Chapter Six
Exponential Functions
An exponential function is a function whose values you calculate by raising
a constant number, called the base, to powers you have selected from your
domain. Examples of exponential expressions include 2x, 3–x, and
.
The graphs of these functions are not hard to get, with good accuracy, using
the elementary properties of exponents. In case you are not as familiar as
you should be with this material, there is a review of the most important
information in the following section.
Review of Exponents
All real numbers are possible exponents. An exponent of 1 is usually not
written, since it indicates the number itself, as in 21 = 2.
An exponent that is a positive integer shows the number of times that you
are using the base number as a factor to form a product.
For example, 42 = 4 @ 4, 53 = 5 @ 5 @ 5, and 75 = 7 @ 7 @ 7 @ 7 @ 7.
If you multiply together two exponential expressions with the same base,
such as 34@ 33, their product contains all the factors of each.
34 @33 = (3 @ 3 @ 3 @ 3) @ (3 @ 3 @ 3) = 3 @ 3 @ 3 @ 3 @ 3 @ 3 @ 3 = 37
When you multiply the product of four 3's by the product of three 3's, the
result is the product of seven 3's. Whenever you multiply together terms
with the same base, the exponent of the product is equal to the sum of the
exponents of each term.
If you wish to divide one exponential expression by another with the same
base, such as
45
, you can calculate the quotient by first finding the number
43
of common factors that appear in both the numerator and the denominator,
and canceling as many as you can.
Note the difference between a
polynomial term, like x3, and an
exponential term, like 3x. For x3,
you have varying input values all
raised to the same third power. In
the exponential case, you have the
constant 3 raised to powers that vary
according to the values you select
from the domain.
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CHAPTER 6: EXPONENTIAL FUNCTIONS
= 4 @ 4 = 42
=
When you divide the product of five 4's by the product of three 4's the
result is two 4's. Whenever you divide exponential terms with the same
base, you form the quotient by subtracting the exponent of the denominator
from the exponent of the numerator to account for the cancellation of the
common factors.
Look at some applications of these statements:
= 23 – 3 = 20. Since
= 1, you have 20 = 1.
A number raised to the zero power is simply the number 1. The only
exception to this is the expression 00, which is undefined.
= 32 – 5 = 3–3. Also,
=
=
=
.
You should understand a negative exponent to represent the result of the
division of a number raised to a power by the same number raised to a
higher power. The outcome of this calculation reduces to 1 divided by
the factors remaining after cancellation is completed. The negative
exponent indicates how many factors of the base remain in the
denominator after you cancel all the corresponding factors in the
numerator.
Note that when you raise a number to a negative exponent, the result is
usually not a negative number.
Rewrite: 3–4, x–6,
Ans.:
=
,
.
,
= 1 @ 52 = 25.
You should now understand the meaning of any integer exponent. Next we
consider rational exponents that are not integers.
If you multiply a number by itself (for instance, a @ a = b), the product (b)
is called the square of each factor a, and each factor (a) is called the square
root of the product b.
Since you add exponents when you multiply, to obtain a product of b = b1
from two identical factors having base b, you must have an exponent of
for each factor. You then have
of b.
, and
is the square root
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CHAPTER 6: EXPONENTIAL FUNCTIONS
Similarly, the product of the three identical factors
is called the cube root of b, written
to the third power)
, since you have to cube (raise
to obtain b.
A rational exponent having the form
is called the (positive) n-th root of
the base. You write the fifth root of 7 as
= 4, and
.
=
, the tenth root of 12 as
,
.
If a rational exponent has a numerator other than 1, you interpret the
numerator as the number of times that you use that root as a factor. Thus
, or the third power of
also see this as the square root of 63 (since
,
). Both approaches
it is probably easier to compute
.
Other examples:
,
. Notice that you can
always produce the same result, and your calculator can use any of these
forms. However, if you are doing the calculation yourself, you will usually
find it easier to take the root first, then raise that to the power given in the
numerator. The root should be a smaller number than the numerator power
of the base, so if you take the root first, then raise your answer to the power,
you will usually have less trouble with the calculation than if you did the
steps in the other order. If you first raise the base to a high power and then
try to take the root of this larger number, you may not recognize the value
of the root. For example, to calculate
Rewrite:
Ans.:
,
,
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CHAPTER 6: EXPONENTIAL FUNCTIONS
Irrational numbers are also used as exponents but are harder to interpret than
rational exponents and you would rarely have to evaluate such an expression.
Your calculator can provide an answer that is accurate enough if you ever
have to evaluate such a number.
Understanding the above paragraphs will give you the background you need
to work with most exponential problems, and enough understanding to graph
most exponential functions.
Since you now know what each exponent indicates, consider the possible
choices of a base. If you use 1 for the base, you have no problems since
every power of 1 is equal to 1, but this is not very interesting. Multiplying
or dividing by any number of 1's never changes the value of an expression.
Similarly, 0 is not a very interesting base, as 0 raised to any positive power
is always 0 (the product of 0's is always 0). Negative (and zero) powers are
undefined (since you cannot to divide by 0).
Negative bases cause special problems, since negative numbers can be
raised to some powers, but not others (for instance
is meaningful but
is not - why?). So you are not likely to
see any negative numbers used as bases.
Positive numbers between 0 and 1 cause no difficulties when used as bases,
but are not necessary to study in detail, because you can write any such
number to the –1 power and obtain a number larger than 1.
For example,
= 2–1 ,
=
, and
=
. You can
obtain any exponential expression with a base between 0 and 1 using a
base greater than 1 by changing the sign of the exponent.
Therefore, you can restrict your attention to expressions that have bases
greater than 1. For most of the remainder of this chapter, you may assume
the base used is a number greater than 1, unless otherwise noted.
In many applications, the most common base used is the number e. The
importance of this number lies in its relation to the rate of natural growth
(or decay) of many populations. The symbol e is the name given to the
number that the function
large.
approaches asymptotically as x becomes
Your calculator gives a value for
2B = 8.82498, which seems
reasonable since B is larger than 3,
and 23 = 8, and for
= 9.7385,
which is consistent with
=
since
= 11.1803,
<
.
CHAPTER 6: EXPONENTIAL FUNCTIONS
The graphs below show that this number is near 2.71828, but e is an
irrational number with an infinite, non-repeating decimal representation.
X interval: [0, 10]
X interval: [10, 100]
x
(1+ 1/x)x
1
2
10
2.593742
100
2.704813
1000
2.716923
10,000
2.718145
105
X interval: [100, 1000]
Be sure you understand that e is not a variable, but the name for a particular
number, like 7 or B. Many exponential functions that describe natural
growth of some kind use e for the base, and most of the following examples
will also. The only disadvantage to using e for the base is the fact that
calculating powers of e is not convenient to do by hand. However, if you
need to calculate a numerical value for a power of e, your calculator can do
it. It will not be necessary for you to do perform such calculations to
produce the graphs.
Graphing
Since you may use any real number, whether rational or irrational, positive
or negative or zero, as the exponent of any base larger than 1, you do not
have to restrict your domain when you analyze exponential functions, unless
the term in the exponent requires some restriction itself.
You can usually find the intercept without too much trouble and then analyze
the end-behavior to obtain the graph.
For example, try f1(x) =
.
This has an intercept at (0, 1) because 20 is equal to 1.
As you move through higher values in the domain from x = 0, the
function instructs you to raise 2 to successively higher powers. When
you raise 2 to higher powers, you obtain larger numbers. Also, these
numbers do not increase at the same rate at all, but increase much more
rapidly as the exponents get larger. The graph is very far from linear
and shows rapid upward curvature, increasing faster and faster as the
values you input get larger.
21 = 2, 22 = 4, 23 = 8, 25 = 32,
210 = 1024 and 220 = 1,048,576!
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CHAPTER 6: EXPONENTIAL FUNCTIONS
When you move in the other direction from x = 0, through increasingly
more negative values, you must recall the meaning of negative
exponents. The negative sign in the exponent does not indicate that the
result is negative, but that you now have a larger power term in the
denominator of the number. Since the result of dividing +1 by a
positive number is positive, you can see that raising 2 to negative
powers always produces positive results. Your graph appears above the
x-axis across the entire domain. However, as your exponents become
more negative, the number you divide by rapidly becomes larger.
Therefore, the values of the function get closer and closer to zero,
although always above zero.
2–1 =
, 2–2 =
=
2–3 =
=
, 2–5 =
2–10 =
=
and
2–20 =
=
.
,
The completed graph shows a curve that is very close to (but above) the
x-axis on the extreme left, increases as it moves to the right, passes
through the intercept (0, 1), and then increases very rapidly as it moves
through the positive x-values.
Graph of f1(x) = ex
Notice that nothing you did in the last example depended on using 2 for the
base. Every step is the same, even the intercept, if you graph
instead of
. The graphs of both these functions look very similar. Naturally the
values of the functions (except at the intercept) differ. If you graph them
both on the same axes, you will see
rise on the right more quickly than
, because raising 3 to a positive power gives you a larger number than
raising 2 to the same power. Also,
gets closer to the x-axis faster on the
left since dividing by larger numbers (powers of 3 instead of powers of 2)
produces smaller function values.
To see the effect of using different bases, use your calculator to graph
several functions on the same axes, just changing the bases used. You
should see a collection of similar-looking graphs differing only in how fast
they rise on the right and decrease on the left, but all passing through the
same intercept.
Next, try f2(x) =
, which is the same as
.
There is no reason to exclude any values from the domain, so the graph
is in one piece across the entire x-axis with the intercept at (0, 1).
Now, if you let the x values increase, the exponent becomes more
negative, and the function values decrease, getting very close to 0.
Graphs of 7x, 3x, 2x, 1.5x.
=
,
CHAPTER 6: EXPONENTIAL FUNCTIONS
On the negative side, the exponent becomes a larger positive value when
you use more negative x-values. Therefore, the further to the left you
go, the higher the function values become, and the faster they increase.
Use your calculator to graph both
and
on the same axes. You
see that the two graphs are symmetric to each other across the y-axis,
because the values for each are the same when you switch the signs.
Graphs of 3x and 3–x.
For another example, look at f3(x) =
.
You do not have to exclude any x-values from the domain, and the
intercept point is again (0, 1), because –02 = 0 in the exponent and
50 = 1.
When you use a large positive value for x, squaring produces an even
larger positive number, and the exponent, –x2, is a large negative
number. Raising 5 to a large negative power gives you a function value
very close to 0, so the graph decreases from the intercept to values very
close to the x-axis (although above it) on the far right side.
Right-half graph of f3(x).
For the left side, you use large negative values, which when squared
produce large positive numbers, so again –x2 becomes a large-sized
negative number, an exponent that gives you a function value that is a
small positive number. Therefore, the graph on the left also is near the
x-axis, and slightly above it.
Since the function produces the same value when you change the sign
of your input value, you can see that you have an even function that has
a symmetric graph. Remember that –x2 instructs you to square your
x-value first, then multiply by –1, which is not the same as (–x)2, which
squares the number after you multiply by –1.
Complete graph of f3(x).
For a trickier example, look at f4(x) =
.
Now you cannot allow x = 0 to remain in your domain, because you
find your exponent by dividing 1 by each x-value in the domain.
Therefore, your graph has no intercept point.
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CHAPTER 6: EXPONENTIAL FUNCTIONS
Positive x-values produce positive function values that get closer to
y = 1 as the x-values increase, since large positive x-values yield values
for the exponent,
, that are close to 0. Positive x-values close to
x = 0 give large positive values for
for
, and very large positive values
. The graph on the positive side shows a curve that is very high
just to the right of the y-axis and approaches the asymptote y = 1
rapidly as you move further to the right.
On the negative side, x-values on the far left produce values for
are close to 0, but negative. Your values for
Right side graph of f4(x).
which
are near, but less
than 1. As the x-values approach x = 0 from the negative side,
becomes very large negatively. Large negative powers of e produce
values that are close to 0 (but positive).
The graph on the left side of the y-axis shows a curve that is just below
its asymptote, y = 1, on the left and approaches the point (0, 0) as you
move toward x = 0. Just as it reaches (0, 0) the graph stops without
reaching the point at the origin, since x = 0 is not in the domain. Just
the single point (0, 0) is missing - this is hard to see in the graph.
Complete graph of f4(x).
Applications
80 – 60
Exponential functions appear frequently in models that describe natural
growth or learning and loss or decay.
The function f5(x) = 80 – 60
= 80 – 60
= 20. What does this
value, 20, represent?
could serve as a model for the number
of words a typist can type per minute, where x represents the amount of
experience the typist has had.
This function has a domain consisting of all the real numbers, but only
values of x equal to or greater than zero make sense in the model, since
the typist would not have negative experience. In such a model, you
might want to limit the graph to the part of the domain for which the
model makes sense. This time, graph the complete function, then see
what part makes sense for the model.
You calculate the intercept as usual.
–.5x = 0;
= e0 = 1; 60
If x = 0, then of course,
= 60 @ 1 = 60;
If you increase the value
CHAPTER 6: EXPONENTIAL FUNCTIONS
of x (which means what in the model?), then –.5x becomes more
negative;
60
gets closer to 0; 60
gets close to 0; and 80 –
approaches 80 – 0 = 80. What is your interpretation of this
value of the function?
Finally, to complete the graph, although this part has no significance in
this model (why?), look at the negative values for x. As these values
become more negative, –.5x becomes more positive;
(in the positive direction); 60
gets larger
is a very large positive number; and
when you subtract increasingly larger positive numbers from 80, your
results start to become increasingly negative.
The graph of the function shows a curve increasing from left to right.
It passes through the intercept (0, 20) because the typist could type 20
words per minute without any experience, then continues to increase to
the right as the typist gains more experience and eventually approaches
y = 80, representing the limit of the typist's abilities.
Functions like this previous one are often useful in fields such as psychology
or industrial management and are called learning curves. They are usually
applied in situations that involve the learning of physical skills. Typically
one starts doing a task with a certain level of skill, which usually improves
rapidly with practice. However, once the person reaches a certain level of
aptitude, the rate of improvement slows. Further improvement may be hard
to discern when the skill has reached a level close to the individual's
maximum performance, which may differ from person to person.
Another example, which could describe the decline of a wildlife
population in a polluted area, is given by the function
f6(t) = 50 + 130
,
which gives the population (hundreds of animals) in terms of t, the
number of years since 1980.
First find the intercept. How do you interpret this point?
Now see what the long-term behavior will be. Very large positive
values represent large periods of time after 1980. The model tells you
what you can expect to happen in the future. How many animals will
manage to survive in the future?
In the other direction, you use increasingly more negative values.
Negative time after 1980 represents time before 1980. What was the
animal population like then?
Graph of f5(x).
109
110
CHAPTER 6: EXPONENTIAL FUNCTIONS
Your graph shows a large population before 1980, although the model
is probably not realistic too far to the left. The intercept shows a
population of 18,000 (180 hundred) animals in the area in 1980, and a
subsequent decline to approximately 5000 (50 hundred) animals. You
must be aware of the size of the numbers produced by the function
before you can obtain the graph on your calculator.
You might use the same kind of function as this last one to describe what
happens after one studies a large quantity of information. You can view the
graph as a "forgetting" curve, noting rapid loss of some material down to
a base of knowledge that is not lost. It might also serve to model the rate
at which a drug passes out of the bloodstream and many other similar types
of models.
If you understand the meaning of the variables of the function, it only takes
some common-sense to interpret the information contained in the model.
Notice that although your calculator can produce all the graphs you need,
you have to understand how the function behaves to set your viewing
window to the optimal dimensions for the complete graph.
For a more sophisticated example, look at f7(x) =
. This
function produces a type of curve called a logistic curve, which serves
as a good model for the growth of populations (among other
applications).
You get the graph of this function by applying the same methods you
always use. First, notice that this function requires a division step.
You must remove from your domain any values of your variable, x, that
give you a value of zero in the denominator. You know that
is
always positive, as all powers of positive bases are. Multiplying
by 6 keeps the value positive. Adding 4 to a positive number cannot
produce a value of zero, so there are no values of x that can make this
denominator zero, and you can use all real numbers for x in this
function. [Notice that if the denominator had been
, there
would be an x-value that would make it zero and you would have to
remove this value from the domain.]
You find the intercept as usual,
point is (0, 2).
. The intercept
Graph of f6(t).
CHAPTER 6: EXPONENTIAL FUNCTIONS
For large positive x-values,
111
gets near zero. Multiplying by 6
keeps the value near 0, and adding 4 produces a denominator near 4.
Dividing 20 by a number near 4 gives you a function value near 5. For
large positive x-values, the function takes on values close to 5.
When you use large negative values for x,
becomes a large
positive number, since the exponent is positive when you multiply –.3
by the negative x-value. Multiplying this already large number by 6 and
adding 4 has little effect, so the function value is the result of dividing
20 by a large positive number. This result is a positive number that is
near zero.
The graph shows an s-shaped logistic curve. It is close to y = 0 for
negative x-values, rises to pass through (0, 2), and continues to increase
rapidly until the function values approach y = 5, where the curve
becomes asymptotic to this line.
This type of model simulates the growth of a population that stays small
until it becomes large enough to grow rapidly. It continues this growth until
the stress caused by deficiencies in natural resources forces the population
to limit its further growth. It serves well for the growth of populations in
areas unaffected by immigration and emigration and for modeling the spread
of diseases or rumors.
Graph of the logistic curve, f7(x).
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CHAPTER 6: EXPONENTIAL FUNCTIONS
Exercises
1. Calculate:
a.
b.
c.
2. Rewrite in simplest form:
a.
b.
c.
d.
e.
3. Sketch the graph of:
a. f(x) =
b. f(x) =
c. f(x) =
d. f(x) = 2 @
–8
e. f(x) =
4. Sketch the graph of f(x) =
a. f(–x)
5.
b. f(x + 3)
Sketch the graph of:
a. f(x) = 8 – 3
b. f(x) = –7 + 4
c. f(x) = –2 + 9
d. f(x) = 6 + 2
e. f(x) =
f. f(x) =
g. f(x) =
h. f(x) =
i. f(x) =
j. f(x) =
. On the same axes, sketch the graph of:
c. 2f(x)
d. –f(x)
e. f(x) + 5
f. 4 – 3f(x – 2)
CHAPTER 6: EXPONENTIAL FUNCTIONS
113
6. A student's knowledge of American history, given as a percentage of the material taught in a course he took,
can be modeled by K(t) = 38
+ 40, where t is the number of months since his final examination.
Sketch this model. How well did he do on his final examination? How much of the material will stay with
him in later life? About how long does it take before he knows less than half the material covered in the
course?
7. A model for the world record in the shot put (given in feet) is given by R(t) = 80 – 34
, where t
represents the number of years since 1900. Sketch the graph of this model. According to the model, what
was the record in 1900? What is the ultimate world record? Give an estimate for what the model predicts
the record would be today.
8. A new medication can remove viruses from the blood stream. A model of how this drug worked on a
volunteer patient is given by B(t) = 90 – 30
, where B gives the count (in thousands) of the viruses
detected in the patient t hours after the drug was administered. Sketch a graph of this model. How many
viruses were present when the drug was taken? What happened? Approximately when were the viruses
eliminated from the patient?
9. Sketch the graphs of:
a. f(x) = x +
b. f(x) =
+
c. f(x) =
–
d. f(x) =
e. f(x) =
f. f(x) =
10.
Find the points of intersection of the graphs of f(x) =
a. g(x) = 3x – 2
2
b. 2x + 4
– 5 and
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CHAPTER 6: EXPONENTIAL FUNCTIONS
Answers
1. a.
=
c.
=
2. a.
=
= 12
=
=
=
= 105 = 100,000
e.
3. a.
b.
d.
= 50
= 2.25
b.
d.
b.
c.
e.
c.
=
=
= 4.5
CHAPTER 6: EXPONENTIAL FUNCTIONS
4. a.
b.
c.
d.
e.
f.
5. a.
b.
c.
d.
e.
115
116
CHAPTER 6: EXPONENTIAL FUNCTIONS
f.
g.
i.
h.
j.
6.
His grade on the final examination, which he took on the last day of the
course, was K(0) = 78%. He will retain about 40% of the material
eventually. After about 7 months he retains about 50% of the material.
7.
The record in 1900 was R(0) = 46 feet. According to the model the
ultimate world record is 80 feet. The model predicts a 1991 record of
slightly above 74 feet.
CHAPTER 6: EXPONENTIAL FUNCTIONS
117
8. When the drug was taken the number of viruses present was B(0) = 90
thousand. The viruses were eliminated rapidly from the patient's system.
After approximately 22 hours the viruses were eliminated.
9. a.
b.
c.
d.
e.
f.
10.
a.
b.
(–.87, –4.61), (2, 4)
(3, 22)
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CHAPTER 6: EXPONENTIAL FUNCTIONS