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3 Functions and Graphs
3.4 Definition of function
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Definition of Function
The notion of correspondence occurs frequently in
everyday life. Some examples are given in the following
illustration.
Illustration: Correspondence
• To each book in a library there corresponds the number
of pages in the book.
• To each human being there corresponds a birth date.
• If the temperature of the air is recorded throughout the
day, then to each instant of time there corresponds a
temperature.
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Definition of Function
The element x of D is the argument of f. The set D is the
domain of the function. The element y of E is the value of f
at x (or the image of x under f) and is denoted by f(x), read
“f of x.”
The range of f is the subset R of E consisting of all
possible values f(x) for x in D.
Note that there may be elements in the set E that are not in
the range R of f.
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Definition of Function
Consider the diagram in Figure 2.
Figure 2
The curved arrows indicate that the elements f(w), f(z), f(x)
and f(a) of E correspond to the elements w, z, x, and a of
D.
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Definition of Function
To each element in D there is assigned exactly one
function value in E; however, different elements of D, such
as w and z in Figure 2, may have the same value in E.
The symbols
f : D  E,
and
signify that f is a function from D to E, and we say that f
maps D into E. Initially, the notations f and f(x) may be
confusing. Remember that f is used to represent the
function.
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Definition of Function
It is neither in D nor in E. However f(x), is an element of the
range R—the element that the function f assigns to the
element x, which is in the domain D.
Two functions f and g from D to E are equal, and we write
f = g provided
For example, if g(x) =
x in , then g = f.
f(x) = g(x)
for every x in D.
(2x2 – 6) + 3 and f(x) = x2 for every
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Example 1 – Finding function values
Let f be the function with domain
every x in .
such that f(x) = x2 for
(a) Find f(– 6), f( ), f(a + b), and f(a) + f(b), where a
and b are real numbers.
(b) What is the range of f ?
Solution:
(a) We find values of f by substituting for x in the equation
f(x) = x2:
f(– 6) = (– 6)2 = 36
f(
)=(
)2 = 3
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Example 1 – Solution
cont’d
f(a + b) = (a + b)2 = a2 + 2ab + b2
f(a) + f(b) = a2 + b2
(b) By definition, the range of f consists of all numbers of
the form f(x) = x2 for x in .
Since the square of every real number is nonnegative,
the range is contained in the set of all nonnegative real
numbers.
Moreover, every nonnegative real number c is a value
of f, since f( ) = ( )2 = c. Hence, the range of f is the
set of all nonnegative real numbers.
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Definition of Function
If a function is defined as in Example 1, the symbols used
for the function and variable are immaterial; that is,
expressions such as f(x) = x2, f(s) = s2, g(t) = t2, and
k(r) = r2 all define the same function.
This is true because if a is any number in the domain, then
the same value a2 is obtained regardless of which
expression is employed.
The phrase f is a function will mean that the domain and
range are sets of real numbers.
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Definition of Function
If a function is defined by means of an expression, as in
Example 1, and the domain D is not stated, then we will
consider D to be the totality of real numbers x such that f(x)
is real. This is sometimes called the implied domain of f.
To illustrate, if f(x) =
then the implied domain is the
set of real numbers x such that
is real—that is,
x – 2  0, or x  2.
Thus, the domain is the infinite interval [2,
).
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Definition of Function
If x is in the domain, we say that f is defined at x or that f(x)
exists. If a set S is contained in the domain, f is defined on
S. The terminology f is undefined at x means that x is not in
the domain of f.
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Example 2 – Finding function values
Let g(x) =
(a) Find the domain of g.
(b) Find g(5), g(–2), g(–a), and –g(a).
Solution:
(a) The expression
is a real number if and
only if the radicand 4 + x is nonnegative and the
denominator 1 – x is not equal to 0.
Thus, exists if and only if
4 + x  0 and 1 – x ≠ 0
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Example 2 – Solution
cont’d
or, equivalently,
x–4
and
x ≠ 1.
We may express the domain in terms of intervals as
[– 4, 1)  (1, ).
(b) To find values of g, we substitute for x:
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Example 2 – Solution
cont’d
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Definition of Function
Graphs are often used to describe the variation of physical
quantities. For example, a scientist may use the graph in
Figure 5 to indicate the temperature T of a certain solution
at various times t during an experiment.
Figure 5
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Definition of Function
The sketch shows that the temperature increased gradually
from time t = 0 to time t = 5, did not change between t = 5
and t = 8, and then decreased rapidly from t = 8 to t = 9.
Similarly, if f is a function, we may use a graph to indicate
the change in f(x) as x varies through the domain of f.
Specifically, we have the following definition.
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Definition of Function
In general, we may use the following graphical test to
determine whether a graph is the graph of a function.
The x-intercepts of the graph of a function f are the
solutions of the equation f(x) = 0. These numbers are
called the zeros of the function.
The y-intercept of the graph is f(0), if it exists.
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Example 3 – Sketching the graph of a function
Let f(x) =
(a) Sketch the graph of f.
(b) Find the domain and range of f.
Solution:
(a) By definition, the graph of f is the graph of the equation
y=
The following table lists coordinates of several points on
the graph.
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Example 3 – Solution
cont’d
Plotting points, we obtain the sketch shown in Figure 7.
Note that the x-intercept is 1 and there is no y-intercept.
Figure 7
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Example 3 – Solution
cont’d
(b) Referring to Figure 7, note that the domain of f consists
of all real numbers x such that x  1 or, equivalently, the
interval [1, ).
The range of f is the set of all real numbers y such that
y  0 or, equivalently, [0, ).
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Definition of Function
The square root function, defined by f(x) =
has a
graph similar to the one in Figure 7, but the endpoint is at
(0, 0).
Figure 7
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Definition of Function
The y-value of a point on this graph is the number
displayed on a calculator when a square root is requested.
This graphical relationship may help you remember that
is 3 and that
is not 3.
Similarly, f(x) = x2, f(x) = x3, and f(x) =
are often referred
to as the squaring function, the cubing function, and the
cube root function, respectively.
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Definition of Function
In general, we shall consider functions that increase or
decrease on an interval I, as described in the following
chart, where x1 and x2 denote numbers in I.
Increasing, Decreasing, and Constant Functions
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Definition of Function
cont’d
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Definition of Function
An example of an increasing function is the identity
function, whose equation is f(x) = x and whose graph is
the line through the origin with slope 1.
An example of a decreasing function is f(x) = – x, an
equation of the line through the origin with slope –1. If
f(x) = c for every real number x, then f is called a constant
function.
We shall use the phrases f is increasing and f(x) is
increasing interchangeably. We shall do the same with the
terms decreasing and constant.
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Example 4 – Using a graph to find domain, range, and where a
function increases or decreases
Let f(x) =
(a) Sketch the graph of f.
(b) Find the domain and range of f.
(c) Find the intervals on which f is increasing or is
decreasing.
Solution:
(a) By definition, the graph of f is the graph of the equation
y=
We know from our work with circles that the graph of
x2 + y2 = 9 is a circle of radius 3 with center at the origin.
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Example 4 – Solution
cont’d
Solving the equation x2 + y2 = 9 for y gives us y = 
It follows that the graph of f is the upper half of the circle, as
illustrated in Figure 8.
Figure 8
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Example 4 – Solution
cont’d
(b) Referring to Figure 8, we see that the domain of f is the
closed interval [–3, 3], and the range of f is the interval
[0, 3].
(c) The graph rises as x increases from –3 to 0, so f is
increasing on the closed interval [–3, 0].
Thus, as shown in the preceding chart, if x1 < x2 in
[–3, 0], then f(x1) < f(x2)
(note that possibly x1 = –3 or x2 = 0)
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Example 4 – Solution
cont’d
The graph falls as x increases from 0 to 3, so f is
decreasing on the closed interval [0, 3].
In this case, the chart indicates that if x1 < x2 in [0, 3], then
f(x1) > f(x2) (note that possibly x1 = 0 or x2 = 3)
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Definition of Function
Of special interest in calculus is a problem of the following
type.
Problem: Find the slope of the secant line through the
points P and Q shown in Figure 9.
Figure 9
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Definition of Function
The slope is mPQ given by
mPQ
The last expression (with h ≠ 0) is commonly called a
difference quotient.
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Example 5 – Simplifying a difference quotient
Simplify the difference quotient
using the function f(x) = x2 + 6x – 4.
Solution:
definition of f
expand numerator
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Example 5 – Solution
cont’d
subtract terms
simplify
factor out h
= 2x + h + 6
cancel h ≠ 0
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Definition of Function
The following type of function is one of the most basic in
algebra.
The graph of f in the preceding definition is the graph of
y = ax + b, which, by the slope-intercept form, is a line with
slope a and y-intercept b.
Thus, the graph of a linear function is a line.
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Definition of Function
Since f(x) exists for every x, the domain of f is
As illustrated in the next example, if a ≠ 0 then the range of
f is also
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Example 6 – Sketching the graph of a linear function
Let f(x) = 2x + 3.
(a) Sketch the graph of f.
(b) Find the domain and range of f.
(c) Determine where f is increasing or is decreasing.
Solution:
(a) Since f(x) has the form ax + b, with a = 2 and b = 3, f is
a linear function.
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Example 6 – Solution
cont’d
The graph of y = 2x + 3 is the line with slope 2 and
y-intercept 3, illustrated in Figure 10.
Figure 10
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Example 6 – Solution
cont’d
(b) We see from the graph that x and y may be any real
numbers, so both the domain and the range of f are
(c) Since the slope a is positive, the graph of f rises as x
increases; that is, f(x1) < f(x2) whenever x1 < x2.
Thus, f is increasing throughout its domain.
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Definition of Function
Many formulas that occur in mathematics and the sciences
determine functions.
For instance, the formula A = r2 for the area A of a circle of
radius r assigns to each positive real number r exactly one
value of A.
This determines a function f such that f(r) = r2, and we
may write A = f(r).
The letter r, which represents an arbitrary number from the
domain of f, is called an independent variable.
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Definition of Function
The letter A, which represents a number from the range of
f, is a dependent variable, since its value depends on the
number assigned to r.
If two variables r and A are related in this manner, we say
that A is a function of r.
In applications, the independent variable and dependent
variable are sometimes referred to as the input variable
and output variable, respectively.
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Definition of Function
As another example, if an automobile travels at a uniform
rate of 50 mi/hr, then the distance d (miles) traveled in time
t (hours) is given by d = 50t, and hence the distance d is a
function of time t.
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Example 8 – Expressing the volume of a tank as a function of its radius
A steel storage tank for propane gas is to be constructed in
the shape of a right circular cylinder of altitude 10 feet with
a hemisphere attached to each end. The radius r is yet to
be determined. Express the volume V (in ft3) of the tank as
a function of r (in feet).
Solution:
The tank is illustrated in Figure 12.
Figure 12
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Example 8 – Solution
cont’d
We may find the volume of the cylindrical part of the tank
by multiplying the altitude 10 by the area r 2 of the base of
the cylinder.
This gives us
volume of cylinder = 10(r 2) = 10r 2.
The two hemispherical ends, taken together, form a sphere
of radius r.
Using the formula for the volume of a sphere, we obtain
volume of the two ends =
r 3.
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Example 8 – Solution
cont’d
Thus, the volume V of the tank is
V=
+ 10r2
This formula expresses V as a function of r.
In factored form,
V(r) =
(4r + 30)
=
(2r + 15)
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Definition of Function
Ordered pairs can be used to obtain an alternative
approach to functions.
We first observe that a function f from D to E determines
the following set W of ordered pairs:
W = {(x, f(x)): x is in D}
Thus, W consists of all ordered pairs such that the first
number x is in D and the second number is the function
value f(x).
In Example 1, where f(x) = x2, W is the set of all ordered
pairs of the form (x, x2).
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Definition of Function
It is important to note that, for each x, there is exactly one
ordered pair (x, y) in W having x in the first position.
Conversely, if we begin with a set W of ordered pairs such
that each x in D appears exactly once in the first position of
an ordered pair, then W determines a function.
Specifically, for each x in D there is exactly one pair (x, y)
in W, and by letting y correspond to x, we obtain a function
with domain D.
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Definition of Function
The range consists of all real numbers y that appear in the
second position of the ordered pairs.
It follows from the preceding discussion that the next
statement could also be used as a definition of function.
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