2
f(x)
Chapter1
FUNCTIONS IN
MODELING
■
LESSON 1.1
FUNCTIONS AS MODELS
■
LESSON 1.3
MODELING LINEAR PATTERNS:
Beginning the Tool Kit of Functions
■
MODELING BEHAVIOR:
EXPLANATIONS AND PATTERNS
Providing Explanations
Mathematical modeling is a process by which a realworld situation is replaced with a mathematical one. If the
real-world situation and the mathematical setting are well
matched, then information obtained in the mathematical
setting is meaningful in the real-world setting. Quite
often, the mathematical setting includes one or more
equations or graphs or other geometric objects that are
called the mathematical model.
LESSON 1.2
CREATING A MATHEMATICAL MODEL
■
CHAPTER INTRODUCTION
LESSON 1.4
EXPANDING THE TOOL KIT
OF FUNCTIONS
Mathematical models are built to explain why things
happen in a certain way. They are also created for the
purpose of making predictions about the future. For
example, you may want to know why a comet travels in
the path that it does and when it will be visible again
from the earth.
Finding Patterns
■
LESSON 1.5
TRANSFORMATIONS OF FUNCTIONS
■
LESSON 1.6
OPERATIONS ON FUNCTIONS
■
A good mathematical model can help the modeler
understand important features of the situation under
investigation. The reverse is also true. Knowing features
of the situation can help you develop a good model.
CHAPTER 1 REVIEW
Corbis
When developing a model, modelers often look for the
existence of regularity or patterns in the real world.
Frequently these patterns involve numbers. Describing
these patterns mathematically helps produce information
that is useful.
Among the simplest patterns are those that relate one
real-world quantity to another. Sometimes these patterns
are more evident when viewed visually, that is, with the
aid of a graph.
3
Number
of Storms
Atlantic Tropical Storm Formation
1885–1996
200
180
160
140
120
100
80
60
40
20
0
Dec 1–15
Nov 17–23
Nov 1–8
Oct 17–23
Oct 1–8
Sept 17–23
Sept 1–8
Aug 17–23
Aug 1–8
July 17–23
July 1–8
June 16–23
June 1–7
S1
Graphs are used so often because they tell a story more easily
than it can be told with words or numbers. The story in
Figure 1.1 is apparent at a glance: The number of tropical
storms in the Atlantic increases to a maximum around the
end of August, then falls off rapidly. There is a moderate
increase in early October, after which the decreasing pattern
resumes.
The following activity asks you to reflect on several familiar
real-world situations in which one quantity is related to a
second. You will be asked to think about how graphs
describing these situations might appear. Look for patterns
and trends as you analyze the possibilities. Focus on features
you consider to be important in the graphical models.
Warm-Up Activity
Note that the graphs in the activity (as others you will be
asked to draw) are shown without numerical scales. They
show qualities that capture the key features of the situation
(patterns and trends), but do not show exact quantities.
For each of the following six scenarios, a context and a figure
showing several graphs are given. After discussing the
context with your partner or group, answer the following
questions for each situation.
FIGURE 1.1.
4
Precalculus: Modeling Our World
a) Examine each of the graphs in the figure. Which graph best
models the situation?
b) What features made you choose that particular graph? What
features made you discount the other graphs?
c) What are the two quantities or variables in the situation?
1. Situation #1: The height of a person over his or her lifetime.
See Figure 1.2.
2. Situation #2: The circumference of a circle as its radius changes.
See Figure 1.3.
3. Situation #3: The height of a ball as it is thrown into the air.
See Figure 1.4.
4. Situation #4: The amount of observable mold on a piece of bread
sitting at room temperature from the time it is baked to several
months later. See Figure 1.5.
5. Situation #5: The daily average low temperature in degrees
Fahrenheit in Fairbanks, Alaska from January 1 to December 31.
See Figure 1.6.
6. Situation #6: The temperature of a cold drink left in a warm room.
See Figure 1.7.
DISCUSSION/REFLECTION
1. List some of the important features of the graphs that helped you
choose the ones that best represented, or modeled, the given
situations.
2. In situations 2 and 6, arrows were drawn on the ends of the graphs
to show that the graphs continue indefinitely. Explain why such
arrows were not used in the other graphs.
3. You identified the two variables in each of the six situations in the
Warm-Up Activity. For which of those situations does it make sense
for either of the variables to be negative? Explain.
5
Precalculus: Modeling Our World
Functions in Modeling
y
y
a
a
b
b
c
c
x
x
FIGURE 1.2.
Possible graphs for situation #1
FIGURE 1.3.
Possible graphs for situation #2
y
a
y
c
a
b
c
b
x
x
FIGURE 1.5.
Possible graphs for situation #4
FIGURE 1.4.
Possible graphs for situation #3
y
y
a
b
a
c
b
c
x
x
FIGURE 1.6.
Possible graphs for situation #5
FIGURE 1.7.
Possible graphs for situation #6
Precalculus: Modeling Our World
Exponentials & Logarithms
Chapter One
f(x)
Functions as Models
LESSON 1.1
n the Warm-Up Activity, graphs were used to represent relationships between two
quantities. In each case, a graph served as a useful tool in detecting a pattern or general
trend.As you will see in this chapter, a graph can also be used to predict a value of one
quantity from a value of the other. Often, that is the goal of mathematical model: to
predict what will happen to one quantity when another changes.That is the central idea
in something mathematicians call a function: to produce exactly one value of one
quantity from a known value of another.
I
FUNCTIONS
DOMAINS AND RANGES
REPRESENTATIONS
EXERCISES 1.1
In this chapter you will begin a study of mathematical functions and their properties.
FUNCTIONS
For the purposes of this book, the variable G is defined as a
function of the variable x if each value of x has a unique (one
and only one) value of G associated with it. G is called the
dependent or response variable and x is called the independent
or explanatory variable. For example, in situation #2 of the
Warm-Up Activity, the circumference of a circle (C) is a function
of the radius (r). Thus the dependent variable is C and the
independent variable is r. That is, the independent (explanatory)
variable “explains” the dependent variable while the dependent
(response) variable “responds” to changes in the independent
variable.
DOMAINS AND RANGES
The set of all values that make sense for the independent or
explanatory variable is called the domain of the function.
Corresponding to the domain, the set of output values that a
function generates from its domain is called the range of the
function.
EXAMPLE 1
1. Determine a domain for each situation in the Warm-Up
Activity.
2. Determine the domain for the function defined by the
equation y = 1_x.
Lesson 1.1
Precalculus: Modeling Our World
Functions in Modeling
3. Determine the domain of the function that assigns the GNP (Gross
National Product) to each year in the 1990’s decade.
SOLUTION:
1. The domain for each of the six situations in the Warm-Up Activity
is determined by the context of the problem. The domains in
situations 1, 3, 4, and 5 are bounded; that is, there are numbers that
are too large or too small to make any sense. Thus, the domains
consist of numbers x for which 0 ≤ x ≤ n where n is some number
meaningful to the context.
TAKE
The domains in situations 2 and 6 consist of all
numbers greater than or equal to zero; there is no
largest reasonable value for the independent
(explanatory) variable.
1
2. The domain of y = x does not include 0, since division
by 0 is impossible. In this case, x ≠ 0 is a restriction on
the domain.
3. The domain is the years from 1990 through 1999: 1990,
1991, 1992, 1993, 1994, 1995, 1996, 1997, 1998, 1999.
An easy way to write domains and ranges is to use interval
notation. A closed interval [a, b] indicates all real numbers x
for which a ≤ x ≤ b. Closed intervals include their endpoints.
An open interval (a, b) indicates all real numbers x for which
a < x < b. Open intervals exclude their endpoints. Half-open
or half-closed intervals are denoted by (a, b] or [a, b). These
notations indicate all real numbers x for which a < x ≤ b (that
is, open at a and closed at b) and a ≤ x < b (open at b and
closed at a) respectively.
[a, ∞) indicates all real numbers x for
which a ≤ x < ∞.
a
(a, ∞) indicates all real numbers x for
which a < x < ∞.
a
(– ∞, a] indicates all real numbers x
for which – ∞ < x ≤ a.
a
(– ∞, a) indicates all real numbers x
for which – ∞ < x < a.
a
(–∞, ∞) indicates all real numbers x
for which – ∞ < x < ∞, namely, all real
numbers.
EXAMPLE 2
Express each of the following in interval notation.
a) –2 < x ≤ 5
If you want to indicate
NOTE
the unboundedness of
x in the positive direction, use the symbol ∞
(infinity).To indicate unboundedness
in the negative direction, use –∞.
Thus,
b) 8 ≥ x ≥ 0
c)
–5
–4
–3
–2
–1
0
7
8
Lesson 1.1
Precalculus: Modeling Our World
Chapter One
SOLUTION:
a) The half-open interval can be expressed as (–2, 5].
b) The closed interval can be expressed as [0, 8].
c) Graphs typically indicate open endpoints with open dots (dots
that are not filled in). Closed endpoints are denoted by solid dots.
Thus, the graph indicates the half-open interval [–5, –1).
Caution: The notation (3, 4) can be interpreted as the point (3, 4) in the
coordinate plane or as the open interval consisting of all numbers strictly
between 3 and 4. Context determines the meaning.
REPRESENTATIONS
Functions can be represented by verbal descriptions, symbolic rules,
tables, graphs, and visual devices such as function machines and arrow
diagrams. A function that represents a given contextual situation is an
example of a mathematical model.
Verbal Descriptions
For simple functions, a verbal description may be all you need. For
example, the doubling function doubles each input value, and the
squaring function squares each input value.
Symbolic Rules
Since functions involve mathematical operations, equations are
frequently the most compact representations. Thus, the doubling
function can be written as y = 2x. The equation y = x2 describes the
squaring function. Both of these functions have unrestricted domains.
One way of writing the equation for a function is to use function
notation. For example, you could write the squaring function as
H(x) = x2. The symbol H(x) is read as “the value of H when x is input,”
or simply “H of x.” It means that H is the name of the function, that x is
the independent variable, that H(x) is the dependent variable, and that
the action is to square the input value.
Lesson 1.1
Precalculus: Modeling Our World
Functions in Modeling
In function notation, the input variable is named in the parentheses
and H(x) is the output variable. However, it is not the letter used
that is important when naming a function; it is the rule for the
function that counts. For example, g(x) = 2x + 5, f(m) = 2m + 5 and
h(t) = 2t + 5 all represent the same function, one that doubles the
input value and adds five. Be aware, too, that notations such as g(x)
and y are used interchangeably. For example, y = 2z + 5 defines the
same function as g, f, and h, above. Function notation is especially
useful when evaluating functions at specific values.
9
TAKE
g(x) is the
NOTE
name of the
dependent
variable and does not indicate
multiplication. So, g(x) does
not mean some variable g
times some variable x.
EXAMPLE 3
If g(x) = 4x – 3, find g(8). Also find the value of x for which g(x) = –9.
SOLUTION:
g(8) = 4(8) – 3 = 32 – 3 = 29.
If g(x) = –9, then 4x – 3 = –9.
Solve this equation:
4x = –9 + 3 = –6
x = –6/4 = –3/2 = –1.5
Visual Representations
Verbal descriptions usually describe functions
as actions, that is, what the functions do to their
inputs.
A more visual representation, such as the function
machines shown in Figure 1.8, may convey the
function-as-an-action idea more clearly.
x
x
double x
square x
D(x)
The function machines in Figure 1.8 represent
D(x) = 2x and S(x) = x2. A value for the independent variable x is selected
from the domain. It then goes into the machine and the rule is applied to
it. The resulting value of the dependent variable (what comes out of the
machine) is a value in the range of the function.
To be a function, each value of the domain must correspond to exactly
one value in the range. For example, for the doubling function, D(3) = 6
S(x)
FIGURE 1.8.
Function machines.
10
Lesson 1.1
Chapter One
Precalculus: Modeling Our World
(and only 6). For the squaring function, S(3) = 9 (and only 9). Note that
S(–3) is also 9, but that is acceptable. Each input needs exactly one output,
but it is permissible for an output value to correspond to more than one
input value.
Arrow diagrams (Figure 1.9) can be thought of as simplified function
machines. Again, they represent functions as operations on numbers.
multiply by 2
10
x-value
square
20
y-value
10
x-value
100
y-value
FIGURE 1.9.
Arrow diagram.
Tables
Tables are another important representation. In tables, the column
headings represent the variables, and the numbers in the columns are the
values of the variables. For example, if Figures 1.10 and 1.11 represent the
doubling function D, then D(1) = 2. Exactly two table columns are needed
to define a function’s input and output, so if a table displays more than
two columns, you need to check which two columns represent the function
you are studying.
FIGURE 1.10.
Spreadsheet table for y = 2x.
FIGURE 1.11.
Calculator lists for y = 2x.
It is sometimes helpful to combine representations, especially with tables.
Arrow diagrams can show how to get from one column of a table to another.
Similar information appears as the formulas for columns of a spreadsheet
(top of Figure 1.10) or calculator lists (bottom left of Figure 1.11).
Functions in Modeling
Lesson 1.1
Precalculus: Modeling Our World
To determine whether a table represents a function, check that each
input value has exactly one output value. This is the case for the tables
shown in Figures 1.10 and 1.11, so each represents a function. Table 1.1
does not describe a function since the input value 1 has two values, 7
and 5, associated with it.
One drawback of a table as a representation of a function is that it may
not display all of the values of the function. If that is the case, care must
be taken to consider all input values in the domain.
11
Input
Output
1
7
4
6
–2
14
5
3
1
5
TABLE 1.1.
Input and output values
(not a function).
Graphs
Perhaps the most familiar representation of functions is as
graphs. In graphing functions, it is customary to use the
horizontal axis for the domain and the vertical axis for the
range. A graph of y = 2x can be constructed from Figure 1.10
by plotting the points and connecting them with a line
(Figure 1.12).
y
•
14
12
•
10
8
•
6
In Figure 1.13, the dot at the left end of the curve indicates
that the graph does not continue further to the left and that
f(–3) = –1. The arrow at the right indicates that the graph does
continue to the right. Thus, the indicated domain of the
function is all numbers greater than or equal to –3, or [–3, ∞).
Similarly, the range is all numbers greater than or equal to –1,
or [–1, ∞).
One advantage of graphs (and tables) over
arrow diagrams, function machines, and
function notation is that both graphs and
tables display many input-output pairs
simultaneously, possibly making important
properties more visible. However, a graph is
not suitable for evaluating a function precisely.
Pairs of values that can be plucked from a table
or calculated from a formula must be estimated
as coordinates of a point on the graph.
4
•
2
x
–2 –1
•
1
–2
2
3
4
5
6
8
9
7
–4
FIGURE 1.12.
y
2
1
x
–4
–3
•
–2 –1
–1
1
2
3
4
5
6
7
–2
As you continue to learn more about functions, you will use all of these
different representations as tools to help you understand and solve realworld problems.
FIGURE 1.13.
Finding the domain and
range from a graph.
1.S.20
12
Lesson 1.1
Chapter One
Precalculus: Modeling Our World
Exercises 1.1
1. Consider the graphs in Figures 1.14–1.17. Use the definition of
function to determine whether each graph is the graph of a
function. Explain your reasoning.
a)
y
y
zz
x
x
FIGURE 1.15.
FIGURE 1.14.
y
c)
d)
x
FIGURE 1.16.
y
x
FIGURE 1.17.
2. One test of whether a graph represents a function is the vertical
line test. This simple test says that if you can find any vertical line
that intersects the graph in more than one place, the graph is not a
function. If no such line exists, then the graph is a function.
Lesson 1.1
Precalculus: Modeling Our World
Functions in Modeling
Consider the graphs shown in Figure 1.18.
Exercises 1.1
y
y
vertical line
vertical line
x
x
FIGURE 1.18.
Graph A
Graph B
Since graph A intersects the vertical line in two places, it does not
pass the vertical line test, and therefore, is not the graph of a
function. But in graph B, no matter where you move the vertical
line, it will never intersect the graph in more than one point, so
this is the graph of a function.
a) Use the definition of function to explain why the vertical line test
is a valid test.
b) Apply the vertical line test to the graphs in Exercise 1. Which
graphs pass the test and are functions, and which do not?
3. In each of Tables 1.2–1.5, x is the independent variable and y the
dependent variable. Which of the tables represent functions and
which do not? Explain.
x
a)
c)
–2
5
0
1
y
–4
–3
0
7
x
2
2
2
2
y
–1
0
1
2
TABLE 1.2.
TABLE 1.4.
b)
d)
x
3
4
3
–4
y
0
8
1
–8
x
–1
0
1
2
y
2
2
2
2
e) Look back at the vertical line test in Exercise 2. State a “table test”
that can be used to identify tables that do not represent functions.
TABLE 1.3.
TABLE 1.5.
13
14
Lesson 1.1
Chapter One
Precalculus: Modeling Our World
f) Use your table test and the definition of function to explain why
y = x2 is a function even though both x = 3 and x = –3 give the
same x-value.
Exercises 1.1
g) Draw arrow diagrams for Table 1.2.
For each function described in Exercises 4–8, determine (a) the value of
f(3), and (b) the value(s) of x for which f(x) = 3.
4. f(x) = 5x + 2
5. f(x) = |x|
6. f(x) = x2 – 1
7. See Figure 1.19.
y
14
8. See Figure 1.20.
12
y
5
10
4
8
3
6
2
4
1
2
x
–6
–5
–4
–3
–2
–1
–1
–2
1
2
3
4
x
–4
–3
–2
–1
1
2
3
4
–2
–3
–4
FIGURE 1.19.
–5
–4
–6
FIGURE 1.20.
9. Return to the six situations in the Warm-Up Activity. What is the
independent variable in each situation? What is the dependent
variable?
10. For each of the following situations, identify two quantities that
vary. Which is the independent variable? Which is the dependent
variable? Use interval notation to indicate a reasonable domain and
range for each situation.
a) The number of leaves on a tree during the year in New England.
b) The amount of time spent studying and the grade earned on
the test.
Precalculus: Modeling Our World
Functions in Modeling
c) Depth of the water in a bath tub with a steady stream of water
running in.
Lesson 1.1
15
Exercises 1.1
d) The height of a candle as it burns down.
e) The number of car accidents in a certain city and the amount of
alcohol consumed per person.
f) The length of a student’s hair over the course of a year.
11. A qualitative graph demonstrates the important features of a graph
without worrying about exact scales. The graphs in the Warm-Up
Activity were qualitative graphs. Sketch a qualitative graph for
each of the situations in Exercise 10.
12. Express each of the following in interval notation.
a) 4 < y ≤ 10
b)
0
1
2
3
4
5
8.0
8.5
9.0
c) –2 ≥ x ≥ –3.5
d)
6.5
7.0
7.5
13. Consider the function f(x) = x + 1 .
x−2
a) Is the point (5, 2) on the graph of f ?
b) Determine the value of f(–4).
c) What is the domain of f ?
14. To chemists and others, solubility in
water is an important property of a
substance. As they investigated this
property, they discovered a pattern in
the relationship between temperature
and solubility, which can be seen by
exploring data such as those in
Table 1.6.
Temperature °C
Grams of potassium chloride (KC l)
per 100 Grams of Water
10
30
19
32
30
36
43
40
50
42
59
45
TABLE 1.6.
a) Does the table represent a function? Explain.
b) Which variable is the independent variable and which is the
dependent variable?
16
Lesson 1.1
Chapter One
Precalculus: Modeling Our World
c) One way to examine data for trends and patterns is to construct a
graph of ordered pairs, which is called a scatter plot. To construct
a scatter plot of the ordered pairs in Table 1.6, plot the data with
the independent variable as the x-coordinate and the dependent
variable as the y-coordinate. Make a scatter plot of the data in
Table 1.6. (The first two ordered pairs are plotted in Figure 1.21
for you.)
Exercises 1.1
TAKE
y
Grams of KCl per 100 grams of water
Figure 1.21
NOTE
shows a
scatter plot
of the number of grams of
KCl that can be dissolved
in 100 grams of water
versus the temperature
of the solution.The placement of the word “versus”
means that the number of
grams of KCl per 100
grams of water is the
dependent (response)
variable and the temperature is the independent
(explanatory) variable.
45
40
35
30
25
20
15
10
5
x
0
0
10
20
30
40
50
FIGURE 1.21
60
Temperature °C
d) Scatter plots with data that fall along a straight line are said to
have a linear form. If the data do not fall along a straight line,
the scatter plot has a nonlinear form. Describe the form of the
scatter plot in (c).
15. Table 1.7 shows data for the temperature of a cup of coffee as it
cools down.
Time (minutes)
TABLE 1.7.
Temperature (°C)
0
1
2
3
4
5
6
7
8
80.0
66.3
55.6
47.3
40.8
35.8
31.8
28.8
26.4
a) What is the independent variable? What is the dependent
variable?
b) Enter the data from Table 1.7 into your calculator lists and create
a scatter plot of the temperature of the coffee versus time.
c) Describe the form of the scatter plot.
d) Describe the pattern you see in the scatter plot.
Precalculus: Modeling Our World
Functions in Modeling
16. Graphs can be characterized by their curvature. This text uses an
informal treatment beginning with this exercise. A graph is concave
up if it bends upward and concave down if it bends downward.
You might think of a graph that is concave up as being shaped like
a cup (or part of a cup) in its upright position, while a graph that
is concave down is shaped like an inverted cup. In Figure 1.22,
graphs (a) and (b) are concave up while (c) and (d) are concave down.
b
a
1 S 48b
FIGURE 1.22.
Illustrations of concavity.
d
c
a) Consider the graph in Figure 1.23. When is the graph concave
up? When is it concave down?
y
10
8
6
4
2
x
– 5 – 4 – 3 – 2 –1
–2
1
2
3
4
FIGURE 1.23.
Lesson 1.1
Exercises 1.1
17
18
Lesson 1.1
Exercises 1.1
Chapter One
Precalculus: Modeling Our World
b) Consider the graph in Figure 1.24. When is the graph concave
up? When is it concave down?
y
70
60
50
40
30
20
10
x
–3
–2
–1
–10
1
2
3
4
5
–20
–30
–40
–50
–60
FIGURE 1.24.