chapter 5: relations and functions - mr....

Chapter 5: Relations and Functions Section 5.1

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Chapter 5: Relations and Functions

Section 5.1: Representing Relations

Terminology:

Set:

A collection of distinct objects.

Element:

A set is one object in the set.

Relation:

A relation associates the elements of one set with the elements of another set.

One way to write a set is to list its elements inside braces. For example, we can

write a set of natural numbers from 1 to 5 as: {1, 2, 3, 4, 5}

NOTE: the numbers 1, 2, 3, 4, and 5 are each considered elements.

The order of the elements in a set do not matter.

Representing Relational Data

Data will be representing in a variety of ways:

1. In a table:

Data is arranged in two columns representing two sets in a relation.

2. In words:

A relation can be described in words.

3. Set of Ordered Pairs:

Data can be represented in a set of coordinates, where each coordinate will

represent an element in the set.

4. Arrow Diagrams:

A diagram which uses two ovals, one for the domain and one for the range, and

uses arrows to relate the elements of the first set to the elements of the second.


85

Ex1. Northern communities can be associated with the territories they are located in.

Consider the relation shown in the table.

Community Territory

Hay River NWT

Iqualuit Nunavut

Nanisivic Nunavut

Old Crow Yukon

Whitehourse Yukon

Yellowknife NWT

(a) Describe this relation in words

(b) Represent this relation in a set of ordered pairs

(c) Represent this relation using an arrow diagram


86

Ex2. Animals can be associated with the class they are in.

Animal Class

Ant Insecta

Eagle Aves

Snake Reptilia

Turtle Reptilia

Whale Mammalia

Butterfly Insecta





87

Ex3. Fruit can be classified according to their most common colour(s)

Fruit Colour

Apple Red

Apple Green

Blueberry Blue

Cherry Red

Pear Green

Huckleberry Blue





88

Ex4. Different Breeds of Dogs can be associated with their mean heights. Consider the

relation represented by this graph. Represent the relation:

(a) As a table

(b) As an arrow diagram

(c) As a set of ordered pairs

(d) Describe the relation in words


89

Ex4. Different towns in British Columbia can be associated with the average time, in

hours, it takes to drive to Vancouver. Express this data as:

(a) As a table

(b) As an arrow diagram

(c) As a set of ordered pairs

(d) Describe the relation in words


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Section 5.2: Properties of Functions

Terminology:

Domain:

i. The set of first elements in a relation.

ii. All possible values of the independent variable in a relation (x)

Range:

i. The set of related second elements of a relation.

ii. All possible values of the dependent variable in a relation (y)

Function:

A special type of relation in which each element in the domain is associated with

exactly one element in the range.

Example 1: Determine if the data given in each situation represents a function or not

using an arrow diagram and a set of ordered pairs.

(a) The relation between the number of wheels and the

vehicles they belong to

ARROW DIAGRAM:

ORDERED PAIRS:

Number of Wheels

Type of Vehicle

2 Bicycle

3 Tricycle

1 Unicycle

4 Car

2 Motorcycle


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(a) The relation between vehicles and the number of

wheels they possess:

ARROW DIAGRAM:

ORDERED PAIRS:

Type of Vehicle

Number of Wheels

Bicycle 2

Unicycle 1

Tricycle 3

Motorcycle 2

Car 4


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Identifying Functions

For each relation below, determine if each relation is a function, justify your reasoning,

and identify the domain and range of each.

(a) A relation that associates given shapes with the number of right angles in a shape:

{(𝑟𝑖𝑔ℎ𝑡 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒, 1), (𝑎𝑐𝑢𝑡𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒, 0), (𝑠𝑞𝑢𝑎𝑟𝑒, 4), (𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒, 4), (𝑟𝑒𝑔𝑢𝑙𝑎𝑟 ℎ𝑒𝑥𝑎𝑔𝑜𝑛, 0)}

(b) A relation that associates a number with a prime factor of that number:

{(4,2), (6,2), (6,3), (8,2), (9,3)}

(c)

is the square of

1

4

9

-3

-2

-1

1

2


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(d)

Properties of Tables

Ex. In a workplace, a person’s gross pay, P dollars, often depends on the number of

hours worked, h.

Hence we would say that P is the dependent variable. Since the number of hours

worked, h, does not depend on the gross pay, P, we say that h is the independent

variable.

In a Table this data will be displayed as shown:

Hours Worked, h

Gross Pay, P ($)

1 12

2 24

3 36

4 48

5 60

has this number of days

January

February

March

28

30

31April

Independent Variable Dependent Variable

Domain Values Range Values

The values of the

independent variable are

listed in the first column

of the table of values.

These elements belong to

the domain.

The values of the

dependent variable are

listed in the second

column of the table of

values. These elements

belong to the range.


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Describing Functions

Ex1. The table shows the masses, m grams, of

different number s of identical marbles, n.

(a) Why is this relation also a function?

(b) Identify the independent variable and the dependent variable. Justify your

choice.

(c) Write the domain and range.

Number of Marbles, n

Mass of Marbles, m (g)

1 1.27

2 2.54

3 3.81

4 5.08

5 6.35

6 7.62


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Ex2. The table shows the costs of student bus

tickets, C dollars, for different numbers of

tickets, n.

(a) Why is this relation also a function?

(b) Identify the independent variable and the dependent variable. Justify your

choice.

(c) Write the domain and range.

Number of Tickets, n

Cost, C ($)

1 1.75

2 3.50

3 5.25

4 7.00

5 8.75


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Functions and Function Notation

If a relation is a function it can be written in what is called function notation.

Ex. A arcade game requires a player to pay a quarter, q, per game. The total cost of

playing in dollars is represented by C. This relation can be written as:

𝐶 = 0.25𝑞

Since this relation is a function, it can be written in function notation:

𝐶(𝑞) = 0.25𝑞

This is read as “C of q is equal to 0.25q.” This notation shows that C is the dependent

variable and that C depends on q.

Function notation is used to solve situational questions such as:

1. “What would the cost be for a person who wishes to play 15 games at the arcade?”

2. “How many games could a person play if they have $13?


92

Ex1. The equation 𝑉 = −0.08𝑑 + 50 represents the volume, V litres, of gas remaining in

a vehicle’s tank after travelling d kilometres. The gas tank is not refilled until its empty.

(a) Describe the function. Write the equation in function notation.

(b) Determine the volume of gas in the tank after travelling 600 km.

(c) Determine the number of kilometres that have been travelled if there is 26 L in

the tank.

Ex2. The equation 𝐶 = 25𝑛 + 1000 represents the cost, C dollars, for a feast following an

Artic sports competition, when n is the number of people attending.

(a) Describe the function. Write the equation in function notation.

(b) Determine the cost when 100 people attend the competition.

(c) Determine how many people attend the competition if the cost is $5000.


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Section 5.3: Interpreting and Sketching Graphs

Interpreting Graphs

Ex1. Each point on this graph represents a bag of popping corn. Explain the answer to

each question.

(a) Which bag is the most expensive?

(b) Which bag is the least expensive?

(c) Which bags have the same mass? What is that mass?

(d) Which bags cost the same? What is that cost?

(e) Which of bags C or D has the better value for their cost?


94

Ex2. Each point on this graph represents a person. Explain your answer to each question

below.

(a) Which person is the oldest? What is their age?

(b) Which person is the youngest? What is their age?

(c) Which two people have the same height? What is that height?

(d) Which two people have the same age? How old are they?

(e) Which person, B or C, is taller for her or his age?


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Describing a Possible Situation for a Graph

Ex1. Describe the journey for each segment of the graph.


96

Ex2. This graph represents a day trip from Athabasca to Kikino in Alberta, a distance of

approximately 140 km. Describe the journey for each segment of the graph.


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Sketching a Graph for a Given Situation

Ex1. Samuel went on bike ride. He accelerated until he reached a speed of 20 km/h, then

he cycled for 30 minutes at approximately 20 km/h. Samuel arrived at the bottom of a

hill, and his speed decreased to approximately 5 km/h for 10 minutes as he cycled up the

hill. He stopped at the top of the hill for 10 minutes.

Sketch a graph of speed as a function of time. Label each section of the graph and

explain what it represents.


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Ex2. At the beginning of a race, Alicia took 2s to reach a speed of 8 m/s. She ran at

approximately 8 m/s for 12s, then slowed down to a stop in 2 seconds.

Sketch a graph of the speed as a function of time. Label each section of the graph and

explain what it represents.


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Section 5.5: Graphs of Relations and Functions

Determining if a Relation is a Function

It is possible to determine if a relation is a function looking at its graph using a simple

test known as the Vertical line test.

Vertical Line Test for a Function

A graph represents a function when no two points on the graph lie on the same

vertical line.

Place a ruler or straight edge vertically on a graph, then slide the ruler across the

graph. If one edge of the implement always intersects the graph at no more than

one point, then the graph represents a function.

CASE 1: IS A FUNCTION

A relation is not a function when it has two or more

ordered pairs with the same first coordinate

(ie domain value). So, when the order pairs of the

relation are plotted on a grid, a vertical line can

be drawn to pass through more than one point.

CASE 2: IS NOT A FUNCTION

A function has ordered pairs with different first

coordinates. So, when the ordered pairs of the

function are plotted on a grid, any vertical line

drawn will only pass through no more than one

point.


100

Identifying Whether a Graph Represents a Function

Which of these graphs represent a function? How do you know?

(a) (b)

(c) (d)


101

Determining Domain and Range

Domain and Range can be recorded in several ways. We will focus on two different ways.

1. Set Notation A formal mathematical way to give values for domain and range:

{ } is the type of brackets used for a set.

∈ means “is an element of”, “is a member of”, or “belongs to”.

| means “such that”

Recall the symbols used for the number sets are

R for real numbers

Q for rational numbers

Q̅ for irrational numbers

I for integers

N for natural numbers

W for whole numbers

Ex. If a graph had a domain that represented all real numbers between 5 and 12,

the domain would be written as {𝑥 | 5 ≤ 𝑥 ≤ 12, 𝑥 ∈ 𝑅}

Ex. If a range would be represented by all integers less than -2, its range would be

written as {𝑦|𝑦 < −2, 𝑦 ∈ 𝐼}

2. Interval Notation: Brackets are used to indicate an interval of the set of real numbers.

] is the type of bracket used if the end number is included.

) is the type of bracket used if the end number is not included.

is the infinity symbol; used if there is no end point.

Ex. If a graph had a domain that represented all real numbers between 5 and 12,

the domain would be written as [5,12]

Ex. If a range would be represented by all integers less than -2, its range would be

written as (−∞, −2)


102

Ex. Determine the domain and range for each function.

(a) (b)

(c) (d)


103

(e) (f)

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 4

- 3

- 2

- 1

1

2

3

4

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

(g) (h)

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5


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(i)

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

Determining the Domain and Range Values from the Graph of a Function

Ex1. Here is the graph of the function 𝑓(𝑥) = −3𝑥 + 7.

(a) Determine the range value when the domain is -2.

(b) Determine the range value when the domain is 0

(c) Determine the domain value when the range is 4

(d) Determine the domain value when the range is 16


105

Ex2. Here is the graph of the function 𝑔(𝑥) = 4𝑥 − 3

(a) Determine the range value when the domain is 3.

(b) Determine the domain value when the range is -7.


106

Section 5.6: Properties of Linear Functions

Terminology:

Linear Function:

Any graph of a straight line that is not a completely vertical line, represents a

function. We call these functions LINEAR FUNCTIONS. In a linear function,

there the change between consecutive x-values is constant and the change

between consecutive y-values is also constant.

Rate of Change: The change that occurs in the dependant variable divided by the change that occurs in the independent variable between two points on a graph.

𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 =𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡ℎ𝑒 𝐷𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒

𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡ℎ𝑒 𝐼𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒

Determining Rate of Change From a Graph

Ex// Determine the rate of change for the car rental cost in the graph below

𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 =𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡ℎ𝑒 𝐷𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒

𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡ℎ𝑒 𝐼𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑉𝑎𝑡𝑟𝑖𝑎𝑏𝑙𝑒=

$20

100𝑘𝑚= $0.20/𝑘𝑚

The rate of change is $0.20/km; that is for each additional 1 km driven, the rental cost increases by 20¢. The rate of change is constant for a linear relation. NOTE: The unit for the rate of change is the unit of the dependent variable over the unit of the independent variable (here the unit was $/𝑘𝑚)

Note: To determine the Rate of Change,

you choose two points on a graph (any two

points) and divide the change in the dependent

variable (how much the graph rises between

the two points) by the change in the independent

variable (the amount the graph changes

horizontally between the points). We usually

express rate of change in terms of a simplified

fraction or a terminating decimal wherever

possible.


107

Ex2. Determine the rate of change for each of the following

(a) (b)

Determining Rate of Change From a Set of Ordered Pairs

Before we can determine the rate of change for a set of ordered pairs, we must first

determine if the ordered pairs represent a Linear Function. This can be done by

examining the difference in consecutive Domain values and the difference in

consecutive Range values. If the change in domain values and the change in range

values are both constant, than the ordered pairs represent a Linear relation.

If it is a linear relation, the Rate of Change can he determined by dividing the change in

Range values by the change in Domain values.

Ex. Determine if the set of ordered pairs represents a linear relation. If it is, than

determine the rate of change.

{(0, 60) , (100 ,80) , (200 ,100), (300, 120), (400,140)}

Check the change in the domain and range values first:

{(0, 60) , (100 ,80) , (200 ,100), (300, 120), (400,140)}

Since the change in the domain values is constantly 100 and the change in the range

values is constantly 20, this is a linear function.

5 10 15 20

5

10

15

20

Internet Usage (GB)

Co

st

($

)

Cell Phone Internet Charges

2 4 6 8 10 12 14 16 18 20

20

40

60

80

100

Time (h)

Di

st

an

ce

(k

m)

Jake's Bike Ride

+100 +100 +100 +100

+20 +20 +20 +20


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Therefore we can calculate the rate of change:

𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 =𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑅𝑎𝑛𝑔𝑒 𝑉𝑎𝑙𝑢𝑒𝑠

𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐷𝑜𝑚𝑎𝑖𝑛 𝑉𝑎𝑙𝑢𝑒𝑠

=20

100

=1

5 𝑜𝑟 0.20

𝐍𝐍𝐍𝐍: 𝐍𝐍 𝐍𝐍𝐍 𝐍𝐍𝐍 𝐍𝐍 𝐍𝐍𝐍𝐍𝐍𝐍𝐍 𝐍𝐍𝐍𝐍𝐍 𝐍𝐍 𝐍𝐍𝐍 𝐍 𝐍𝐍𝐍𝐍𝐍𝐍 𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍,𝐍𝐍 𝐍𝐍𝐍𝐍𝐍𝐍 𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍 𝐍𝐍𝐍 𝐍𝐍𝐍𝐍 𝐍𝐍 𝐍𝐍𝐍𝐍𝐍𝐍.

Ex. Determine if the set of ordered pairs represents a linear relation. If it is, than

determine the rate of change

(a) {(4, 7), (8, 16), (12, 25), (16, 34), (20, 43)}

(b) {(10, 100), (18, 90), (25, 80), (31, 70), (36, 60)}

(c) {(9, 30), (7, 33), (5, 36), (3, 39), (1, 42)}


109

Determining Rate of Change From a Table of Values

This is done in exactly the same way that we deal with sets of ordered pairs. You must

first analyze the change in domain and range values to determine if the table represents

a linear function. Like before, if it is a constant change, than it is a linear function. Then,

if it is linear, we use the changes in domain and range to determine the rate of change.

Ex. Determine if the table represents a linear function. If it does, calculate its rate of

change.

Ex. Determine if each table represents a linear function. Identify its Rate of Change:

Time (s) Height (m)

0 100

5 88

10 76

15 64

20 52

25 40

Number of Movies Rented Cost ($)

0 7.50

2 15.00

4 22.50

6 30.00

8 37.50

10 45.00

Distance (km)

Cost ($)

0 10

100 60

200 110

300 160

400 210

500 260

Time (h) Cost ($)

0 7

2 15

4 22

6 29

8 37

10 44

+2

+2

+2

+2 +2

+2

+7.50

+7.50

+7.50

+7.50

+7.50

Since there is a constant change in both the

independent and dependent variables, this is a

Linear Function.

The Rate of Change is:

𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 =𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐷𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡

𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐼𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡

=$7.50

2 𝑚𝑜𝑣𝑖𝑒𝑠

= $3.75/𝑚𝑜𝑣𝑖𝑒𝑠

The rate of change is $3.75 per movie


110

Section 5.6: Properties of Linear Functions

Terminology:

Horizontal Intercept:

The point where a graph intersects the horizontal axis (or x-axis). This

point of intersection occurs when the dependent variable (y) equals zero.

Vertical Intercept:

The point where a graph intersects the vertical axis (or y-axis). This

point of intersection occurs when the independent variable (x) equals zero.

Determining Information from a Graph

Ex1. The graphs below shows the temperature, T, in degrees Celsius, as a function of

time, t, in hours, for two locations.


Horizontal axis has coordinates ________

Horizontal intercept = __________.

The horizontal intercept represents________________________________.

Vertical Intercept:

Vertical axis has coordinates ________

Vertical intercept = __________.

The vertical intercept represents________________________________.

Domain:

There is a minimum time value of 0 hours and a maximum value of 12 hours:

DOMAIN:

Range:

There is a minimum temperature shown as -5oC and a maximum temp of 10oC:

RANGE :


111

Rate of Change:

Ex2. The graphs below shows the temperature, T, in degrees Celsius, as a function of

time, t, in hours, for two locations.


Horizontal axis has coordinates ________

Horizontal intercept = __________.

The horizontal intercept represents________________________________.

Vertical Intercept:

Vertical axis has coordinates ________

Vertical intercept = __________.

The vertical intercept represents________________________________.

Domain:

There is a minimum time value of 0 hours and a maximum value of 12 hours:

DOMAIN:

Range:

There is a minimum temperature shown as -5oC and a maximum temp of 10oC:

RANGE :

Rate of Change:


112

Matching a Graph to its Rate of Change and Vertical Intercept

Ex1. Which of the following has a rate of change of 1

2 and a vertical intercept of 6? Justify

your answer.

GRAPH A

GRAPH B


113

Solving a Problem involving Linear Functions:

Ex. This graph shows the cost of publishing a school yearbook for College Louis-Riel in

Winnipeg

Using the graph answer the following:

(a) The budget for publishing costs is $4200. What is the maximum number of

yearbooks that can be printed?

(b) If the school needs 125 year books ordered, what will be the total cost for

publishing?


114

Ex2. This graph shows the total cost for a house call by an electrician for up to 6 hours.

(a) The electrician worked for 5 hours, how much did he get paid?

(b) The electrician was paid $190, for how many hours did he work?


115

Sketching a Graph Using Horizontal and Vertical Intercepts

To sketch the graph of a linear function we must determine the Horizontal and Vertical

Intercepts. This is done by plugging in 𝑥 = 0 to determine the value of the y-intercept

and plugging in 𝑓(𝑥) = 0 to determine the x-intercept. Once both are determined, we

plot both and draw a line through the graph that passes through these two points. This

will be the graph of the linear function.

(a) Sketch the graph of the linear function 𝑓(𝑥) = −2𝑥 + 7

(b) Sketch a graph of the linear function 𝑓(𝑥) = 4𝑥 − 3

x- 10 - 5 5 10

y

- 10

- 5

5

10

x- 10 - 5 5 10

y

- 10

- 5

5

10


116

(c) Sketch the graph of the linear function 𝑓(𝑥) = −1

2𝑥 + 4

(d) Sketch the graph of the linear function 𝑓(𝑥) =2

3𝑥 − 2

x- 10 - 5 5 10

y

- 10

- 5

5

10

x- 10 - 5 5 10

y

- 10

- 5

5

10

chapter 5: relations and functions - mr....

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