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2A Set notation 2B Relations and graphs 2C Domain and range 2D Types of relations (including (functions) 2E Function notation and special types of function 2F Circles 2G Functions and modelling

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2

syllabus

syllabus

rref

efer

erence

ence

Topic:

• Introduction to functions

In this

In this

cha

chapter

pter

2A Set notation

2B Relations and graphs

2C Domain and range

2D Types of relations (including

functions)

2E Function notation and

special types of function

2F Circles

2G Functions and modelling

Relations and

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Introduction

Mathematical models are used in a wide variety of contexts.

The amount of erosion varies with wave energy.

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Climate modellers investigate links between deforestation and greenhouse warming.

A medical scientist investigates whether there are links between the incidence of cancer and the presence of microwave radiation.

As mathematicians develop models they use the concept of a function. • Erosion is a function of wave energy.

• Money invested is a function of interest rates. • Greenhouse warming as a function of deforestation.

• The incidence of cancer is a function of microwave radiation.

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Set notation

Set notation is used in mathematics in the same way that symbols are used to represent language statements.

Definitions

1. A set is a collection of things.

2. The symbol {. . .} refers to a set of something.

3. Anything contained in a set; that is, a member of a set, is referred to as an element of the set.

(a) The symbol ∈ means ‘is an element of’, for example, 6 ∈ {2, 4, 6, 8, 10}. (b) The symbol ∉ means ‘is not an element of’, for example, 1 ∉ {2, 4, 6, 8, 10}. 4. A capital letter, for example, A, B or C etc. is often used to refer to a particular set

of things.

5. The symbol ⊂ means ‘is a subset of’, for example, if BA, then all of the elements of set B are contained in set A.

6. The symbol ⊄ means ‘is not a subset (or is not contained in)’.

7. The symbol ∩ means ‘intersection’, for example, AB is the set of elements common to sets A and B.

8. The symbol ∪ means ‘union’, for example, AB is the set of all elements belonging to either set A or B.

9. The symbol A \ B denotes all of the elements of A which are not an element of B. 10. The symbol ∅ means the null set. It implies that there is nothing in the set, or that

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Sets of numbers

Certain letters are ‘reserved’ for important sets that arise frequently in the study of mathematics.

1. R is the set of real numbers; that is, any number you can think of. 2. N is the set of natural numbers; that is, {1, 2, 3, 4, 5, . . .}. 3. J is the set of integers; that is, {. . ., −3, −2, −1, 0, 1, 2, 3, . . .}.

4. Q is the set of rational numbers (that is, numbers which can be expressed as fractions in the form where a and b are integers).

5. Q′ is the set of numbers which are not rational (that is, cannot be expressed as a ratio of two whole numbers). These numbers are called irrational, for example, π,

, etc.

Note that N JQR; that is,

If A= {1, 2, 4, 8, 16, 32}, B= {1, 2} and C= {1, 2, 3, 4}, find: a A∩B b A∪C c A \ B d {3, 4} ∩B

e whether or not: i 8 A ii BA iii CA.

THINK WRITE

a The elements that A and B have in common are 1 and 2.

a AB={1, 2}

b The elements that belong to either A or C

are 1, 2, 3, 4, 8, 16 and 32.

b AC={1, 2, 3, 4, 8, 16, 32}

c The elements of A which are not an element of B are 4, 8, 16 and 32.

c A \ B={4, 8, 16, 32}

d {3, 4} and B have no common elements. d {3, 4} ∩B = ∅

e i 8 is an element of A. e i Yes. 8 ∈A

ii All elements of B belong to A. ii Yes. BA

iii 3 is an element of C but not A. iii No. CA

1

WORKED

E

xample

a b

---3

1

1 – 2 3

4 2 3 7 – 5 33 — 51

2 3 4 0

–1 –2 –3

... ...

...

...

R

Q

J

N

Q'

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Set notation

1 If A = {2, 4, 6, 8, 10, 12, 14}, B = {1, 3, 5, 7, 9, 11, 13}, C = {4, 5, 6, 7} and D = {6, 7, 8}, find:

2 If A = {−3, −2, −1, 0, 1, 2, 3}, B = {0, 1, 2, 3} and C = {−3, 2, 3, 4}, find:

3 If F = {a, e, i, o, u}, G = {a, b, c, d, e, f, g, h, i} and H = {b, c, d, f, g, h}, find:

4

Given that A ⊂ B, then A ∩ B is equivalent to:

5

Given that C ⊂ B ⊂ A, then it follows that

a A ∪ B ∪ C is equivalent to:

b (A \ B) ∩ C is equivalent to:

6 Answer true (T) or false (F) to each of the following statements relating to the number sets N, J, Q and R.

a A ∩ B b A ∩ C c A ∩ C ∩ D

d A ∪ B e C ∪ D f A \ C

g C \ D.

a A ∩ B ∩ C b A \ B c A \ (B ∪ C)

d A \ (B ∩ C) e A ∪ C.

a F ∩ G ∩ H b G ∩ H c G \ H

d H \ F e (F ∪ H) \ G.

A B BC {1, 2} D A ∪ B E A

A B B C C A D A ∪ B E B ∪ C

A B BC C D A ∩ B E B \ C

a ∈ R b −4 ∈ N c 6.4217 ∈ Q

d ∈ Q e 1.5 ∈ J f {5, 10, 15, 20} ⊂ J

g {5, 10, 15, 20} ⊂ N h J \ N = {. . ., −3, −2, −1} i J ∩ N = N

j Q ⊂ N k Q ∩ J = ∅ l (J ∪ Q) ⊂ R

remember

1. {. . .} refers to a set of something.

2. ∈ means ‘is an element of’.

3. ∉ means ‘is not an element of’.

4. ⊂ means ‘is a subset of’.

5. ⊄ means ‘is not a subset (or is not contained in)’.

6. ∩ means ‘intersection with’.

7. ∪ means ‘union with’.

8. \ means ‘excluding’.

9. ∅ refers to ‘the null, or empty set’.

remember

2A

W WORKEDORKED E Example

1

m

multiple choiceultiple choice

m

multiple choiceultiple choice

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Relations and graphs

A relation is a set of ‘ordered pairs’ of values or ‘variables’. Consider the following. The cost of hiring a boat depends on the number of hours for which it is hired. We can say that a relation exists between the number of hours and the cost. The table below outlines the relation.

Since the cost depends upon the number of hours, the cost is said to be the dependent variable, while the number of hours is called the independent variable. The information in the table can be represented by a graph, which usually gives a better indication of how two variables are related. When graphing a relation, the independent

variable is displayed on the horizontal (or x) axis and the dependent variable is

displayed on the vertical (or y) axis. So we can plot the set of points {(3, 50), (4, 60), (5, 70), (6, 80), (7, 90), (8, 100)}. The points are called (x, y) ordered pairs, where x is the first element and y is the second element.

This graph clearly shows that the cost increases as the number of hours of hire increases. The relation appears to be linear. That is, a straight line could be drawn that passes through every point. However, the dots are not joined as the relation involves ‘integer-valued’ numbers of hours and not minutes or seconds. The number of hours can be referred to as a discrete independent variable.

Discrete variables include names and numbers of things; that is, things that can be counted (values are natural numbers or integers).

Some variables are referred to as continuous variables. Continuous variablesinclude

height, weight and volume; that is, things that can be measured (values are real numbers). If a relationship exists between the variables we may try to find a rule and then write this rule in mathematical terms. In our example, the relationship appears to be that for each extra hour of hire the cost increases by $10 after an initial cost of $20.

Cost = 10 × number of hours + 20 Using x and y terms, this is written as

y= 10x+ 20

Number of hours of hire 3 4 5 6 7 8

Cost ($) 50 60 70 80 90 100

Cost of boat hire ($) 0 1 2 3 4 5 6 7 8

50 40 60 70 80 90 100

Number of hours y

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Sketch the graph by plotting selected x-values for the following relations and state whether each is discrete or continuous.

a y=x2, where x {1, 2, 3, 4} b y= 2x+ 1, where xR

THINK WRITE

a Use the rule to calculate y and state the ordered pairs by letting x = 1, 2, 3 and 4.

a Whenx= 1, y = 12

= 1 (1, 1)

x= 2, y = 22

= 4 (2, 4)

x= 3, y = 32

= 9 (3, 9)

x= 4, y = 42

= 16 (4, 16) Plot the points (1, 1), (2, 4), (3, 9) and

(4, 16) on a set of axes.

Do not join the points as x is a discrete variable (whole numbers only).

It is a discrete relation as x can be only whole number values.

b Use the rule to calculate y. Select values of x, say x = 0, 1 and 2 (or find the intercepts). State the ordered pairs.

b Whenx= 0, y = 2(0) + 1

= 1 (0, 1)

x= 1, y = 2(1) + 1

= 3 (1, 3)

x= 2, y = 2(2) + 1

= 5 (2, 5) Plot the points (0, 1), (1, 3) and (2, 5)

on a set of axes.

Join the points with a straight line, continuing in both directions as x is a continuous variable (any real number).

It is a continuous relation as x can be any real number.

1

2 y

x

0 1 2 3 4 1

4 8 12 16

3

1

2 y

x

0

–2 –1 1 2 1

2 3 4 5

–3 –2 –1

y = 2x + 1

3

2

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remember

1. The independent variable (for example, x) is shown on the horizontal axis of a graph.

2. The dependent variable (for example, y) is shown on the vertical axis of a graph.

3. Discrete variables are things which can be counted. Graph points are not joined.

4. Continuous variables are things which can be measured. Graph points may be joined.

remember

The pulse rate of an athlete, R beats per

minute, t minutes after the athlete finishes a workout, is shown in the table below.

a Plot the points on a graph.

b Estimate the athlete’s pulse rate

after 3 minutes.

t 0 2 4 6 8

R 180 150 100 80 70

THINK WRITE

a Draw a set of axes with t on the horizontal axis and R on the vertical axis because heart rate is dependent on the time.

Plot the points given in the table.

b Join the points with a smooth curve since t (time) is a continuous variable.

b

Construct a vertical line up from t= 3 until it touches the curve.

From this point draw a horizontal line back to the vertical axis.

Estimate the value of R where this line touches the axis.

When t= 3, the pulse rate is approximately 125 beats per minute.

1

2

1

0 12 34 56 78 20

80 60 40 100 120 140 160 180

t (min)

R

(beats/min)

2

3

4

3

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Relations and graphs

Questions 1, 2, and 3 refer to the following information.

A particular relation is described by the following ordered pairs: {(0, 4), (1, 3), (2, 2), (3, 1)}.

1

The graph of this relation is represented by:

2

The elements of the dependent variable are:

3

The rule for the relation is correctly described by:

4

During one week, the number of people travelling on a particular train, at a certain time, progressively increases from Monday through to Friday. Which graph below best represents this information?

A B C

D E

A {1, 2, 3, 4} B {1, 2, 3} C {0, 1, 2, 3, 4}

D {0, 1, 2, 3} E {1, 2}

A y = 4 − x, x ∈ R B y = x − 4, x ∈ N C y = 4 − x, x ∈ N

D y = x − 4, x ∈ J E y = 4 − x, x ∈ {0, 1, 2, 3}

A B C

2B

m

multiple choiceultiple choice

y

x

0 1 2 3 4 1

2 3 4

y

x

0 1 2 3 4 1

2 3 4

y

x

0 1 2 3 4 1

2 3 4

y

x

0 1 2 3 4 1

2 3 4

y

x

0 1 2 3 4 1

2 3 4

m

multiple choiceultiple choice

m

multiple choiceultiple choice

m

multiple choiceultiple choice

0 M T W T F

Number of people

0 M T W T F

Number of people

0 M T W T F

(11)

5 State whether each of the following relations has discrete (D) or continuous (C) variables.

a {(–4, 4), (–3, 2), (–2, 0), (–1, –2), (0, 0), (1, 2), (2, 4)}

b The relation which shows the air pressure at any time of the day.

c d

e The relation which shows the number of student absences per day during term 3 at your school.

f The relation describing the weight of a child from age 3 months to one year.

6 Sketch the graph representing each of the following relations, and state whether each is discrete or continuous.

a

b {(0, 0), (1, 1), (2, 4), (3, 9)}

c y =−x2, where x ∈ {−2, −1, 0, 1, 2}

d y = x − 2, where x ∈ R

e y = 2x + 3, where x ∈ J

f y = x2+ 2, where −2 ≤ x ≤ 2 and x ∈ R

7 The table at right shows the temperature of a cup of coffee, T°C, t minutes after it is poured.

a Plot the points on a graph.

b Join the points with a smooth curve.

c Explain why this can be done.

d Use the graph to determine how long it takes the coffee to reach half of its initial temperature.

8 A salesperson in a computer store is paid a base salary of $300 per week plus $40 commission for each computer she sells. If n is the number of computers she sells per week and P dollars is the total amount she earns per week, then:

a copy and complete the table on the next page.

D E

Day Mon Tues Wed Thur Fri Sat Sun

Cost of petrol (c/L) 68 67.1 66.5 64.9 67 68.5 70

0 M T W T F

Number of people

0 M T W T F

Number of people

y

x

0

y

x

0

EXCEL Spreadshe

et Plotting relations W

WORKEDORKED E Example

2

W WORKEDORKED E Example

3 t (min) 0 2 4 6 8

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b plot the information on a graph.

c explain why the points cannot be joined together.

9 The speed of an aircraft, V km/h, t seconds after it starts to accelerate down the runway, is shown in the following table.

a Plot a graph which represents the information shown in the table.

b Use the graph to estimate the speed after: i 2.5 s ii 4.8 s.

10 The cost, C dollars, of taking n students on an excursion to the zoo is $50 plus $6 per student.

a Complete a table using 15 ≤ n ≤ 25.

b Plot these points on a graph.

c Explain why the dots can or cannot be joined.

To plot points rather than a continuous graph based on a known formula, follow these steps:

n 0 1 2 3 4 5 6

P

t 0 1 2 3 4 5

V 0 30 80 150 240 350

Graphics Calculator

Graphics Calculator

tip!

tip!

Plotting points

CASIO

Plotting

points 1. Press , EDIT and 1:Editselect and

enter the x-coordinates of the points to be plotted in L1. Enter the y-coordinates in L2 as shown below. Delete any values in the table that are not required (such as those from pre-vious calculations). (Note: To clear a com-plete list, scroll to the list heading and press followed by .

2. Press [STAT

PLOT], select 1:Plot 1

by pressing and ensure the options are selected as shown in the screen below:

3. Adjust WINDOW set-tings or ZOOM for a suitable view of the plotted points. Press

[FORMAT] and

select GridOn and

LabelOn with ,

followed by . Press to check coordinates.

STAT

CLEAR ENTER

2nd

ENTER

2nd

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Domain and range

Domain and range

A relation may be described by: 1. a listed set of ordered pairs 2. a graph or

3. a rule.

The set of all first elements of a set of ordered pairs is known as the domain and the set of all second elements of a set of ordered pairs is known as the range. Alter-natively, the domain is the set of independent values and the range is the set of dependent values.

If a relation is described by a rule, it should also specify the domain. For example: 1. the relation {(x, y): y = 2x, x ∈ {1, 2, 3}} describes the set of ordered pairs {(1, 2),

(2, 4), (3, 6)}

2. the domain is the set X = {1, 2, 3}, which is given

3. the range is the set Y = {2, 4, 6}, and can be found by applying the rule y = 2x to the domain values.

If the domain of a relation is not specifically stated, it is assumed to consist of all real numbers for which the rule has meaning. This is referred to as theimplied domain of a relation. For example:

1. {(x, y): y = x3} has the implied domain R.

2. {(x, y): y = } has the implied domain x ≥ 0, where x ∈ R.

Interval notation

If a and b are real numbers and a < b, then the following intervals are defined with an accompanying number line:

The closed circle indicates that the number is included and the open circle indicates that the number is not included.

(a, b) implies a < x < b or (a, b] implies a < x ≤ b or

(a, ∞) implies x > a or [a, ∞) implies x ≥ a or

(−∞, b) implies x < b or (−∞, b] implies x ≤ b or

[a, b) implies a ≤ x < b or [a, b] implies a ≤ x < b or x

x

a b a b x

x

a a x

x

b b x

x

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Describe each of the following subsets of the real numbers using interval notation.

a b c

THINK WRITE

a The interval is x < 2 (2 is not included). a (−∞, 2)

b The interval is −3 ≤ x < 5 (3 is included). b [−3, 5)

c The interval is both 1 ≤ x < 3 and x ≥ 5 (1 is included, 3 is not).

c [1, 3) ∪ [5, ∞) x

–4 0 2 –3 0 5 x 0 1 3 5 x

4

WORKED

E

xample

Illustrate the following number intervals on a number line.

a (2, 10] b [1, )

THINK WRITE

a The interval is −2 < x ≤ 10 (−2 is not included, 10 is).

a

b The interval is x ≥ 1 (1 is included). b

x

10 0

–2

x

0 1

5

WORKED

E

xample

State the domain and range of each of the following relations.

a {(1, 2), (2, 5), (3, 8), (4, 11)}

b

c d

Mass (kg) 10 15 20 25 30

Cost per kg ($) 3.5 3.2 3.0 2.8 2.7 y

x

0

y

x

0

–4 4

–4 4

6

(15)

THINK WRITE

a The domain is the set of first elements of the ordered pairs.

a Domain = {1, 2, 3, 4}

The range is the set of second elements of the ordered pairs.

Range = {2, 5, 8, 11}

b The domain is the set of independent values in the table, that is, the mass values.

b Domain = {10, 15, 20, 25, 30}

The range is the set of dependent values in the table, that is, the cost values.

Range = {2.7, 2.8, 3.0, 3.2, 3.5}

c The domain is the set of values that the graph covers horizontally.

c Domain =R

The range is the set of values that the graph covers vertically.

Range = [0, ∞)

d The domain is the set of values that the graph covers horizontally.

d Domain = [−4, 4]

The range is the set of values that the graph covers vertically.

Range = [−4, 4] 1

2

1

2

1

2

1

2

For each relation given, sketch its graph and state the domain and range using interval notation.

a {(x, y): y= } b {(x, y): y=x2 4, x [0, 4]}

Continued over page

THINK WRITE

a The rule has meaning for x ≥ 1 because if x < 1, y = .

a

Therefore, calculate the value of y when x = 1, 2, 3, 4 and 5, and state the coordinate points.

Whenx= 1, y =

= 0 (1, 0).

x= 2, y =

= 1 (2, 1)

x= 3, y = (3, )

x= 4, y = (4, )

x= 5, y =

= 2 (5, 2) Plot the points on a set of axes.

Join the points with a smooth curve starting from x = 1, extending it beyond the last point. Since no domain is given we can assume x ∈ R (continuous). Place a closed circle on the point (1, 0) and put an arrow on the other end of the curve.

x1

1

negative number

2 0

1

2 2

3 3

4

3 y

x

0 1 2 3 4 5 1

–1 2

y = x – 1

4

5

7

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THINK WRITE

The domain is the set of values covered horizontally by the graph, or implied by the rule.

Domain = [1, ∞)

The range is the set of values covered vertically by the graph.

Range = [0, ∞)

b Calculate the value of y when x = 0, 1, 2, 3 and 4, say, as the domain is [0, 4]. State the coordinate points.

b Whenx= 0, y = 02− 4

= −4 (0, −4)

x= 1, y = 12− 4

= −3 (1, −3)

x= 2, y = 22− 4

= 0 (2, 0)

x= 3, y = 32− 4

= 5 (3, 5)

x= 4, y = 42− 4

= 12 (4, 12) Plot these points on a set of axes.

Join the dots with a smooth curve from x = 0 to x = 4.

Place a closed circle on the points (0, −4) and (4, 12).

The domain is the set of values covered by the graph horizontally.

Domain = [0, 4]

The range is the set of values covered by the graph vertically.

Range = [−4, 12]

Verify that the graphs are correct using a graphics calculator. 6

7

1

2 y

x

0 1 2 3 4 2

–2 –4 6 4 8 10 12 y = x

2 – 4, x [0, 4] 3

4

5

6

remember

1. The domain of a relation is the set of first elements of an ordered pair. 2. The range of a relation is the set of second elements of an ordered pair. 3. The implied domain of a relation is the set of first element values for which a

rule has meaning.

4. In interval notation a square bracket means the end point is included in a set of values, whereas a curved bracket means the end point is not included.

a b

(a, b]

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Domain and range

1 Describe each of the following subsets of the real numbers using interval notation.

2 Illustrate each of the following number intervals on a number line.

a [−6, 2) b (−9, −3) c (−∞, 2]

d [5, ∞) e (1, 10] f (2, 7)

g (−∞, −2) ∪ [1, 3) h [−8, 0) ∪ (2, 6] i R \ [1, 4]

j R \ (−1, 5) k R \ (0, 2] l R \ [−2, 1)

3 Describe each of the following sets using interval notation.

4

Consider the set described by R \ {x: 1 ≤ x < 2}.

a It is written in interval notation as:

b It is represented on a number line as:

5

The domain of the relation graphed at right is:

a b

c d

e f

g h

a {x: −4 ≤ x < 2} b {x: −3 < x ≤ 1} c {y: −1 < y < }

d {y: − < y ≤ } e {x: x > 3} f {x: x ≤ −3}

g R h R+ ∪ {0} i R \ {1}

j R \ {−2} k R \ {x: 2 ≤ x ≤ 3} l R \ {x: −2 < x < 0}

A (−∞, 1) ∪ (2, ∞) B (−∞, 1] ∪ [2, ∞) C (−∞, 1) ∪ (2, ∞]

D (−∞, 1] ∪ (2, ∞) E (−∞, 1) ∪ [2, ∞)

A B

C D

E

A [−4, 4] B (−4, 7) C [−1, 7]

D (−4, 4) E (−1, 7)

2C

W

WORKEDORKED E Example

4

–2 0 1 0 5

4 0

–3 –8 0 9

0

–1 0 1

3 0

–5 –2 –3 0 1 2 4

W WORKEDORKED E Example

5

3

1 2

--- 1 2

---m

multiple choiceultiple choice

2 1

0 0 1 2

2 1

0 0 1 2

2 1 0

m

multiple choiceultiple choice

SkillS

HEET

2.1

y

x

0 3 7

4

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6

The range of the relation {(x, y): y = 2x + 5, x ∈ [–1, 4]} is:

7 State i the domain and ii the range of each of the following relations.

a {(3, 8), (4, 10), (5, 12), (6, 14), (7, 16)}

b {(1.1, 2), (1.3, 1.8), (1.5, 1.6), (1.7, 1.4)}

c

d

e y = 5x − 2, where x is an integer greater than 2 and less than 6.

f y = x2− 1, x ∈ R

8 State the domain and range of each of the following relations. Use a graphics calcu-lator to view more of each graph if required.

9 For each relation given, sketch its graph and state the domain and range using interval notation.

Verify that the graphs are correct with a graphics calculator.

10 State the implied domain for each relation defined by the following rules:

A [7, 13] B [3, 13] C [3, ∞) D R E R \ (7, 13)

Time (min) 3 4 5 6

Distance (m) 110 130 150 170

Day Monday Tuesday Wednesday Thursday Friday

Cost ($) 25 35 30 35 30

a b c

d e f

g h i

a {(x, y): y = 2 − x2} b {(x, y): y = x3+ 1, x ∈ [−2, 2]}

c {(x, y): y = x2+ 3x + 2} d {(x, y): y = x2− 4, x ∈ [−2, 1]}

e {(x, y): y = 2x − 5, x ∈ [−1, 4)} f {(x, y): y = 2x2− x − 6}

a y = 10 − x b y = 3 c y = −

d y = x2+ 3 e y = f y = 10 − 7x2 m

multiple choiceultiple choice

W WORKEDORKED E Example 6a, b W WORKEDORKED E Example 6c, d y x 0 2 –3 y x 0 2 y x 0 2 –2 2 y x 0 1

y = x – 1 y

x 0 4 y x 0 –3 y x 0

y = 1 x y x 0 1 y x 0 –2 W WORKEDORKED E Example 7 Wor

kSHEET

2.1

x 16x2

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---Types of relations (including functions)

One-to-one relations

A one-to-one relation exists if for any x-value there is only one corresponding y-value and vice versa. For example:

1. {(1, 1), (2, 2), (3, 3), (4, 4)} 2.

One-to-many relations

A one-to-many relation exists if for any x-value there is more than one y-value, but for any y-value there is only one x-value. For example:

1. {(1, 1), (1, 2), (2, 3), (3, 4)} 2.

Many-to-one relations

A many-to-one relation exists if there is more than one x-value for any y-value but for any x-value there is only one y-value. For example:

1. {(−1, 1), (0, 1), (1, 2)} 2.

Interesting relations

This investigation deals with graphs of different relations and will require the use of graphing software such as GrafEq, DERIVE™ or Graphmatica to produce quick, accurate graphs. A demonstration version of GrafEq is on your Maths Quest CD-ROM.

Use a program such as GrafEq to produce a graph of each of the following. Sketch each graph into your workbook, and label each with its equation.

1 x2 + 2y2 = 9

2 x3 + y3 = 1

3 sin (x2 + y2) = 1

4 x2 – y2 = 1

5 7x2 – 6 xy + 13y2 = 16

6 x4 = x2– y2

7 x2 + y2 < 25

8 x2 + y2 > 25

9 9 < x2 + y2 < 36

10 x sin x + y sin y < 1 The graph of y2(1 − x) = x2(x + 1) produced by GrafEq. 3

y

x

1 –1

–1 1

y

x

0

y

x

0

y

x

(20)

Many-to-many relations

A many-to-many relation exists if there is more than one x-value for any y-value and vice versa. For example:

1. {(0, −1), (0, 1), (1, 0), (−1, 0)} 2. y

x

0

y

x

0

What type of relation does each graph represent?

a b c

THINK WRITE

a For some x-values there is more than one y-value. A line through some x-values shows that 2 y-values are available:

a One-to-many relation.

For any y-value there is only one x-value. A line through any y-value shows that only one x-value is available:

b For any x-value there is only one y-value. b One-to-one relation. For any y-value there is only one x-value.

c For any x-value there is only one y-value. c Many-to-one relation. For some y-values there is more than one

x-value. y

x

0

y

x

0

y

x

0

1

y

x

0

x = –1

F

2

y

x

0

y = 1

1

2

1

2

8

(21)

Functions

Relations which are one-to-one or many-to-one are called functions. That is, a function is a relation where for any x-value there is only one y-value. For example:

1. 2.

Vertical line test

A function is determined from a graph if a vertical line drawn anywhere on the graph cannot intersect with the curve more than once.

y

x

0

y

x

0

State whether or not each of the following relations are functions.

a {(2, 1), (1, 0), (0, 1), (1, 2)}

b c

THINK WRITE

a For each x-value there is only one y-value. (Or, a plot of the points would pass the vertical line test.)

a Function

b It is possible for a vertical line to intersect with the curve more than once.

b Not a function

c It is not possible for any vertical line to intersect with the curve more than once.

c Function y

x

0

y

x

0

9

WORKED

E

xample

remember

1. A function is a relation which does not repeat the first element in any of its

ordered pairs. That is, for any x-value there is only one y-value (one-to-one or

many-to-one relations.)

2. Vertical line test: The graph of a function cannot be crossed more than once by any vertical line.

y

x

0

y

x

0

remember

(22)

Types of relations

(including functions)

1 What type of relation does each graph represent?

a b c

d e f

g h i

j k l

2 Use the vertical line test to determine which of the relations in question 1 are functions.

3

Which of the following relations is not a function?

A {(5, 8), (6, 9), (7, 9), (8, 10), (9, 12)}

B C y2= x D y = 8x − 3 E

2D

W WORKEDORKED E Example

8

y

x

0

y

x

0

y

x

0

y

x

0

y

x

0

y

x

0

y

x

0

y

x

0

y

x

0

y

x

0

y

x

0

y

x

0

W WORKEDORKED E Example

9

m

multiple choiceultiple choice

y

x

0

y

x

(23)

4

Consider the relation y ≥ x + 1.

a The graph which represents this relation is:

b This relation is:

c The domain and range are respectively:

5 Which of the following relations are functions? State the domain and range for each function.

A B

C D

E Note: The shaded side

indicates the region not required.

A one-to-one

B one-to-many

C many-to-one

D many-to-many

E a function

A R and R+ (R+ stands for positive real numbers.)

B R and R

C R and R− (R− stands for negative real numbers.)

D R+ and R

E R− and R

a {(0, 2), (0, 3), (1, 3), (2, 4), (3, 5)} b {(−3, −2), (−1, −1), (0, 1), (1, 3), (2, −2)}

c {(3, −1), (4, −1), (5, −1), (6, −1)} d {(1, 2), (1, 0), (2, 1), (3, 2), (4, 3)}

e {(x, y): y = 2, x ∈ R} f {(x, y): x = −3, y ∈ J}

g y = 1 − 2x h y > x + 2

i x2+ y2= 25 j y = , x ≥ −1

k y = x3+ x l x = y2+ 1 m

multiple choiceultiple choice

y

x

0 1

–1

y

x

0 1

–1

y

x

0 1

1

y

x

0 1

1

y

x

0 1

–1

(24)

Function notation

Consider the relation y = 2x, which is a function.

The y-values are determined from the x-values, so we say ‘y is a function of x’, which is abbreviated to y = f(x).

So, the rule y = 2x can also be written as f(x) = 2x.

Evaluating functions

For a given function y = f(x), the value of y when x = 1 is written as f(1) or the value of y when x = 5 is written as f(5) etc.

If x = 1, then y= f(1)

= 2 × 1

= 2

If x = 2, then y= f(2)

= 2 × 2

= 4, and so on.

If f(x) =x2− 3, find:

a f(1) b f(2) c f(a) d f(2a).

THINK WRITE

a Write the rule. a f(x)=x2− 3

Substitute x = 1 into the rule. f(1)= 12− 3

Simplify. = 1 − 3

= −2

b Write the rule. b f(x) =x2− 3

Substitute x = −2 into the rule. f(−2)= (−2)2− 3

Simplify. = 4 − 3

= 1

c Write the rule. c f(x)=x2− 3

Substitute x = a into the rule. f(a)=a2− 3

d Write the rule. d f(x) =x2− 3

Substitute x = 2a into the rule. f(2a)= (2a)2− 3

Simplify the expression if possible. = 22a2− 3

= 4a2− 3 1

2

3

1

2

3

1

2

1

2

3

10

(25)

Special types of function

One-to-one functions

As we have already seen, one-to-one relations and many-to-one relations are functions. A one-to-one function has, at most, one y-value for any x-value and vice versa. The graph of a relation is a function if any vertical line crosses the curve at most once. Similarly, a one-to-one function exists if any horizontal line crosses the curve at most once. For example:

A function which is not one-to-one A one-to-one function y

x

0

y

x

0

Which of the following functions are one-to-one?

a {(0, 1), (1, 2), (2, 3), (3, 1)} b {(2, 3), (3, 5), (4, 7)} c f(x) = 3x

THINK WRITE

Check whether each function has, at most, one y-value for any x-vaue and vice versa.

a When x= 0 and x= 3, y= 1. It is not a one-to-one function.

b There is only one x-value for each y-value. It is a one-to-one function.

Sketch the graph of f(x) = 3x. Check whether both a vertical line and a horizontal line crosses only once.

c

It is a one-to-one function.

Write a statement to answer the question. The functions are one-to-one for b and c. 1

2 y

x

0 3

1

f(x)

3

11

WORKED

E

xample

Which of the following graphs show a one-to-one function?

a b c

THINK WRITE

If a function is one-to-one, any vertical or horizontal line crosses the graph only once.

Only b shows a one-to-one function. y

x

0

y

x

0

y

x

0

12

(26)

Hybrid functions

A hybrid, mixed, or piecewise defined function is a function which

has different rules for different subsets of the domain. For example:

is a hybrid function which obeys the

rules y=x+ 1 if x∈ (−∞, 0] and y=x2 if x∈ (0, ∞). The graph of f(x) is shown at right.

f x( ) x+1,

x2,   

= for x≤0

for x>0

y

x 0

1

–1

f(x)

13

WORKED

Example

THINK WRITE

a (Calculate and plot points as shown or use a graphics calculator.)

a If x= −1, y=x

= −1. Sketch the graph of y=x for the

domain (−∞, 0).

If x= 0, y=x

= 0. On the same axes sketch the graph of

y=x+ 1 for the domain [0, 2).

If x= 0, y =x+ 1 = 1. If x= 2, y =x+ 1

= 3. On the same axes sketch the graph of

y= 5 −x for the domain [2, ∞). Use a graphics calculator to assist with the graphing if necessary.

If x= 2, y= 5 −x

= 3. If x = 5, y= 5 −x

= 0.

b The range is made up of (or is the union of) two sections, (−∞, 0) with [1, ∞).

b ran f= (−∞, 0) ∪ [1, ∞)

1

2

3

y

x 0

1 1 2 3 4 5

1 2 3

–1

f(x)

a Sketch the graph of f x( ) b State the range of f. x,

x+1, 5x,

     =

x<0 x<2 x≥2

remember

1. A function is one-to-one if for each x-value there is only one y-value and vice versa.

2. A many-to-one function may be ‘converted to’ a one-to-one function by restricting the domain.

(27)

Function notation and special

types of function

1 a If f(x) = 3x + 1, find i f(0), ii f(2), iii f(−2) and iv f(5) respectively.

b If g(x) = , find i g(0), ii g(−3), iii g(5) and iv g(−4) respectively.

c If g(x) = 4 − , find i g(1), ii g , iii g and iv g respectively.

d If f(x) = (x + 3)2, find i f(0), ii f(−2), iii f(1) and iv f(a) respectively.

e If h(x) = , find i h(2), ii h(4), iii h(−6) and iv h(12) respectively.

2 Find the value (or values) of x for which each function has the value given.

3 Given that find:

4 Which of the following functions are one-to-one? Use a graphics calculator to obtain the graph of the function where appropriate.

5 Consider the relations below and state:

i which of them are functions ii which of them are one-to-one functions.

a f(x) = 3x − 4, f(x) = 5 b g(x) = x2− 2, g(x) = 7

c f(x) = , f(x) = 3 d h(x) = x2− 5x + 6, h(x) = 0

e g(x) = x2+ 3x, g(x) = 4 f f(x) = , f(x) = 3

a f(2) b f(−5) c f(2x)

d f(x2) e f(x + 3) f f(x − 1)

a {(1, −1), (2, 1), (3, 3), (4, 5)} b {(−2, 1), (−1, 0), (0, 2), (1, 1)}

c {(x, y): y = x2+ 1, x ∈ [0, ∞)} d {(x, y): y = 3 − 4x}

e {(x, y): y = 3 − 2x2} f f(x) = x3− 1

g y = x2, x ≤ 0 h g(x) =

a b c d

e f g h

2E

Mathca

d

Function notation

SkillS

HEET

2.2

SkillS

HEET

2.3

W WORKEDORKED E Example

10 x+4

1 x --- 1 2 --- 

  –1 2

--- 

  –1 5 ---    24 x ---1 x

---8–x

f x( ) 10 x

---= –x,

W WORKEDORKED E Example

11

1x2

(28)

6

Consider the following hybrid function:

a The graph which correctly represents this function is:

b The range of this hybrid function is:

7 a Sketch the graph of the function

b State the range of f.

8 a Sketch the graph of the function

b State the range of g.

c Find i g(−1) ii g(0) iii g(1).

9 a Sketch the graph of the function

b State the range of z.

c Find i f(−3) ii f(−2) iii f(1) iv f(2) v f(5).

i j k l

A B C

D E

A R B R \ {−1} C (−1, ∞) D [0, ∞) E R+ y x 0 y x 0 x y 0 x y 0 m

multiple choiceultiple choice

f x( ) –x, x,

  

= x<1

x≥1

x y

0 1 2 1

–1

x y

0 1 2 1 –1 x y 0 1 1 –1 x y 0 1 1 –1 x y 0 1 1 –1 W WORKEDORKED E Example 13 Mathc ad Hybrid functions

f x( ) 1 x ---,

x+1,

    

= x<0

x≥0

g x( ) x

2+1,

2–x,

  

= x≥0

x<0

f x( )

x–2,

x24,

x+2,

     =

x<–2

2≤ ≤x 2 –

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10 Specify the rule for the function represented by the graph at right.

11 The graph of the relation {(x, y): x2+y2= 1, x≥ 0} is shown at right.

From this relation, form 2 one-to-one functions and state the range of each.

The following screen dumps show how a two-piece hybrid or piecewise defined function may be entered and displayed on the TI–83 and TI–89 graphics calculators.

Inequality signs (<, >, ≤, ≥) are found under the [TEST] menu on the TI–83. On the TI–89 keypad, the < and > signs are above keys on the bottom row, and the ≤ and ≥ signs are found by holding down the green diamond key and pressing < or >.

TI-83 screens TI-89 screens

How could a ‘three-piece’ function be entered on each calculator?

x y

0 1

1 2 3

–1 –1 –2

–2

f(x)

Work

SHEET

2.2

x y

0 1

1

–1

Graphics Calculator

Graphics Calculator

tip!

tip!

Piecewise defined functions

CASIO

(30)

A special relation

You are familiar with the shape of the graph of y = x2 (a parabola), but what about the relation x2+ y2= 25?

If x = 0 and y = 5 are substituted into the rule, we get

x2+ y2= 02+ 52= 25 = RHS, so (0, 5) lies on the graph of x2+ y2= 25. What other points lie on the graph?

Below is a table of coordinates. Twelve of the coordinate pairs listed lie on the graph of x2+ y2= 25.

1 Copy and complete the table below.

2 Use the table to plot the graph of x2+ y2= 25. (Use a smooth curve to join points.)

3 Use graphing software (for example, Graphmatica) or one of the Maths Quest CD-ROM files listed opposite to explore the effect of a on the graph of x2+ y2= a2. Try values of a such as 1, 3, 9, 12, 36, 50 and 100.

4 Investigate graphs of relations of the form (x − h)2+ (y − k)2= a2, for example, (x − 1)2+ (y + 3)2= 16. How is the equation related to features of the graph?

Coordinate pairs

x y x2 y2 x2+y2

On the graph of x2+y2=25?

0 5 0 25 25 Yes

4 2 16 4 20 No

3 0

0 8

3 4

4 3

7 7

–4 −3

–5 0

–4 3

1 5

3 −4

9 0

4 −3

0 −5

6 −6

−3 4

−2 −5

5 0

−3 −4

Mathc

ad

Circular relations

E

XCEL Spreadshe

et

(31)

Circles

A circle is a many-to-many relation.

The rule that defines a circle with its centre at (0, 0) and of radius r is x2+ y2= r2

The graph of this circle is shown at right.

The vertical-line test clearly verifies that the circle graph is not a function.

Solving the equation for y we have y2= r2− x2, so or .

These two relations represent two semicircles that together make a complete circle. is the ‘upper semicircle’ (above the x-axis).

is the ‘lower semicircle’ (below the x-axis).

x y r r –r –r y = r2x2

y = r2x2

y = r2x2

y = r2x2

x y

r

r –r

y = r2x2

x y

–r r –r

y = – r2x2

Sketch the graphs of the following relations.

a x2+y2= 16 b x2+y2= 9, 0 x 3 c y=

THINK WRITE

a This relation is a circle of centre (0, 0) and radius = .

a

On a set of axes mark x- and y-intercepts of −4 and 4. Draw the circle.

b This relation is part of a circle of centre (0, 0) and radius = .

b

Since the domain is [0, 3], on a set of axes mark y-intercepts −3 and 3 and x-intercept 3.

Draw a semicircle on the right-hand side of the y-axis.

c This relation is an ‘upper semicircle’ (as y > 0) of centre (0, 0) and radius = .

c

On a set of axes mark the x-intercepts of − and and y-intercepts of .

Draw a semicircle above the x-axis.

8x2

1

16 = 4

y x 0 –4 4 –4 4 2 3 1

9 = 3

y x 0 3 –3 3 2 3 1 8 y x 0 8

– 8 8

2

8 8

8 3

14

(32)

General equation of a circle

The general equation of a circle with centre (h, k) and radius r is (x − h)2+ (y − k)2= r2.

The domain is [h − r, h + r]. The range is [k − r, k + r].

Note: When using a graphics calculator to plot circle graphs ensure that the upper and lower values are entered as separate equations in Y1 and Y2 and then use and

select 5:ZSquare to show the graph in true proportion.

y

x

0

k + r

k

k – r

h – r h h + r

Range

(h, k)

Domain

(xh)2 + (yk)2 = r2

Sketch the graphs of the following circles. State the domain and range of each.

a x2+ (y 3)2= 1 b (x+ 3)2+ (y+ 2)2= 9

THINK WRITE

a This circle has centre (0, 3) and radius 1.

a

On a set of axes mark the centre and four points; 1 unit (the radius) left and right of the centre, and 1 unit (the radius) above and below the centre.

Draw a circle which passes through these four points.

State the domain. Domain is [−1, 1]. State the range. Range is [2, 4]. b This circle has centre (−3, −2) and

radius 3.

b

On a set of axes mark the centre and four points; 3 units left and right of the centre, and 3 units above and below the centre.

Draw a circle which passes through these four points.

State the domain. Domain is [−6, 0]. State the range. Range is [−5, 1].

1 y

x

0 –1 1

4

2 3

x2 + (y – 3)2 = 1 2

3

4 5

1 y

x

0 –3

–2

–5 1 –6

(x + 3)2 + (y + 2)2 = 9 2

3

4 5

15

WORKED

E

xample

ZOOM

remember

1. The general equation of a circle with centre (h, k) and radius r is

(xh)2+ (yk)2=r2.

2. An ‘upper semicircle’ with centre (0, 0) and radius r is .

3. A ‘lower semicircle’ with centre (0, 0) and radius r is .

y = r2x2

y = r2x2

(33)

Circles

1 State the equation of each of the circles graphed below.

2 State the domain and range of each circle in question 1.

3 Sketch the graph of each of the following relations.

4 Sketch the graph of each of the following relations and state whether it is a function or not.

5

Consider the circle below.

a The equation of the circle is:

b The range of the relation is:

a b c d

e f g h

a x2+ y2= 4 b x2+ y2= 16 c x2+ y2= 49

d x2+ y2= 7 e x2+ y2= 12 f x2+ y2=

a b c

d e f

g h

A x2+ (y − 2)2= 4 B (x − 2)2+ y2= 16 C (x + 2)2+ y2= 16

D (x − 2)2+ y2= 4 E (x + 2)2+ y2= 4

A R B [−2, 2] C [0, 4] D [2, 4] E [−2, 1]

2F

Mathca

d

Circle graphs

EXCEL Spreadshe

et Circle graphs y x 0 –3 3 –3 3 y x 0 –1 1 –1 1 y x 0 –5 5 –5 5 y x 0 –10 10 –10 10 y x 0 6

– 6

– 6 6

y

x

0 –2 2

–2 2

2 2 2 2

y x 0 –3 3 3 y x 0 –4 4 –4 W WORKEDORKED E Example 14 1 4

---y = ± 81x2 y = 4x2 y = 1x2

y 1

9 ---x2

= y 1

4 ---x2

= y = 5x2

y = ± 10x2 x2+y2 = 3, 3 ≤ ≤x 0

m

multiple choiceultiple choice

(34)

6

Consider the equation (x + 3)2+ (y − 1)2= 1.

a The graph which represents this relation is:

b The domain of the relation is:

7 Sketch the graph of the following circles. State the domain and range of each.

8 Express the relation x2 +y2= 36 as two functions and state the largest domain and range of each.

9 Express the relation x2+ (y − 2)2= 9 as two functions stating the largest domain and range of each.

10 Circular ripples are formed when a water drop hits the surface of a pond. If one ripple is represented by the equation x2+ y2= 4 and then 3 seconds later by x2+ y2= 190, where the length of measurements are in centimetres,

a find the radius (in cm) of the ripple in each case

b calculate how fast the ripple is moving outwards. (State your answers to 1 decimal place.)

A B C

D E

A [−3.5, −2.5] B (−4, −2) C R

D [2, 4] E [−4, −2]

a x2+ (y + 2)2= 1 b x2+ (y − 2)2= 4

c (x − 4)2+ y2= 9 d (x − 2)2+ (y + 1)2= 16

e (x + 3)2+ (y + 2)2= 25 f (x − 3)2+ (y − 2)2= 9

g (x + 5)2+ (y − 4)2= 36 h (x − )2+ (y + )2= m

multiple choiceultiple choice

y

x

0 –3

–2 1 4

–6

y

x

0 3 1

2

2 4

y

x

0 –3

1 2

–4 –2

y

x

0 –1 –2

3 2 4

y

x

0 –3

2

1

–3.5 –2.5

W WORKEDORKED E Example

15

1 2

--- 3 2 --- 9

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---Functions and modelling

When using functions to model rules in real-life situations the domain usually has prac-tical restrictions imposed on it. For example, the area of a circle is determined by the function A(r) =πr2.

For a circle to be drawn the radius needs to be a positive number. Hence, the domain is (0, ∞) or R+.

The table describes hire rates for a removal van.

Hours of hire (h) Cost ($C)

Up to 3 200

Over 3 up to 5 300

Over 5 up to 8 450

THINK WRITE

a The cost is $200 if 0 < h ≤ 3. a The cost is $300 if 3 < h ≤ 5.

The cost is $450 if 5 < h ≤ 8.

State the cost function C(h).

b Sketch a graph with 3 horizontal lines over the appropriate section of the domain.

b 1

2

3

4 C h( )

200, 300,

450,

     =

0<h≤3 3<h≤5

5<h≤8

C ($)

h (hours) 0 1 2 3 4 5 6 7 8

250 300

150 200

50 100 350 400 450

16

WORKED

E

xample

a Express the cost as a hybrid function.

b Sketch the graph of the function.

remember

When using functions to model situations:

1. form an equation involving one variable and sketch a graph 2. use the graph to determine domain and range etc.

(36)

Functions and modelling

1 The cost of hiring a paper recycling removalist is described in the following table:

a State the cost function, $C, in terms of the time, t hours, for hiring up to 6 hours.

b Sketch the graph of the function.

2 The charge for making a 10-minute STD call on the weekend is listed below.

a State the cost function in terms of the distance.

b Sketch the graph of the function.

3 A car travels at a constant speed of 60 km/h for 1 hours, stops for half an hour then travels for another 2 hours at a constant speed of 80 km/h, reaching its destination.

a Construct a function that describes the distance travelled by the car, d (km), at time, t hours.

b State the domain and range of this function.

c Calculate the distance travelled after: i 1 hour ii 3 hours.

4 At a fun park, a motorised toy boat operates for 5 minutes for every dollar coin placed in a meter. The meter will accept a maximum of 120 one-dollar coins.

a Write a rule which gives the time of boat operation, B hours, in terms of the number of dollar coins, n.

b Sketch the graph of the function and state the domain and range.

c How much is in the meter when the boat has operated for 450 minutes?

5 The tax for Australian residents who earn a taxable income between $20 700 and $38 000 is $3060 plus 34 cents for every dollar earned over $20 700.

a Write a rule for the tax payable, $T, for a taxable income, $x, where 20 701 ≤x≤ 38 000.

b Sketch a graph of this function.

c Calculate the tax paid on an income of $32 000.

Hours of hire Cost

Up to 1 $40 Over 1 up to 2 $70 Over 2 up to 4 $110 Over 4 up to 6 $160

Distance d (km) Up to 50

km

50 to 100 km

100 to 200 km

200 to 700 km

Over 700 km

Cost $C 0.40 0.60 0.80 1.70 2.00

2G

WORKED

Example

16

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---6 The maximum side length of the rectangle shown is 10 metres.

a Write a function which gives the perimeter, P metres, of the rectangle.

b State the domain and range of this function.

7 A rectangular swimming pool is to have a length 4 metres greater than its width.

a Write a rule for the area of the pool, A m2, as a function of the width, x metres.

b State the domain and range if the maximum side length is 12 metres.

8 Timber increases in value (appreciates) by 2% each year. If a consignment of timber is currently worth $100 000:

a Express the value of the timber, P dollars, as a function of time, t, where t is the number of years from now.

b What will be the value of the timber in 10 years?

9 The number of koalas remaining in a parkland t weeks after a virus strikes is given by the

function koalas per

hectare.

a How many koalas per hectare were there before the virus struck?

b How many koalas per hectare are there 13 weeks after the virus struck?

c How long after the virus strikes are there 23 koalas per hectare?

d Will the virus kill off all the koalas? Explain why.

10 A school concert usually attracts 600 people at a cost of $10 per person. On average, for every $1 rise in admission price, 50 less people attend the concert. If T is the total amount of takings and n is the number of $1 increases:

a write the rule for the function which gives T in terms of n

b sketch the graph of T versus n

c find the admission price which will give the maximum takings.

(x + 4) m (x – 1) m

N t( ) 15 96 t+3

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Set notation

• {. . .} refers to a set of something.

•∈ means ‘is an element of’.

•∉ means ‘is not an element of’.

•⊂ means ‘is a subset of’.

•⊄ means ‘is not a subset (or is not contained in)’.

•∩ means ‘intersection with’.

•∪ means ‘union with’.

• \ means ‘excluding’.

•∅ refers to ‘the null, or empty set’.

• {(a, b), (c, d), . . .} is a set of ordered pairs.

• A relation is a set of ordered pairs.

N refers to the set of natural numbers.

J refers to the set of integers.

Q refers to the set of rational numbers.

R refers to the set of real numbers.

Relations and graphs

• The independent variable (domain) is shown on the horizontal axis of a graph.

• The dependent variable (domain) is shown on the vertical axis of a graph.

• Discrete variables are things which can be counted.

• Continuous variables are things which can be measured.

Domain and range

• The domain of a relation is the set of first elements of a set of ordered pairs.

• The range of a relation is the set of second elements of a set of ordered pairs.

• The implied domain of a relation is the set of first element values for which a rule has meaning.

• In interval notation a square bracket means that the end point is included in a set of values, whereas a curved bracket means that the end point is not included.

s

ummary

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Types of relations (including functions)

• A function is a relation which does not repeat the first element in any of its ordered pairs. That is, for any x-value there is only one y-value.

• The graph of a function cannot be crossed more than once by any vertical line.

Function notation

f(x) = . . . is used to describe ‘a function of x’. To evaluate the function, for example, when x= 2, find f(2) by replacing each occurrence of x on the RHS with 2.

• Functions are completely described if the domain and the rule are given.

Special types of function

• A function is one-to-one if for each x-value there is only one y-value and vice versa.

• A many-to-one function may be ‘converted to’ a one-to-one function by restricting the domain.

• A hybrid function obeys different rules for different subsets of the domain.

Circles

• The general equation of a circle with centre (h, k) and radius r is

(xh)2+ (yk)2=r2

• An ‘upper semicircle’ with centre (0, 0) and radius r is y= .

• A ‘lower semicircle’ with centre (0, 0) and radius r is y= − .

Functions and modelling

• When using functions to model situations:

1. form an equation involving one variable and sketch a graph

2. use the graph to determine domain and range etc.

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1

If A= {−2, −1, 0, 1, 2, 3} and B= {−2, 0, 2, 4, 6} then AB is:

2

Which of the following statements is false?

3

The rule describing the relation shown is:

A y= 2x

B y= 2x, x∈ {1, 2, 3, 4}

C y= 2x, xN

D y=

E y= 2x, xR+

4

Which one of the relations graphed below is continuous?

A {−2, −1, 0, 1, 2, 3, 4, 6} B {−2, 0, 2}

C {−1, 1, 3, 4, 6} D {−1, 1, 3}

E

A JQ B 3.142 ∈Q

C π∈R D {0, 1, 2, 3} ∈N

E (NJ) =J

A B C

D E

CHAPTER

review

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multiple choiceultiple choice

2A

2A

mmultiple choiceultiple choice

m

multiple choiceultiple choice

2B

y

x

0 1 2 3 4 2

4 6 8

x

2

---m

multiple choiceultiple choice

2B

y

x

0

y

x

0

y

x

0

y

x

0

y

x

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5 The total number of cars that have entered a car park during the first 5 hours after opening is shown in the table below.

a Plot these points on a graph.

b Explain why the dots cannot be joined.

c Estimate the number of cars in the park 2 hours after the car park opens.

6 a Sketch the graph of the relation {(x, y): y= 1 −x2, x∈ [−3, 3]}.

References

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