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3 Functions, Function Notation, and the Graph of a Function

# 3 Functions, Function Notation, and the Graph of a Function

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cob19545_ch01_117-133.qxd

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College Algebra G&M—

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CHAPTER 1 Relations, Functions, and Graphs

If the relation is pointwise-defined or given as a set of individual and distinct plotted points, we need only check that no two points have the same first coordinate with a

different second coordinate. This gives rise to an alternative definition for a function.

Functions (Alternate Definition)

A function is a set of ordered pairs (x, y), in which each first component

is paired with only one second component.

EXAMPLE 2

Identifying Functions

Two relations named f and g are given; f is pointwise-defined (stated as a set of

ordered pairs), while g is given as a set of plotted points. Determine whether each

is a function.

f: 1Ϫ3, 02, 11, 42, 12, Ϫ52, 14, 22, 1Ϫ3, Ϫ22, 13, 62, 10, Ϫ12, (4, Ϫ5), and (6, 1)

Solution

The relation f is not a function, since Ϫ3 is paired

with two different outputs: 1Ϫ3, 02 and 1Ϫ3, Ϫ22 .

g

5

y

(0, 5)

(Ϫ4, 2)

The relation g shown in the figure is a function.

Each input corresponds to exactly one output,

otherwise one point would be directly above the

other and have the same first coordinate.

(3, 1)

(Ϫ2, 1)

Ϫ5

5

x

(4, Ϫ1)

(Ϫ1, Ϫ3)

Ϫ5

Now try Exercises 11 through 18 ᮣ

The graphs of y ϭ x Ϫ 1 and x ϭ ͿyͿ from Section 1.1 offer additional insight into

the definition of a function. Figure 1.40 shows the line y ϭ x Ϫ 1 with emphasis on the

plotted points (4, 3) and 1Ϫ3, Ϫ42. The vertical movement shown from the x-axis to a

point on the graph illustrates the pairing of a given x-value with one related y-value.

Note the vertical line shows only one related y-value (x ϭ 4 is paired with only y ϭ 3).

Figure 1.41 gives the graph of x ϭ ͿyͿ, highlighting the points (4, 4) and (4, Ϫ4). The

vertical movement shown here branches in two directions, associating one x-value with

more than one y-value. This shows the relation y ϭ x Ϫ 1 is also a function, while the

relation x ϭ ͿyͿ is not.

Figure 1.41

Figure 1.40

5

y yϭxϪ1

y

x ϭ ԽyԽ

(4, 4)

5

(4, 3)

(2, 2)

(0, 0)

Ϫ5

5

x

Ϫ5

5

x

(2, Ϫ2)

(Ϫ3, Ϫ4)

Ϫ5

Ϫ5

(4, Ϫ4)

This “vertical connection” of a location on the x-axis to a point on the graph can

be generalized into a vertical line test for functions.

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Section 1.3 Functions, Function Notation, and the Graph of a Function

Vertical Line Test

A given graph is the graph of a function, if and only if every

vertical line intersects the graph in at most one point.

Applying the test to the graph in Figure 1.40 helps to illustrate that the graph of

any nonvertical line must be the graph of a function, as is the graph of any pointwisedefined relation where no x-coordinate is repeated. Compare the relations f and g from

Example 2.

EXAMPLE 3

Using the Vertical Line Test

Use the vertical line test to determine if any of the relations shown (from Section 1.1)

are functions.

Solution

Visualize a vertical line on each coordinate grid (shown in solid blue), then mentally

shift the line to the left and right as shown in Figures 1.42, 1.43, and 1.44 (dashed

lines). In Figures 1.42 and 1.43, every vertical line intersects the graph only once,

indicating both y ϭ x2 Ϫ 2x and y ϭ 29 Ϫ x2 are functions. In Figure 1.44, a

vertical line intersects the graph twice for any x 7 0 [for instance, both (4, 2) and

14, Ϫ22 are on the graph]. The relation x ϭ y2 is not a function.

Figure 1.42

Figure 1.43

y

(4, 8)

(Ϫ2, 8)

y ϭ x2 Ϫ 2x

5

Figure 1.44

y

y y ϭ ͙9 Ϫ x2

(0, 3)

5

x ϭ y2

(4, 2)

(2, ͙2)

5

(Ϫ1, 3)

(Ϫ3, 0)

(3, 3)

(0, 0)

(3, 0)

Ϫ5

5

x

Ϫ5

5

(2, 0)

(0, 0)

Ϫ5

5

(1, Ϫ1)

Ϫ2

x

Ϫ5

Ϫ5

x

(2, Ϫ͙2)

(4, Ϫ2)

Now try Exercises 19 through 30 ᮣ

EXAMPLE 4

Using the Vertical Line Test

Use a table of values to graph the relations defined by

a. y ϭ ͿxͿ

b. y ϭ 1x,

then use the vertical line test to determine whether each relation is a function.

Solution

WORTHY OF NOTE

For relations and functions, a good

way to view the distinction is to

consider a mail carrier. It is possible

for the carrier to put more than one

letter into the same mailbox (more

than one x going to the same y), but

quite impossible for the carrier to

place the same letter in two different

boxes (one x going to two y’s).

a. For y ϭ ͿxͿ, using input values from x ϭ Ϫ4 to x ϭ 4 produces the following

table and graph (Figure 1.45). Note the result is a V-shaped graph that “opens

upward.” The point (0, 0) of this absolute value graph is called the vertex.

Since any vertical line will intersect the graph in at most one point, this is the

graph of a function.

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CHAPTER 1 Relations, Functions, and Graphs

y ‫ ͦ ؍‬xͦ

x

y ‫ ͦ ؍‬xͦ

Ϫ4

4

Ϫ3

3

Ϫ2

2

Ϫ1

1

0

0

1

1

2

2

3

3

4

4

Figure 1.45

y

5

Ϫ5

5

x

Ϫ5

b. For y ϭ 1x, values less than zero do not produce a real number, so our graph

actually begins at (0, 0) (see Figure 1.46). Completing the table for nonnegative

values produces the graph shown, which appears to rise to the right and remains

in the first quadrant. Since any vertical line will intersect this graph in at most

one place, y ϭ 1x is also a function.

Figure 1.46

y

y ‫ ؍‬1x

x

5

y ‫ ؍‬1x

0

0

1

1

2

12 Ϸ 1.4

3

13 Ϸ 1.7

4

Ϫ5

5

x

2

Ϫ5

A. You’ve just seen how

we can distinguish the graph

of a function from that of a

relation

Now try Exercises 31 through 34 ᮣ

B. The Domain and Range of a Function

Vertical Boundary Lines and the Domain

WORTHY OF NOTE

On a number line, some texts will

use an open dot “º” to mark the

location of an endpoint that is not

included, and a closed dot “•” for

an included endpoint.

In addition to its use as a graphical test for functions, a vertical line can help determine

the domain of a function from its graph. For the graph of y ϭ 1x (Figure 1.46), a vertical line will not intersect the graph until x ϭ 0, and then will intersect the graph for

all values x Ն 0 (showing the function is defined for these values). These vertical

boundary lines indicate the domain is x Ն 0.

Instead of using a simple inequality to write the domain and range, we will often

use (1) a form of set notation, (2) a number line graph, or (3) interval notation. Interval notation is a symbolic way of indicating a selected interval of the real numbers.

When a number acts as the boundary point for an interval (also called an endpoint),

we use a left bracket “[” or a right bracket “]” to indicate inclusion of the endpoint. If

the boundary point is not included, we use a left parenthesis “(” or right parenthesis “).”

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Section 1.3 Functions, Function Notation, and the Graph of a Function

EXAMPLE 5

121

Using Notation to State the Domain and Range

Model the given phrase using the correct inequality symbol. Then state the result in

set notation, graphically, and in interval notation: “The set of real numbers greater

than or equal to 1.”

Solution

WORTHY OF NOTE

Since infinity is really a concept and

not a number, it is never included

(using a bracket) as an endpoint for

an interval.

Let n represent the number: n Ն 1.

• Set notation: 5n |n Ն 16

[

• Graph:

Ϫ2 Ϫ1

0

1

2

3

4

• Interval notation: n ʦ 31, q 2

5

Now try Exercises 35 through 50 ᮣ

The “ʦ” symbol says the number n is an element of the set or interval given. The

“ q ” symbol represents positive infinity and indicates the interval continues forever to

the right. Note that the endpoints of an interval must occur in the same order as on the

number line (smaller value on the left; larger value on the right).

A short summary of other possibilities is given here for any real number x. Many

variations are possible.

Conditions (a Ͻ b)

x is greater than k

x is less than

or equal to k

x is less than b

and greater than a

x is less than b and

greater than or equal to a

x is less than a or

x is greater than b

Set Notation

5x| x 7 k6

5x |x Յ k6

5x |a 6 x 6 b6

5x| a Յ x 6 b6

5x| x 6 a or

x 7 b6

Number Line

Interval Notation

x ʦ 1k, q 2

)

k

x ʦ 1Ϫq, k4

[

k

)

)

a

b

[

)

a

b

x ʦ 1a, b2

x ʦ 3 a, b2

)

)

a

b

x ʦ 1Ϫq, a2 ´ 1b, q2

For the graph of y ϭ ͿxͿ (Figure 1.45), a vertical line will intersect the graph (or its

infinite extension) for all values of x, and the domain is x ʦ 1Ϫq, q2 . Using vertical

lines in this way also affirms the domain of y ϭ x Ϫ 1 (Section 1.1, Figure 1.5) is

x ʦ 1Ϫq, q 2 while the domain of the relation x ϭ ͿyͿ (Section 1.1, Figure 1.6)

is x ʦ 30, q 2 .

Range and Horizontal Boundary Lines

The range of a relation can be found using a horizontal “boundary line,” since it will

associate a value on the y-axis with a point on the graph (if it exists). Simply visualize a

horizontal line and move the line up or down until you determine the graph will always

intersect the line, or will no longer intersect the line. This will give you the boundaries of

the range. Mentally applying this idea to the graph of y ϭ 1x (Figure 1.46) shows the

range is y ʦ 3 0, q 2. Although shaped very differently, a horizontal boundary line shows

the range of y ϭ ͿxͿ (Figure 1.45) is also y ʦ 30, q2.

EXAMPLE 6

Determining the Domain and Range of a Function

Use a table of values to graph the functions defined by

3

a. y ϭ x2

b. y ϭ 1

x

Then use boundary lines to determine the domain and range of each.

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