Functions And Their Graphs
Difference Equations
to
Differential Equations
Section 2.1
Functions And Their Graphs
Since functions are the basic building blocks out of which mathematicians construct models
of the physical world, it is essential that any student of mathematics have a firm grasp of
the concept. In particular, one must be careful to distinguish between a given function and
a notational or graphical representation for it. A function is a type of relationship, a mental
concept that cannot be seen or touched. Although pictures and symbolic representations
of a function are extremely important in understanding its behavior, the student must
always keep in mind the distinction between the function itself and its representations.
Modern methods for giving a formal definition of a function, developed in the latter
part of the 19th century, are based on set-theoretic ideas. We will not go into the details
necessary to make such a precise definition, but rather aim at an intuitive understanding
of the basic concept. For us, a function is a special type of relationship between two
quantities. We often think of this relationship to be one of dependence. That is, if the
value of one quantity, say y, is determined by the value of another quantity, say x, then we
say that y is a function of x. For example, if x represents the height from which a certain
rock is dropped and y represents the velocity with which the rock strikes the ground, then
the value of y will depend on the value of x and we say that velocity y is a function of height
x. Note here that if y is the terminal velocity of the object, then there are many different
values of x which yield the same value of y, namely, any value of x which gives the object
sufficient time to reach its terminal velocity before striking the ground. On the other hand,
for a given value of x, there is only one related value of y. It is this latter property that
makes the relationship between height and impact velocity a function. For any quantities
represented by y and x, in order to say that y is a function of x we require that every
value of x be related to exactly one value of y. Such a relationship often arises through
some physical dependency, a cause creating a deterministic effect, but the definition does
not require such a link between the quantities in question. A number of examples should
help clarify this concept.
Example Sequences are example of functions. That is, if {xn } is a sequence with n =
1, 2, 3, . . ., then every value of n determines exactly one value xn . For example, the area
of a regular polygon inscribed in a unit circle is a function of the number of sides. Also, a
difference equation, such as
xn+1 = 1.02xn ,
n = 0, 1, 2, . . ., makes xn a function of n. For example, the size of a certain population of
owls will be a function of the number of years from some starting date.
Example
The area of a circle is a function of the radius of the circle.
Example
The distance of the earth from the sun is a function of the time of year.
1
Copyright c by Dan Sloughter 2000
2
Example
Functions And Their Graphs
Section 2.1
The temperature at a certain fixed point in space is a function of time.
In mathematical terminology, if y is a function of x, then we call x the independent
variable and y the dependent variable. Also, the domain of this function is the set of
permissible values for x and the range is the set of all values of y which correspond to some
value of x.
Example Recall that the nth term of the sequence which gives the area of a regular
n-sided polygon inscribed in a unit circle is
2π
n
An = sin
,
2
n
n = 3, 4, 5, . . . The domain of this function is the set of integers {3, 4, 5, . . .}. The range
can be specified only by saying that it is the set of numbers
n
2π
sin
n = 3, 4, 5, . . . .
2
n
Before proceeding further, we should recall the notation for intervals of real numbers.
Given any real numbers a and b, we have
(a, b) = {x | a < x < b},
(2.1.1)
(a, b] = {x | a < x ≤ b},
(2.1.2)
[a, b) = {x | a ≤ x < b},
(2.1.3)
[a, b] = {x | a ≤ x ≤ b},
(2.1.4)
(a, ∞) = {x | x > a},
(2.1.5)
[a, ∞) = {x | x ≥ a},
(2.1.6)
(−∞, b) = {x | x < b},
(2.1.7)
(−∞, b] = {x | x ≤ b}.
(2.1.8)
and
Moreover, we call intervals of the form (2.1.1), (2.1.5), and (2.1.7) open intervals and
intervals of the form (2.1.4), (2.1.6), and (2.1.8) closed intervals.
Example If we let d specify the distance from the sun to the earth and t specify the
time of year, then the function that relates d and t has domain
{t | 0 ≤ t ≤ 8760} = [0, 8760],
where t is specified in hours, and range
{d | 91.4 ≤ d ≤ 94.6} = [91.4, 94.6],
where d is specified in millions of miles.
Section 2.1
Functions And Their Graphs
3
Before learning much about a specific function, a mathematician must represent the
function in some concrete form. This can be done in many ways. For example, we might
construct a table of values for the function. Such a table might have two rows, one for
values of the independent variable and one for the corresponding values of the dependent
variable. For example, if T is the temperature, in degrees Fahrenheit, at the Kalispell
airport weather station at time t, measured in hours past midnight, on August 3, 1999,
then our table might look like the following:
Time (t)
Temperature (T )
0
68
1
66
2
64
3
62
4
61
5
59
6
60
7
64
8
68
9
70
Time (t)
Temperature (T )
10
74
11
76
12
78
13
80
14
84
15
84
16
82
17
81
18
79
19
77
Time (t)
Temperature (T )
20
74
21
70
22
69
23
67
24
66
In other words, this table provides a complete listing of the values of the dependent variable
T which correspond to each value of the independent variable t.
Of course, if the domain of the function contains a large number of points, it might not
be practical to represent the function using a table. Indeed, most of the functions which
we will consider in this course have an infinite number of points in their domain, rendering
complete representations using tables impossible. Moreover, even with a limited amount
of data, it is hard to understand much about the underlying function by looking at a table.
One alternative to a tabular representation of a function is a graphical representation. If y
is a function of x, the graph of this function is the set of all points in the Cartesian plane
with coordinates (x, y). If the domain of the function has only a finite number of points,
as in the preceding example, then its graph is just a set of points in the plane, as we see
in Figure 2.1.1. However, if we were able to plot this function for all values of t between
0 and 24, then its graph would become a curve passing through the points given by the
table. With the given data, we could approximate this curve by plotting the given points
and then connecting successive points by straight lines, as in Figure 2.1.2. In either form,
the graph gives a good pictorial representation of the function. From this picture, we can
easily identify such things as the high and low temperatures for the day, as well as the
time at which they occurred, or the time of day when the temperature was changing most
rapidly.
It would be hard to overestimate the importance of graphs in studying functions; we will
in fact spend much time in this course considering graphs. However, the most concise, and
at the same time most complete, representation for a function is a formula which expresses
the values of the dependent variable in terms of values of the independent variable. For a
given function it may not be possible to find such a formula. For example, the function
which gives the temperature at the Kalispell airport for any given time during the day of
August 3, 1999, is not expressible by a formula; the only way we can compute values for
this function is to record the temperatures as they occur. On the other hand, for a circle
of radius r and area A, the formula A = πr2 gives us an explicit means for computing
values of the dependent variable A for any given value of the independent variable r. The
4
Functions And Their Graphs
Section 2.1
100
80
60
40
20
5
10
15
20
Figure 2.1.1 Plot of temperature data for Kalispell
100
80
60
40
20
5
10
15
20
Figure 2.1.2 Plot of temperature date for Kalispell with lines connecting data points
existence of a formula for a function enables us to perform mathematical computations
which, at best, could only be approximated otherwise. At the same time, it is important
to remember that a function is an abstract object; it is not itself a formula or a number or
a graph, but a relationship which exists between quantities specified by numbers. We need
to keep this in mind, even as we proceed to work more and more with functions through
their representations using formulas and graphs.
Example If V represents the volume and r the radius of a sphere, then V is a function
of r and the formula
4
V = πr3
3
expresses this relationship. Note that the domain of this function is the open interval
(0, ∞), even though negative values of r can be substituted into the formula without any
problems. This emphasizes that the function is determined by the underlying relationship
between V and r. Here we also have the range equal to (0, ∞).
Section 2.1
Example
Functions And Their Graphs
5
Suppose the quantity y is related to the quantity x by the formula
y=√
1
.
1 − x2
Since, by convention, the square root notation refers to the positive square root of a given
number, this relationship makes y a function of x. If we are given no further information
about this function, then we should ascribe to it the largest possible domain and range.
In this case, the domain is the interval (−1, 1) (that is, values of x for which 1 − x2 is
positive) and the range is [1, ∞) (that is, the possible results from dividing 1 by numbers
in the interval (0, 1]).
At this point, we have used notation for the dependent and independent variables of
a function, but not for the function itself. As with variables, it is common to use letters
to designate functions. For example, we frequently use f to denote a function, in which
case f stands for the function itself, a relationship, while expressions like f (x), f (2), and
f (s) denote particular values of the function. That is f (x), f (2), and f (s) represent values
of the dependent variable which correspond to the values x, 2, and s, respectively, of the
independent variable.
Example
The expression
f (x) =
1
x2
tells us that f represents a function which associates the value
1
x2
to a given value x of the independent variable. Hence, for example,
f (2) =
1
,
4
f (−1) = 1,
f (s) =
and
f (z + 1) =
1
,
s2
1
.
(z + 1)2
Note that the domain of f is {x | x 6= 0}. That is, f is defined for every real number
except 0. The range of f is (0, ∞).
Example Suppose S is the function which gives the temperature at the Kalispell airport
on August 3, 1990. If we measure time in of hours since midnight, then we know, for
example, that S(2) = 64 and S(19) = 77. However, if we let t represent the independent
variable for this function, namely, the number of hours since midnight, then we do not
6
Functions And Their Graphs
Section 2.1
have a general formula to express S(t). For example, we cannot compute S(7.5) or S(3.2),
let alone even consider what S(π) might be.
It often happens that the output from one function is used as input for another function.
For example, suppose a pebble is dropped in a pond and the resulting circular wave has a
radius of 20t centimeters after t seconds. Then if r is the radius of the wave and A is the
area inside the wave, we have r = 20t and A = πr2 . But the area inside the wave is also a
function of time, which may be expressed as
A = π(20t)2 = 400πt2 .
That is, the area of the circle is a function of the radius, which in turn is a function of time.
The function that we arrive at, namely, A as a function of t, is called the composition of
the two original functions. In the notation which uses letters to denote functions, we have
the following definition.
Definition If f and g are two functions, then the composition of f and g is the function
f ◦ g whose value at x is given by
f ◦ g(x) = f (g(x)).
Example
If f (x) =
√
x and g(x) = x2 + 1, then
f ◦ g(x) = f (g(x)) = f (x2 + 1) =
and
(2.1.9)
p
x2 + 1
√
g ◦ f (x) = g(f (x)) = g( x) = x + 1.
Note that f ◦ g and g ◦ f , as in this example, are not usually the same function.
Classes of functions
The simplest type of functions are those which involve only multiplication and addition.
In particular, functions of the form
p(x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ,
(2.1.10)
where a0 , a1 , . . . , an are constants and n is a nonnegative integer, are called polynomials.
If an 6= 0, the degree of the polynomial is n. For example,
q(x) = 3x2 − 13x + 3,
f (t) = 21t34 + 18t2 − π,
and
g(s) =
1 3
s + s + 12 − 4s5
2
Section 2.1
Functions And Their Graphs
7
are polynomials, of degrees 2, 34, and 5, respectively, whereas
h(x) =
3
x
is not a polynomial. In a sense polynomials are the building blocks for a large family of
important functions in calculus. In this regard, one of the major goals of this text is to
show how polynomials may be used to approximate more complicated functions.
Functions which may be written in the form of a polynomial divided by a polynomial
are called rational functions. For example,
f (x) =
and
g(s) =
3x2 − 4x + 1
x4 + 1
12
1
+ 2
s
s − 3s + 1
are both rational functions, the latter because it may be rewritten as a polynomial divided
by a polynomial if all the √
terms are put over the common denominator s(s2 − 3s + 1).
The function f (x) = x is neither a polynomial nor a rational function because x is
raised to a power which is not an integer. Functions which permit addition, multiplication,
division, and rational numbers for powers are called algebraic functions. Thus, for example,
3
g(t) = t 2 + 2t2 − 3
is neither a polynomial nor a rational function, but is an algebraic function. Similarly,
h(s) =
p
s2 + 3s + 2
is an algebraic function. We should note that every polynomial is also a rational function
and every rational function is also an algebraic function.
Functions which are not algebraic are called transcendental. The trigonometric functions are examples of transcendental functions. We shall discuss them in detail in the next
section.
Graphs of functions
We are now in a position to say more about the graphs of functions. With the notation we
have now, the graph of a function f is the set of all points (x, f (x)) in the plane, where x
is in the domain of f . For example, you should recall from previous work that the graph
of y = x2 is a parabola opening about the x-axis with its vertex at (0, 0). Also, you should
recall the shapes of the graphs of such functions as y = x2 , y = x3 , y = x4 , and y = x5 .
See Figures 2.1.3, and 2.1.4.
Moreover, given a function f and a constant c, you may recall that the graph of
y = f (x) + c is the graph of f shifted c units vertically (upward if c > 0 and downward if
c < 0), the graph of y = f (x − c) is the graph of f shifted c units horizontally (to the right
8
Functions And Their Graphs
-10
Section 2.1
20
20
15
15
10
10
5
5
5
-5
10
-10
-5
10
5
-5
-5
Figure 2.1.3 Graphs of y = x2 and y = x4
-10
20
20
15
15
10
10
5
5
-5
-5
5
10
-10
-5
-5
-10
-10
-15
-15
-20
-20
5
10
Figure 2.1.4 Graphs of y = x3 and y = x5
if c > 0 and to the left if c < 0), and the graph of y = −f (x) is the graph of f reflected
about the x-axis. Hence, for example, the graph of
y = x2 − 3
is a parabola, opening upward about the x-axis, with its vertex at (0, −3); the graph of
y = (x + 2)2 − 3
is a parabola, opening upward about the line x = −2, with vertex at (−2, −3); and the
graph of
y = −(x + 2)2 + 3
is a parabola, opening downward about the line x = −2, with vertex at (−2, 3). See Figure
2.1.5.
Drawing graphs of functions whose basic shapes are not already known to us can be
a difficult problem. If the domain of a function f is finite, then drawing its graph is
only a matter of plotting some points in the plane. However, most of the functions we
will encounter in this course will have domains containing an infinite number of points;
drawing the graphs of such functions requires much more than plotting a few points. The
Section 2.1
-10
Functions And Their Graphs
9
10
10
10
5
5
5
5
-5
10
-10
5
-5
10
-10
5
-5
-5
-5
-5
-10
-10
-10
10
Figure 2.1.5 Graphs of y = x2 − 3, y = (x + 2)2 − 3, and y = −(x + 2)2 + 3
problem is that no matter how many points we plot, we still do not know how the function
is behaving at the other points. For example, if we want to graph a function f on the
interval [0, 1], we might first plot the points
(0, f (0)), (0.1, f (0.1)), (0.2, f (0.2)), . . . , (1.0, f (1.0)).
Next, to guess at the behavior of the function between the plotted points, we might join
successive points by straight lines. Of course, this will only give us an approximation to the
true curve, the accuracy of which will depend on the actual behavior of the curve between
the plotted points, something about which we frequently have very little information. This
is similar to the problem we had with plotting the graph of a temperature function earlier.
However, here we can get help if we have a formula for f ; for in that case we can try
plotting more points, say
(0, f (0)), (0.05, f (0.05)), (0.10, f (0.10)), . . . , (1.00, f (1.00)),
or
(0, f (0)), (0.01, f (0.01)), (0.02, f (0.02)), . . . , (1.00, f (1.00)).
If the graph of f is a reasonably smooth curve, we will be able to approximate it as well as
we like by plotting a sufficient number of points. This raises two questions: How do we know
that we have plotted a sufficient number of points? And, given that a sufficient number
of points will most likely be a large number, how do we actually plot them all? Of course,
the latter question is answered by using a computer. In fact, this approach to graphing a
function is unreasonable without access to a computer, or at least a calculator. Computers
also provide help in answering the first question. We start by plotting a reasonable number
of points, say 100 or so. If we have reason to doubt the accuracy of the resulting graph,
perhaps because the curve is not as smooth as we expected it to be, we can double the
number of points and plot it again. Because any computer, and in fact many calculators,
do this type of work rapidly, it is reasonable to plot the same function several times until
we are comfortable with the picture. In Section 3.9 we will learn how to use some of the
techniques of calculus to better understand the geometry of the graph of a function. This
will help us identify whether or not the output from a computer is an accurate depiction
of the graph.
Example
Figure 2.1.6 compares the results of plotting the points
(0, f (0)), (0.1, f (0.1)), (0.2, f (0.2)), . . . , (1.0, f (1.0))
10
Functions And Their Graphs
1.5
1.5
1
1
0.5
0.5
0.2
0.4
0.6
0.8
1
Section 2.1
0.2
-0.5
-0.5
-1
-1
-1.5
-1.5
0.4
0.6
0.8
1
Figure 2.1.6 Plot of y = sin(30x), using first 11 points and then 101 points
with plotting the points
(0, f (0)), (0.01, f (0.01)), (0.02, f (0.02)), . . . , (1.00, f (1.00))
(joining successive points with straight lines) for the function f (x) = sin(30x). Clearly,
the second plot a dramatic improvement over the first.
When using a computer software package or a calculator to graph a function, there
are a couple of issues you should keep in mind. First, as we have just noted, plotting
a sufficient number of points is crucial to obtaining a good approximation of the graph.
Some programs will ask you to specify the number of points to plot, others will plot a
predetermined number of points, and still others will determine the number of points to
plot based on an estimate of the number of points necessary to provide an accurate picture
for the particular function. In the latter two cases it should still be possible to override
the program’s decision and specify your own choice for the number of points to plot. If
plotting more points significantly changes the look of the graph, then you should be wary
of the original plot and consider plotting even more points. Second, the computer will plot
the function in a rectangle, called a window. The horizontal scale for this window will be
the interval over which you want to graph the function. The vertical scale may be chosen
by you or by the computer program. If possible, it is usually best first to let the program
choose the vertical scale for the window and then to adjust it as necessary to provide a
good picture of the graph. If the vertical scale is too small, you may miss part of the
graph; if the vertical scale is too large, the interesting features of the graph may be too
small to be visible. For example, Figure 2.1.7 shows the graph of y = sin(x) on the interval
[−4, 4], first with the vertical window scale being the interval [−0.5, −0.5] and next with
the vertical scale changed to [−20, 20]. Certainly, a vertical window scale on the order of
[−1.5, 1.5], as shown in Figure 2.1.8, is a more appropriate choice for this graph.
Section 2.1
Functions And Their Graphs
11
20
0.4
15
10
0.2
5
-4
2
-2
4
-4
2
-2
-5
-0.2
4
-10
-15
-0.4
-20
Figure 2.1.7 Graph of y = sin(x), first with vertical window [−0.5, 0.5] and then [−20, 20]
1.5
1
0.5
-4
2
-2
4
-0.5
-1
-1.5
Figure 2.1.8 Graph of y = sin(x) with vertical window [−1.5, 1.5]
Problems
1. In each of the following, x and y denote certain variable quantities. Discuss whether or
not y is a function of x. If y is a function of x, can you write a formula that describes
the relationship? Also, find the domain and range of each function.
(a) x = speed of a train; y = distance the train travels in two hours
(b) x = height above sea level; y = atmospheric pressure
(c) x = time of the year; y = distance from the earth to the moon
(d) x = temperature at the Great Falls airport; y = time of the day
(e) x = length of the side of a square; y = area of the square
(f) x = area of a circle; y = circumference of the circle
(g) x = weight of a letter; y = first class postage for the letter
(h) x = OPEC price for a barrel of oil; y = Dow Jones Industrial Average
12
Functions And Their Graphs
Section 2.1
2. A projectile was shot vertically into the air. The height h of the projectile was measured
at 20 different times t. The following table gives the results, where t is in seconds and
h is in meters.
Time (t)
Height (h)
0.00
0.00
0.25
11.6
0.50
22.1
0.75
31.2
1.00
39.2
1.25
46.0
1.50
51.5
1.75
55.7
2.00
58.8
2.25
60.6
Time (t)
Height (h)
2.75
60.6
3.00
58.8
3.25
55.7
3.50
51.5
3.75
46.0
4.00
39.2
4.25
31.2
4.50
22.1
4.75
11.6
5.00
0.00
2.50
61.3
(a) Graph this data.
(b) Graph this data with lines connecting successive points. Do you think this is a
reasonable approximation to the graph of h as a function of t?
3. Identify the domain of each of the following functions.
(a) f (x) = x2 − 6x
√
(c) f (t) = t2 + t − 6
(e) f (s) =
41
s2 − 9
(g) z(s) = √
1
s2 + 6s + 1
√
x2 − 9
3
(d) h(x) = 2
x + 6x + 8
3
(f) y(t) = √
3 − t2
4
(h) f (x) =
x(x2 + 1)
(b) g(x) =
4. For each of the following pairs of functions, find f ◦ g(x), g ◦ f (x), f ◦ g(3), and f ◦ g(3),
when they are defined.
(a) f (x) = 4x + 12, g(x) = 5x − 2
√
(b) f (x) = x2 − 12, g(x) = x
1
(c) f (x) = 6x − x2 , g(x) =
x−9
5. (a) If the graph of f is a straight line with slope 3 and the graph of g is a straight line
with slope 4, show that the graph of f ◦ g is a straight line with slope 12.
(b) If the graph of f is a straight line with slope m and the graph of g is a straight
line with slope n, show that the graph of f ◦ g is a straight line with slope mn.
6. Graph each of the following functions on the given interval.
(a) f (x) = x2 + 6x + 1 on [−10, 5]
(b) f (x) = x5 + x4 + x3 + x2 + x + 1 on [−5, 5]
(c) f (x) = x5 + 8x4 + x3 + x2 + x + 1 on [−5, 5]
(d) f (x) = x5 + 8x4 + x3 + x2 + x + 1 on [−10, 10]
1
(e) g(t) =
on [−10, 10]
1 + t2
t
(f) f (t) =
on [−10, 10]
1 + t2
Section 2.1
Functions And Their Graphs
13
t2
on [−10, 10]
1 + t2
x2 − 1
(h) g(x) = 4
on [−5, 5]
x +1
7. Another problem in graphing using a computer or a calculator arises when the function
in question is not defined at some of the points in the interval of interest. For example,
try graphing
1
f (x) =
x
(g) g(t) =
on the interval [−5, 5]. There are several problems which may arise. First, if the
computer program tries to evaluate f at x = 0, you may get an error message. In
this case you may have to change the number of points being plotted so that x = 0 is
missed. Second, if the program evaluates f for values of x very close to 0, the output
from the function will be very large. The result might be that the vertical scale of your
graphing window is much too large. Hence, you may wish to change the scale of the
vertical axis. Another problem that may occur arises because the graph of f has two
pieces; that is, the part of the graph to the left of the y-axis is not connected to the
part of the graph to the right of the y-axis. If your graphing program simply connects
points as it moves from left to right, it will connect points on opposite sides of the
y-axis which should not be connected. This may be hard to avoid with some software,
but you should be aware of the problem and, consequently, interpret your results with
care.
8. In light of the remarks in Problem 7, try graphing the following functions.
1
on [−4, 4]
1 − s2
x3
(b) g(x) =
on [−4, 4]
1 − x2
(a) h(s) =
9. Recall that bxc denotes the largest integer less than or equal to x and dxe denotes the
smallest integer greater than or equal to x. Let f (x) = bxc and g(x) = dxe.
(a) What is the domain of f ? What is the range of f ?
(b) What is the domain of g? What is the range of g?
(c) Graph f and g on the interval [−5, 5].
(d) Graph h(x) = bx2 c on the interval [−2, 2].
10. We say a function f is periodic if there exists a constant T such that f (t + T ) = f (t)
for every value of t in the domain of f .
(a) Is it possible for a polynomial to be periodic?
(b) Are any of the functions in Problem 1 periodic?
(c) Suppose x represents the number of days since January 1, 1950, and y represents
the amount of rainfall in Spokane on day x. Do you think y is a periodic function
of x? If not, might it in some way be close to periodic?