of p(x)=0.8x and q(x)=0.5x shown in the figure at right.
Checkpoint 4.17.
- State the ranges of the functions \(f\) and \(g\) in Example 4.15 on the domain \([-2, 2]\text{.}\)
- State the ranges of the functions \(p\) and \(q\) shown in the Note above on the domain \([-2, 2]\text{.}\) Round your answers to two decimal places.
- \(\displaystyle f: \left[\dfrac{1}{9}, 9\right];~~g: \left[\dfrac{1}{16}, 16\right]\)
- \(\displaystyle p: [0.64, 1.56];~~q: [0.25, 4]\)
Subsection Transformations of Exponential Functions
In Chapter 2, we considered transformations of the basic graphs. For instance, the graphs of the functions y=x2β4 and y=(xβ4)2 are shifts of the basic parabola, y=x2. In a similar way, we can shift or stretch the graph of an exponential function while the basic shape is preserved.Example 4.18.
Use your calculator to graph the following functions. Describe how these graphs compare with the graph of \(h(x) = 2^x\text{.}\)
- \(\displaystyle f (x) = 2^x + 3\)
- \(\displaystyle g(x) = 2^{x+3}\)
Enter the formulas for the three functions as shown below. Note the parentheses around the exponent in the keying sequence for \(Y_3 = g(x).\)
\(Y_1 = 2 \) ^
X
\(Y_2 = 2 \) ^
X +
3
\(Y_3 = 2 \) ^
(
X +
3 )
The graphs of \(h(x) = 2^x\text{,}\) \(f(x) = 2^x + 3\text{,}\) and \(g(x) = 2^{x+3}\) in the standard window are shown below.
The graph of \(f(x) = 2^x + 3\text{,}\) shown in figure (b), has the same basic shape as that of \(h(x) = 2^x\text{,}\) but it has a horizontal asymptote at \(y = 3\) instead of at \(y = 0\) (the \(x\)-axis). In fact, \(f(x) = h(x) + 3\text{,}\) so the graph of \(f\) is a vertical translation of the graph of \(h\) by \(3\) units. If every point on the graph of \(h(x) = 2^x\) is moved \(3\) units upward, the result is the graph of \(f (x) = 2^x + 3\text{.}\)
First note that \(g(x) = 2^x+3 = h(x + 3)\text{.}\) In fact, the graph of \(g(x) = 2^{x+3}\) shown in figure (c) has the same basic shape as \(h(x) = 2^x\) but has been translated \(3\) units to the left.
Reflections of Graphs.
The graph of y=βf(x) is the reflection of the graph of y=f(x) about the x-axis.
The graph of y=f(βx) is the reflection of the graph of y=f(x) about the y-axis.
Checkpoint 4.19.
Which of the functions below have the same graph? Explain why.
- \(\displaystyle f (x) =\left(\dfrac{1}{4}\right)^x\)
- \(\displaystyle g(x) = -4^x\)
- \(\displaystyle h(x) = 4^{-x}\)
(a) and (c)
Subsection Comparing Exponential and Power Functions
Exponential functions are not the same as the power functions we studied in Chapter 3. Although both involve expressions with exponents, it is the location of the variable that makes the difference.Power Functions vs Exponential Functions.
Generalformulaandm |
Power Functions |
Exponential Functions |
General formula |
h(x)=kxp |
f(x)=abx |
Description |
variable base and constant exponent |
constant base and variable exponent |
Example |
h(x)=2x3 |
f(x)=2(3x) |
Example 4.20.
Compare the power function \(h(x) = 2x^3\) and the exponential function \(f(x) = 2(3^x)\text{.}\)
First, compare the values for these two functions shown in the table.
The scaling exponent for \(h(x)\) is \(3\text{,}\) so that when \(x\) doubles, say, from \(1\) to \(2\text{,}\) the output is multiplied by \(2^3\text{,}\) or \(8\text{.}\)
On the other hand, we can tell that \(f\) is exponential because its values increase by a factor of \(3\) for each unit increase in \(x\text{.}\) (To see this, divide any function value by the previous one.)
\(x\) | \(h(x)=2x^3\) | \(f(x)=2(3^x)\) |
\(-3\) | \(-54\) | \(\dfrac{2}{27}\) |
\(-2\) | \(-16\) | \(\dfrac{1}{4}\) |
\(-1\) | \(-2\) | \(\dfrac{2}{3}\) |
\(0\) | \(0\) | \(2\) |
\(1\) | \(2\) | \(6\) |
\(2\) | \(16\) | \(18\) |
\(3\) | \(54\) | \(54\) |
As you would expect, the graphs of the two functions are also quite different. For starters, note that the power function goes through the origin, while the exponential function has \(y\)-intercept \((0, 2)\)as shown at left below.
From the table, we see that \(h(3) = f(3) = 54\text{,}\) so the two graphs intersect at \(x = 3\text{.}\) (They also intersect at approximately \(x = 2.48\text{.}\)) However, if you compare the values of \(h(x) = 2x^3\) and \(f(x) = 2(3^x)\) for larger values of \(x\text{,}\) you will see that eventually the exponential function overtakes the power function, as shown at right above.
The relationship in Example 4.20 holds true for all increasing power and exponential functions: For large enough values of x, the exponential function will always be greater than the power function, regardless of the parameters in the functions. The figure at left shows the graphs of f(x)=x6 and g(x)=1.8x. At first, f(x)>g(x), but at around x=37, g(x) overtakes f(x), and g(x)>f(x) for all x>37.
Checkpoint 4.21.
Which of the following functions are exponential functions, and which are power functions?
- \(\displaystyle F(x) = 1.5^x\)
- \(\displaystyle G(x) = 3x^{1.5}\)
- \(\displaystyle H(x) = 3^{1.5x}\)
- \(\displaystyle K(x) = (3x)^{1.5}\)
Exponential: (a) and (c); power: (b) and (d)
Subsection Exponential Equations
An exponential equation is one in which the variable is part of an exponent. For example, the equationExample 4.22.
Solve the following equations.
- \(\displaystyle 3^{x-2} = 9^3\)
- \(\displaystyle 27 \cdot 3^{-2x} = 9^{x+1}\)
- Using the fact that \(9 = 3^2\text{,}\) we write each side of the equation as a power of \(3\text{:}\)\begin{equation*} \begin{aligned}[t] 3^{x-2} \amp = \left(3^2\right)^3 \\ 3^{x-2} \amp = 3^6 \end{aligned} \end{equation*}Now we equate the exponents to obtain\begin{equation*} \begin{aligned}[t] x - 2 \amp = 6 \\ x \amp = 8 \end{aligned} \end{equation*}
- We write each factor as a power of \(3\text{.}\)\begin{equation*} 3^3 \cdot 3^{-2x} = \left(3^2\right)^{x+1} \end{equation*}We use the laws of exponents to simplify each side:\begin{equation*} 3^{3-2x} = 3^{2x+2} \end{equation*}Now we equate the exponents to obtain\begin{equation*} \begin{aligned}[t] 3 - 2x \amp = 2x + 2 \\ -4x =\amp -1 \end{aligned} \end{equation*}The solution is \(x = \dfrac{1}{4}\text{.}\)
Checkpoint 4.23.
Solve the equation \(~~2^{x+2} = 128\text{.}\)
Example 4.24.
During the summer a population of fleas doubles in number every \(5\) days. If a population starts with \(10\) fleas, how long will it be before there are \(10,240\) fleas?
Let \(P\) represent the number of fleas present after \(t\) days. The original population of \(10\) is multiplied by a factor of \(2\) every \(5\) days, or
We set \(P = \alert{10,240}\) and solve for \(t\text{:}\)
We equate the exponents to get \(10 = \dfrac{t}{5}\text{,}\) or \(t = 50\text{.}\) The population will grow to \(10,240\) fleas in \(50\) days.
Checkpoint 4.25.
During an advertising campaign in a large city, the makers of Chip-Oβs corn chips estimate that the number of people who have heard of Chip-Oβs increases by a factor of \(8\) every 4 days.
- If 100 people are given trial bags of Chip-O's to start the campaign, write a function, \(N(t)\text{,}\) for the number of people who have heard of Chip-O's after \(t\) days of advertising.
- Use your calculator to graph the function \(N(t)\) on the domain \(0 \le t \le 15\text{.}\)
- How many days should the makers run the campaign in order for Chip-O's to be familiar to \(51,200\) people? Use algebraic methods to find your answer and verify on your graph.
- \(\displaystyle N(t)=100 \cdot 8^{t/4}\)
- 12 days
Checkpoint 4.26.
Use the graph of \(y = 5^x\) to find an approximate solution to \(5^x = 285\text{,}\) accurate to two decimal places.
\(x \approx 3.51\)
Subsection Section Summary
ΒΆSubsubsection Vocabulary
Look up the definitions of new terms in the Glossary.Exponential function
Base
Exponential equation
Subsubsection CONCEPTS
-
An exponential function has the form
f(x)=abx, where b>0 and bβ 1, aβ 0 Quantities that increase or decrease by a constant percent in each time period grow or decay exponentially.
Properties of Exponential Functions f(x)=abx, a>0.
Domain: all real numbers.
Range: all positive numbers.
If b>1, the function is increasing and concave up; if 0<b<1, the function is decreasing and concave up.
The y-intercept is (0,a). There is no x-intercept.
The graphs of exponential functions can be transformed by shifts, stretches, and reflections.
Reflections of Graphs.
The graph of y=βf(x) is the reflection of the graph of y=f(x) about the x-axis.
The graph of y=f(βx) is the reflection of the graph of y=f(x) about the y-axis.
Exponential functions f(x)=abx have different properties than power functions f(x)=kxp.
We can solve some exponential equations by writing both sides with the same base and equating the exponents.
We can use graphs to find approximate solutions to exponential equations.
Subsubsection STUDY QUESTIONS
Give the general form for an exponential function. What restrictions do we place on the base of the function?
Explain why the output of an exponential function f(x)=bx is always positive, even if x is negative.
How are the graphs of the functions f(x)=bx and g(x)=(1b)x related?
How is an exponential function different from a power function?
Delbert says that 8(12)x is equivalent to 4x. Convince him that he is mistaken.
Explain the algebraic technique for solving exponential equations described in this section.
Subsubsection SKILLS
Practice each skill in the Homework problems listed.Describe the graph of an exponential function: #1β14
Graph transformations of exponential functions: #15β18, 53β60
Evaluate exponential functions: #19β22
Find the equation of an exponential function from its graph: #23β26
Solve exponential equations: #27β44
Distinguish between power and exponential functions: #45β52, 65, and 66
Exercises Homework 4.2
ΒΆFind the y-intercept of each exponential function and decide whether the graph is increasing or decreasing.
Sketch the functions on the same set of axis with a domain of [β3,3]. Be sure to label your functions. Describe the similarities and differences between the two graphs.
Match each function with its graph.
For Problems 9β12,
Use a graphing calculator to graph the functions on the domain [β5,5].
Give the range of the function on that domain, accurate to hundredths.
In each group of functions, which have identical graphs? Explain why using algebra and the properties of exponents.
For Problems 15β18,
Use the order of operations to explain why the two functions are different.
Complete the table of values and graph both functions on the same set of axes.
Describe each as a transformation of y=2x or y=3x.
15.
f(x)=2xβ1, g(x)=2xβ1
x | y=2x | f(x) | g(x) |
β2 | 0000 | 0000 | 0000 |
β1 | |||
0 | |||
1 | |||
2 |
16.
f(x)=3x+2, g(x)=3x+2
x | y=3x | f(x) | g(x) |
β2 | 0000 | 0000 | 0000 |
β1 | |||
0 | |||
1 | |||
2 |
17.
f(x)=β3x, g(x)=3βx
x | y=3x | f(x) | g(x) |
β2 | 0000 | 0000 | 0000 |
β1 | |||
0 | |||
1 | |||
2 |
18.
f(x)=2βx, g(x)=β2x
x | y=2x | f(x) | g(x) |
β2 | 0000 | 0000 | 0000 |
β1 | |||
0 | |||
1 | |||
2 |
For the given function, evaluate each pair of expressions. Are they equivalent?
23.
The graph of f(x)=P0bx is shown in the figure.
Read the value of P0 from the graph.
Make a short table of values for the function by reading values from the graph. Does your table confirm that the function is exponential?
Use your table to calculate the growth factor, b.
Using your answers to parts (a) and (c), write a formula for f(x).
24.
The graph of g(x)=P0bx is shown in the figure.
Read the value of P0 from the graph.
Make a short table of values for the function by reading values from the graph. Does your table confirm that the function is exponential?
Use your table to calculate the decay factor, b.
Using your answers to parts (a) and (c), write a formula for g(x).
25.
For several days after the Northridge earthquake on January 17, 1994, the area received a number of significant aftershocks. The red graph shows that the number of aftershocks decreased exponentially over time. The graph of the function S(d)=S0bd, shown in black, approximates the data. (Source: Los Angeles Times, June 27, 1995)
Read the value of S0 from the graph.
Find an approximation for the decay factor, b, by comparing two points on the graph. (Some of the points on the graph of S(d) are approximately (1,82), (2,45), (3,25), and (4,14).)
Using your answers to (a) and (b), write a formula for S(d).
26.
The frequency of a musical note depends on its pitch. The graph shows that the frequency increases exponentially. The function F(p)=F0bp gives the frequency as a function of the number of half-tones, p, above the starting point on the scale
Read the value of F0 from the graph. (This is the frequency of the note A above middle C.)
Find an approximation for the growth factor, b, by comparing two points on the graph. (Some of the points on the graph of F(p) are approximately (1,466), (2,494), (3,523), and (4,554).)
Using your answers to (a) and (b), write a formula for F(p).
The frequency doubles when you raise a note by one octave, which is equivalent to 12 half-tones. Use this information to find an exact value for b.
Solve each equation algebraically.
37.
Before the advent of antibiotics, an outbreak of cholera might spread through a city so that the number of cases doubled every 6 days.
Twenty-six cases were discovered on July 5. Write a function for the number of cases of cholera t days later.
Use your calculator to graph your function on the interval 0 \le t\le 90\text{.}
When should hospitals expect to be treating 106,496 cases? Use algebraic methods to find your answer, and verify it on your graph.
38.
An outbreak of ungulate fever can sweep through the livestock in a region so that the number of animals affected triples every 4 days.
A rancher discovers 4 cases of ungulate fever among his herd. Write a function for the number of cases of ungulate fever t days later.
Use your calculator to graph your function on the interval 0 \le t\le 20\text{.}
If the rancher does not act quickly, how long will it be until 324 head are affected? Use algebraic methods to find your answer, and verify it on your graph.
39.
A color television set loses 30\% of its value every 2 years.
Write a function for the value of a television set t years after it was purchased if it cost \$700 originally.
Use your calculator to graph your function on the interval 0 \le t\le 20\text{.}
How long will it be before a \$700 television set depreciates to \$343\text{?} Use algebraic methods to find your answer, and verify it on your graph.
40.
A mobile home loses 20\% of its value every 3 years.
A certain mobile home costs \$20,000\text{.} Write a function for its value after t years.
Use your calculator to graph your function on the interval 0 \le t\le 30\text{.}
How long will it be before a \$20,000 mobile home depreciates to \$12,800\text{?} Use algebraic methods to find your answer, and verify it on your graph.
Use a graph to find an approximate solution accurate to the nearest hundredth.
Decide whether each function is an exponential function, a power function, or neither.
Decide whether the table could describe a linear function, a power function, an exponential function, or none of these.
Fill in the tables. Graph each pair of functions in the same window. Then answer the questions below.
Give the range of f and the range of g\text{.}
For how many values of x does f (x) = g(x)\text{?}
Estimate the value(s) of x for which f (x) = g(x)\text{.}
For what values of x is f (x)\lt g(x)\text{?}
Which function grows more rapidly for large values of x\text{?}
For Problems 53β60, state the domain and range of each transformation, its intercept(s), and any asymptotes.
53.
f(x)=3^x
y = f (x) - 4
y = f (x - 4)
y = -4 f (x)
54.
g(x)=4^x
y = g(x) +2
y = g(x +2)
y = 2g(x)
55.
h(t)=6^t
y = -h(t)
y = h(-t)
y = -h(-t)
56.
j(t)=\left(\dfrac{1}{3} \right)^t
y = j(-t)
y = -j(t)
y = -j(-t)
57.
g(x)=2^x
y = g(x-3)
y = g(x-3)+4
58.
f(x)=10^x
y = f(x+5)
y = f(x+5)-20
59.
N(t)=\left(\dfrac{1}{2} \right)^t
y = -N(t)
y = 6-N(t)
60.
P(t)=0.4^t
y = -P(t)
y = 8-P(t)
For Problems 61β64,
Describe the graph as a transformation of y = 2^x\text{.}
Give an equation for the function graphed.
Match the graph of each function to its formula. In each formula, a\gt 0 and b \gt 1\text{.}
67.
The function f (t) describes a volunteer's heart rate during a treadmill test.
The heart rate is given in beats per minute and t is in minutes. (See Section 2.2 to review functions defined piecewise.) (Source: Davis, Kimmet, and Autry, 1986)
-
Evaluate the function to complete the table.
t 3.5 4 8 10 15 f(t) \hphantom{0000} \hphantom{0000} \hphantom{0000} \hphantom{0000} \hphantom{0000} Sketch the graph of the function.
The treadmill test began with walking at 5.5 kilometers per hour, then jogging, starting at 12 kilometers per hour and increasing to 14 kilometers per hour, and finished with a cool-down walking period. Identify each of these activities on the graph and describe the volunteer's heart rate during each phase.
68.
Carbon dioxide (\text{CO}_2) is called a greenhouse gas because it traps part of the Earth's outgoing energy. Animals release \text{CO}_2 into the atmosphere, and plants remove \text{CO}_2 through photosynthesis. In modern times, deforestation and the burning of fossil fuels both contribute to \text{CO}_2 levels. The figure shows atmospheric concentrations of \text{CO}_2\text{,} in parts per million, measured at the Mauna Loa Observatory in Hawaii.
The red curve shows annual oscillations in \text{CO}_2 levels. Can you explain why \text{CO}_2 levels vary throughout the year?
The blue curve shows the average annual \text{CO}_2 readings. By approximately how much does the \text{CO}_2 level vary from its average value during the year?
In 1960, the average \text{CO}_2 level was 316.75 parts per million, and the average level has been rising by 0.4\% per year. If the level continues to rise at this rate, what \text{CO}_2 readings can we expect in the year 2100?
For part (a): Why would photosynthesis vary throughout the year?