4a set notation 4b relations and graphs 4c 4d 4e relations ...mathsbooks.net/jacplus books/11...

4A Set notation 4b Relations and graphs 4c Domain and range 4d Types of relations (including functions) 4e Power functions (hyperbola, truncus and

square root function) 4F Function notation 4G Special types of function (including

hybrid functions) 4H Inverse relations and functions 4I Circles 4J Functions and modelling

4

170

AREAS oF STudy

Use of the notation • y = f (x) for describing the rule of a function and evaluation of f (a), where a is a real number or a symbolic expressionGraphs of power functions • y = xn for n ∈ N and n = -1, -2, 1Graphs of power functions

1Graphs of power functions

2and transformations of these to

the form y = a(x + b)n + c where a, b and c ∈ R

‘The vertical line test’ and its use to determine • whether a relation is a functionGraphs of relations including those specifi ed by • conditions or constraints, such as arcs of circles Graphs of inverse relations• Use of inverse functions to solve equations•

eBookpluseBookplus

Digital doc10 Quick Questions

Relations, functions and transformations

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

4A

Maths Quest 11 Mathematical Methods CAS for the Casio ClassPad

171Chapter 4 Relations, functions and transformations

defi nitions 1. A set is a collection of things. 2. The symbol {. . .} refers to a set. 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 is often used to refer to a particular set of things. 5. The symbol ⊂ means ‘is a subset of’, so, if B ⊂ A, then all of the elements of set B are

contained in set A. For example, {2, 4} ⊂ {2, 4, 6, 8, 10}. 6. The symbol ⊄ means ‘is not a subset (or is not contained in)’, for example {2, 3} ⊄ {2, 4, 6,

8, 10}. 7. The symbol ∩ means ‘intersection’, so, A ∩ B is the set of elements common to sets A and

B. For example, {1, 2, 3} ∩ {2, 4, 6} is {2}. 8. The symbol ∪ means ‘union’, so, A ∪ B is the set of all elements belonging to either

set A or B. For example, {1, 2, 3} ∪ {2, 4, 6} is {1, 2, 3, 4, 6}.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 the set is empty.

WoRkEd ExAMPlE 1

If A = {1, 2, 4, 8, 16, 32}, B = {1, 2} and C = {1, 2, 3, 4}, fi nd:a A ∩ B b A ∪ Cc A \ B d {3, 4} ∩ Be whether or not: i 8 ∈ A i i B ⊂ A i i i C ⊂ A.

Think WRiTE

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

a {1, 2}

b The elements that belong to either A or C are 1, 2, 3, 4, 8, 16 and 32.

b {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 {4, 8, 16, 32}

d {3, 4} and B have no common elements. d ∅e i 8 is an element of A.

i i All elements of B belong to A. i i i 3 is an element of C but not A.

e i Yes. 8 ∈ A i i Yes. B ⊂ A i i i No. C ⊄ A

Sets of numbersCertain 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. Z 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 ab 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, p,

rational (that is, cannot be expressed as a ratio of two 3 , etc.

Note that N ⊂ Z ⊂ Q ⊂ R, that is,

172

1

1–2 3–

4 2–3

7–5

33—51

2 340

–1–2 –3

...

...

...

...

R

Q

Z

N

Q'π3

{. . .} refers to a set.1. ∈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’.\ means ‘excluding’.8. ∅9. refers to ‘the null, or empty set’.

REMEMBER

Set notation 1 WE1 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:a A ∩ B b A ∩ C c A ∩ C ∩ Dd A ∪ B e C ∪ D f A \ Cg C \ D.

2 If A = {-3, -2, -1, 0, 1, 2, 3}, B = {0, 1, 2, 3} and C = {-3, 2, 3, 4}, find:a A ∩ B ∩ C b A \ B c A \ (B ∪ C)d A \ (B ∩ C) e A ∪ C.

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:a F ∩ G ∩ H b G ∩ H c G \ Hd H \ F e (F ∪ H) \ G.

4 MC Given that A ⊂ B, then A ∩ B is equivalent to:A B b ∅ c {1, 2}d A ∪ B e A

5 MC Given that C ⊂ B ⊂ A, then it follows thata A ∪ B ∪ C is equivalent to:

A B b C c Ad A ∪ B e B ∪ C

b (A \ B) ∩ C is equivalent to:A B b ∅ c Cd A ∩ B e B \ C

ExERCiSE

4A



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

a 7 ∈ R b -4 ∈ N c 6.4217 ∈ Qd 5 ∈ Q e 1.5 ∈ Z f {5, 10, 15, 20} ⊂ Zg {5, 10, 15, 20} ⊂ N h Z \ N = {. . ., -3, -2, -1} i Z ∩ N = Nj Q ⊂ N k Q ∩ Z = ∅ l (Z ∪ Q) ⊂ R

Relations and graphsA relation is a set of ‘ordered pairs’ of values or ‘variables’.

Consider the following relation. The cost of hiring a trailer depends on the number of hours for which it is hired. The table below outlines this relation.

Number of hours of hire 3 4 5 6 7 8

Cost ($) 50 60 70 80 90 100

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.

Cos

t of

trai

ler

hire

($)

10 2 3 4 5 6 7 8

5040

60708090

100

Number of hours

y

x

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 dependent 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 variables include 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

4B

174

Using x and y terms, this is written as

y = 10x + 20

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 x ∈ R

Think WRiTE/dRAW

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

a When x = 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)

2 Plot the points (1, 1), (2, 4), (3, 9) and (4, 16) on a set of axes.

y

x0 1 2 3 4

4

8

12

16

3 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 1 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 When x = 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)

2 Plot the points (0, 1), (1, 3) and (2, 5) on a set of axes.

y

x0−2 −1 1 2

12345

−3−2−1

y = 2x + 1

3 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.

WoRkEd ExAMPlE 2



The pulse rate of an athlete, R beats per minute, t minutes after the athlete finishes a workout, is shown in the table below.

t 0 2 4 6 8

R 180 150 100 80 70

a Plot the points on a graph.b Estimate the athlete’s pulse rate

after 3 minutes.

Think WRiTE/dRAW

a 1 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.

a & b

0 2 4 6 81 3 5 7

20

806040

100120140160180

t (min)

R (

beat

s/m

in)2 Plot the points given in the table.

b 1

2

3

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

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.

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

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

WoRkEd ExAMPlE 3

Note: In any defined domain, for example, -3 ≤ x ≤ 3, the variable (x) is assumed continuous in that domain unless otherwise stated.

The independent variable (for example, 1. x) is shown on the horizontal axis of a graph.The dependent variable (for example, 2. y) is shown on the vertical axis of a graph.Discrete variables are things which can be counted. Graph points are not joined.3. Continuous variables are things which can be measured. Graph points may be joined.4.

REMEMBER

Relations and graphsQuestions 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)}.

ExERCiSE

4B

176

1 MC The graph of this relation is represented by:A y

x0 1 2 3 4

1

2

3

4b y

x0 1 2 3 4

1

2

3

4c y

x0 1 2 3 4

1

2

3

4

d y

x0 1 2 3 4

1

2

3

4e y

x0 1 2 3 4

1

2

3

4

2 MC The elements of the dependent variable are:A {1, 2, 3, 4} b {1, 2, 3} c {0, 1, 2, 3, 4} d {0, 1, 2, 3} e {1, 2}

3 MC The rule for the relation is correctly described by:A y = 4 - x, x ∈ R b y = x - 4, x ∈ N c y = 4 - x, x ∈ Nd y = x - 4, x ∈ Z e y = 4 - x, x ∈ {0, 1, 2, 3}

4 MC 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

0 M WT T F

Num

ber

of p

eopl

e b

0 M WT T F

Num

ber

of p

eopl

e c

0 M WT T F

Num

ber

of p

eopl

e

d

0 M WT T F

Num

ber

of p

eopl

e e

0 M WT T F

Num

ber

of p

eopl

e

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 y

x0

d y

x0



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 WE2 Sketch the graph representing each of the following relations, and state whether each is discrete or continuous.a Day Mon Tues Wed Thur Fri Sat Sun

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

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 ∈ Re y = 2x + 3, where x ∈ Zf y = x2 + 2, where -2 ≤ x ≤ 2

7 WE3 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 below.

n 0 1 2 3 4 5 6

P

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.

t 0 1 2 3 4 5

V 0 30 80 150 240 350

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.

domain and rangedomain and rangeA relation can be described by:1. a listed set of ordered pairs2. a graph or3. a rule.

t (min) 0 2 4 6 8

T (°C) 80 64 54 48 44

4C

178

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 given3. 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 the implied domain of a relation. For example:{(x, y): y = x3} has the implied domain R.{(x, y): y = x} has the implied domain x ≥ 0.

interval notationIf a and b are real numbers and a < b, then the following intervals are defined with an accompanying number line.(a, b) implies a < x < b or (a, b] implies a < x ≤ b or

xa b xa b

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

xa xa

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

xb xb

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

xa b xa b

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

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

x−4 20

b

x−3 50c

x5310

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, ∞)

WoRkEd ExAMPlE 4



Illustrate the following number intervals on a number line.a (-2, 10] b [1, ∞)

Think WRiTE/dRAW

a The interval is -2 < x ≤ 10 (-2 is not included, 10 is). ax100−2

b The interval is x ≥ 1 (1 is included). bx0 1

WoRkEd ExAMPlE 5

State the domain and range of each of the following relations.a {(1, 2), (2, 5), (3, 8), (4, 11)}b

Weight (kg) 10 15 20 25 30

Cost per kg ($) 3.5 3.2 3.0 2.8 2.7

c y

x0

d y

x0−4 4

−4

4

Think WRiTE

a 1 The domain is the set of first elements of the ordered pairs.

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

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

Range = {2, 5, 8, 11}

b 1 The domain is the set of independent values in the table, that is, the weight values.

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

2 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 1 The domain is the set of values that the graph covers horizontally.

c Domain = R

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

Range = [0, ∞)

d 1 The domain is the set of values that the graph covers horizontally.

d Domain = [-4, 4]

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

Range = [-4, 4]

WoRkEd ExAMPlE 6

180

For each relation given, sketch its graph and state the domain and range using interval notation.a {(x, y): y = x −−−−− 1} b {(x, y): y = x2 - 4, x ∈ [0, 4]}

Think WRiTE/dRAW

a 1 The rule has meaning for x ≥ 1 becauseif x < 1, y = negativgativga e numbenumbenum r.ber.be

a

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

When x = 1, y = 0 = 0 (1, 0). x = 2, y = 1 = 1 (2, 1) x = 3, y = 2 (3, 2) x = 4, y = 3 (4, 3) x = 5, y = 4 = 2 (5, 2)

3 Plot the points on a set of axes. y

x0 1 2 3 4 5

1

−1

2y = x − 1

4 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).

5 Place a closed circle on the point (1, 0) and put an arrow on the other end of the curve.

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

Domain = [1, ∞)

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

Range = [0, ∞)

b 1 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 When x = 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)

2 Plot these points on a set of axes. y

x0 1 2 3 4

2

−2−4

64

81012

y = x2 − 4, x ∈ [0, 4]

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

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

WoRkEd ExAMPlE 7eBookpluseBookplus

Tutorialint-0287

Worked example 7



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

Domain = [0, 4]

6 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 or other technology.

The domain of a relation is the set of first elements of an ordered pair.1. The range of a relation is the set of second elements of an ordered pair.2. The implied domain of a relation is the set of first element values for which a 3. rule has meaning.In interval notation a square bracket means the end point is included in a set of values, 4. whereas a curved bracket means the end point is not included.

a b

(a, b]

REMEMBER

domain and range 1 WE4 Describe each of the following subsets of the real numbers using interval notation.

a

−2 10

b

50

c

40−3

d

90−8

e

0−1

f 0 1

g

30−5 −2

h

4210−3

2 WE5 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.a {x: -4 ≤ x < 2} b {x: -3 < x ≤ 1} c {y: -1 < y < 3}d {y:

−12 < y ≤

1

2} 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}

4 MC Consider the set described by R \ {x: 1 ≤ x < 2}.a It is represented on a number line as:

A

210

b

210

ExERCiSE

4C

182

c

210

d

210

e

210b It is written in interval notation as:

A (-∞, 1) ∪ (2, ∞) b (-∞, -1) ∪ [2, ∞) c (-∞, 1) ∪ (2, ∞]d (-∞, 1] ∪ (2, ∞) e (-∞, 1) ∪ [2, ∞)

5 MC The domain of the relation graphed at right is:A [-4, 4] b (-4, 7) c [-1, 7]d (-4, 4) e (-1, 7)

6 MC The range of the relation {(x, y): y = 2x + 5, x ∈ [-1, 4]} is:A [7, 13] b [3, 13] c [3, ∞) d R e R \ (7, 13)

7 WE6a, b State i the domain and i i 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

Time (min) 3 4 5 6

Distance (m) 110 130 150 170

d Day Monday Tuesday Wednesday Thursday Friday

Cost ($) 25 35 30 35 30

e y = 5x - 2, where x is an integer greater than 2 and less than 6f y = x2 - 1, x ∈ R

8 WE6c, d State the domain and range of each of the following relations. Use a CAS calculator to view more of each graph if required.

a y

x0

2

−3

b y

x0

2y = 2ex

c y

x0

2

−2 2

d y

x0 1

y = x − 1e y

x0

4 y = 4e−x2

f y

x0

−3

g y

x0

y = 1–x

h y

x0

1

i y

x0

−2

eBookpluseBookplus

Digital docSkillSHEET 4.1

Domain and range

y

x0 3 7

4

−4

−1



9 WE7 For each relation given, sketch its graph and state the domain and range using interval notation.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}Verify that the graphs are correct with a CAS calculator.

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

a y = 10 - x b y x= 3

c y x= −−

16 2 d y = x2 + 3

e yx

= 1 f y = 10 - 7x2

Types of relations (including functions)one-to-one relationsA one-to-one relation exists if, for any x-value, there is only one corresponding y-value and vice versa.

For example:

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

one-to-many relationsA 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, 2), (2, 3), (3, 4)}

Many-to-one relationsA 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), (0, 1), (1, 2)}

Many-to-many relationsA many-to-many relation exists if there is more than one x-value for any y-value and vice versa.

For example:

{(0, -1), (0, 1), (1, 0), (-1, 0)}

eBookpluseBookplus

Digital docWorkSHEET 4.1

eBookpluseBookplus

Digital docInvestigation

Interestingrelations

4d

y

x0

y

x0

y

x0

y

x0

y

x0

184

What type of relation does each graph represent?a y

x0

b y

x0

c y

x0

Think WRiTE

a 1 For some x-values there is more than one y-value. A line through some x-values shows that 2 y-values are available:

y

x0x = −1

F

a One-to-many relation.

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

y

x0

y = 1

b 1 For any x-value there is only one y-value. b One-to-one relation.

2 For any y-value there is only one x-value.

c 1 For any x-value there is only one y-value. c Many-to-one relation.

2 For some y-values there is more than one x-value.


Tutorialint-0288

Worked example 8

FunctionsRelations 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 at most one y-value. For example:

y

x0

y

x0

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



State whether or not each of the following relations are functions.a {(-2, 1), (-1, 0), (0, -1), (1, -2)}b y

x0

c y

x0

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

WoRkEd ExAMPlE 9

A function is a relation which does not repeat the first element in any of its ordered 1. pairs. That is, for any x-value there is at most one y-value (one-to-one or many-to-one relations).Vertical line test: The graph of a function cannot be crossed more than once by any 2. vertical line.

y

x0

y

x0

Function Not a function

REMEMBER

Types of relations (including functions) 1 WE8 What type of relation does each graph represent?

a y

x0

b y

x0

c y

x0

ExERCiSE

4d

186

d y

x0

e y

x0

f y

x0

g y

x0

h y

x0

i y

x0

j y

x0

k y

x0

l y

x0

2 WE9 Use the vertical line test to determine which of the relations in question 1 are functions.

3 MC Which of the following relations is not a function?A {(5, 8), (6, 9), (7, 9), (8, 10), (9, 12)}

b y

x0

c y2 = x d y = 8x - 3 e y

x0

4 MC Consider the relation y ≥ x + 1.a The graph which represents this relation is:

A y

x0

1

−1

b y

x0

1

−1

c y

x0

1

1

d y

x0

1

1

e y

x0

1

–1

Note: The shaded side indicates the region not required.



b This relation is:A one-to-oneb one-to-manyc many-to-oned many-to-manye a function

c The domain and range are respectively:A R and R+

b R and Rc R and R-

d R+ and Re R- and R

5 Which of the following relations are functions? State the domain and range for each function.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 ∈ Z}g y = 1 - 2x h y > x + 2

i x2 + y2 = 25 j y x x= + ≥ −1 1,k y = x3 + x l x = y2 + 1

Power functions (hyperbola, truncus and square root function)Power functions are functions of the form f (x) = xn, n ∈ R. The value of the power, n, determines the type of function. We saw earlier that when n = 1, f (x) = x, and the function is linear. When n = 2, f (x) = x2, and the function is quadratic. When n = 3, f (x) = x3, and the function is cubic. When n = 4, f (x) = x4, and the function is quartic.

Other power functions are:when • n = -1, f (x) = x-1, and the power function produces the graph of a hyperbolawhen • n = -2, f (x) = x-2, and the power function produces the graph of a truncuswhen • n f x x= =1

2

12, ( ) , and the function is the square root function.

Under a sequence of transformations of f (x) = xn, n ∈ R, the general form of a power function is f (x) = a(x - b)n + c (where a, b, c and n ∈ R).

The hyperbola• The graph shown is called a hyperbola, and is given by the equation y

x= 1 .

This can also be represented as the power function • y = x-1.

The graph exhibits asymptotic behaviour. That is, as • x becomes very large, the graph approaches the x-axis but never touches it. As x becomes very small (approaches 0), the graph approaches the y-axis, but never touches it.So the line • x = 0 (the y-axis) is a vertical asymptote, and the line y = 0 (the x-axis) is the horizontal asymptote.Both the domain and the range of the function are all real numbers, except 0; that is, • R \ {0}.The graph of• y = 1

x can be subject to a number of transformations.

Consider • ya

x bc y a x b c=

−+ = − +−or ( ) .1

4E

eBookpluseBookplus

Interactivityint-0263

Domain and range

x = 0

0y = 0

x

y

188

dilation• The value a is a dilation factor. It dilates the graph from

the x-axis. For example, the graph of • y

x= 3 compared to the basic

graph of yx

= 1 is shown at right.

ReflectionIf • a is negative, the graph of the basic hyperbola is reflected in the x-axis. If • x is replaced with -x, the graph of the basic hyperbola is reflected in the y-axis.

For example, the graphs of • yx

=+1

1 and y

x= − +

1

1 are reflections of each other across the

y-axis.

−1

y

x

−2

12

−2 10 2−1

y = 1x + 1

x = −1

−1

y

x

−2

123

−2 10 2−1

y = 1−x + 1

x = 1

TranslationHorizontal translation

The value • b translates the graph b units horizontally, that is, parallel to the x-axis. If b > 0, the graph is translated to the right, and if b < 0, the graph is translated to the left.

For example, the graph with equation • yx

=+1

1 is a basic hyperbola translated one unit to the

left. This graph has a vertical asymptote of x = -1 and domain R \ {-1}, and a horizontal asymptote of y = 0. If a basic hyperbola is translated one unit to the right, it becomes • y

x=

−1

1, with a vertical

asymptote of x = 1 and domain R \ {1}. Hence, the equation of the vertical asymptote is • x = b and the domain is R \ {b}. The horizontal asymptote and the range remain the same, that is, x = 0 and R \ {0} respectively.

Vertical translationThe value • c translates the graph c units vertically, that is, parallel to the y-axis. If c > 0, the graph is translated upward, and if c < 0, the graph is translated c units downward. The graph with equation • y

x= −1 1 is a basic hyperbola translated one unit down.

This graph has a horizontal asymptote of y = -1, a range of R \ {-1} and a vertical asymptote of x = 0.If a basic hyperbola is translated two units up, it becomes •y

x= +1 2, with a horizontal asymptote of y = 2 and a range

of R \ {2}. Hence, the equation of the horizontal asymptote is y = c, with a vertical asymptote of x = 0, and the range is R \ {c}.

−1

y

x

−2−3

123

−2−3 10 2

(1, 3)y =

(1, 1)

3−1

3–x

y = 1–x

x = 0

y = 0

y = c

x = b

0

y = ax − b

+ c

c

b

y

x



The graph of ya

x bc=

−+ or y = a(x - b)-1 + c shows the combination of these transformations.

WoRkEd ExAMPlE 10

Sketch the graph of yx

=====+++++

+++++31

2, clearly showing the intercepts with the axes and the position of the asymptotes.

Think WRiTE/dRAW

1 Compare the given equation with ya

x bc=

x b−x b+ , and

state the values of a, b and c.a = 3, b = -1, c = 2

2 Write a short statement about the effects of a, b and con the graph of y

x= 1 .

The graph of yx

= 1 is dilated by the factor of 3 from the x-axis (a = 3), translated 1 unit to the left (b = 1) and 2 units up (c = 2).

3 Write the equations of the asymptotes. The horizontal asymptote is at y = c. The vertical asymptote is at x = b.

Asymptotes: x = -1; y = 2

4 Find the value of the y-intercept by letting x = 0. y-intercept: x = 0

y = +30 1+0 1+

2

= 3 + 2= 5

Point (0, 5)

5 Find the value of the x-intercept by making y = 0. x-intercept: y = 0

03

12=

++

x

− =+

23

1x-2(x + 1) = 3

-2x2x2 - 2 = 3-2x2x2 = 5x =

−52

Point −( )5

20,

6 To sketch the graph:(a) draw the set of axes and label them(b) use dotted lines to draw the asymptotes

The asymptotes are x = -1 and y = 2.(c) mark the intercepts with the axes

The intercepts are y = 5 and x =− 5

2.

(d) treat the asymptotes as your new set of axes, sketch the graph of the hyperbola. (As a is positive, the graph is not refl ected in the x-axis.)

−2

y

x

−4−6

246

−2−3 10 2 3−1−3

x = −1

y = 2

(0, 5)

22

1−1−−

y = 3x + 1

+ 2

5 , 0)(−2

The truncusThe graph shown is known as a • truncus. The equation of the graph is given by:

yx

= 12

190

This can also be represented as the power function • y = x-2.The function is undefined for • x = 0. Hence, the equation of the vertical asymptote is x = 0 and the domain of the function is R \ {0}.We can also observe that the graph approaches the • x-axis very closely, but never touches it. So y = 0 is the horizontal asymptote.Since the whole graph of the truncus is above the • x-axis, its range is R+ (that is, all positive real numbers).Similarly to the graphs of the functions, discussed in the previous sections, the graph of•

yx

= 12

can undergo various transformations.

Consider the general formula • ya

x bc=

−+

( )2 or y = a(a - b)-2 + c.

dilationThe value • a is the dilation factor. It dilates the graph from the x-axis. The dilation factor does not affect the domain, range or asymptotes.

Consider the graph of • yx

= 42

.

y

x

246

−2−3 10 2

y =

3−1

4x2

x = 0

y = 0

ReflectionIf • a is negative, the graph of a basic truncus is reflected in the x-axis. The range becomes R- (that is, all negative real numbers).If • x is replaced with -x, the graph of the basic truncus is reflected in the y-axis. The effect of this reflection cannot be seen in the basic graph, but it becomes more obvious if the graph has been translated horizontally first.For example, the graphs of • y

x=

−1

2 2( ) and y

x= − −

1

2 2( ) are reflections across the y-axis.

The vertical asymptote changes from x = 2 to x = -2, and the domain changes from R \ {2} to R \ {-2}.

−1

y

x

−2

123

−2 10 2−1

x = 2

y = 1(x − 2)2

y

x−2−3 0−1

x = −2

y = 1

(−x − 2)2


The value • b translates the graph b units horizontally. If b > 0, the graph is translated to the right, and if b < 0, the graph is translated left. For example, the graph of the equation • y

x=

−1

3 2( ) results from translating a basic truncus

three units to the right. The vertical asymptote is x = 3 and the domain is R \ {3}.

x = 0

y = 0 0 x

y

y = 1x2



If a basic truncus is translated two units to the left, it becomes • yx

=+12 2( )

, where the vertical asymptote is x = -2 and the domain is R \ {-2}. Hence, the equation of the vertical asymptote is • x = b, and the domain is R \ {b}. The range is still R+, and the equation of the horizontal asymptote is y = 0.

Vertical translationThe value • c translates the graph c units vertically. If c > 0, the graph is translated upward, and if c < 0, the graph is translated c units downward.

For example, the graph with equation • yx

= +11

2 results when a basic truncus is translated one

unit upward. The horizontal asymptote is y = 1, and the range is (1, ∞).

If a basic truncus is translated one unit down, it becomes • yx

= −11

2, with y = -1 as the

horizontal asymptote and (-1, ∞) as the range.Hence, the equation of the horizontal asymptote is • y = c, and the range is (c, ∞).

Note: If a is positive (see graph below), the whole graph of the truncus is above the line y = c (the horizontal asymptote), and hence its range is y > c, or (c, ∞).

If • a is negative, the whole graph is below its horizontal asymptote, and therefore the range is y < c, or (-∞, c).• The graph of y

a

x bc=

−+

( )2 or y = a(x - b)-2 + c shows the

combination of these transformations.

WoRkEd ExAMPlE 11

Sketch the graph of yx

==−−

++12

32( )

, clearly showing the position of the asymptotes and the

intercepts with the axes.

Think WRiTE

1 Write the general formula for the truncus. ya

x bc=

−+

( )2

2 Identify the values of a, b and c. a = 1, b = 2, c = 3

3 Write a short statement about the transformations the graph of y

x= 1

2 should undergo in order to be changed

into the one in question.

The graph of yx

= 12

is translated 2 units to

the right and 3 units up.

4 Write the equations of the asymptotes (y = c and x = b). Asymptotes: x = 2 and y = 3

5 Find the x-intercept. Since you cannot get the square root of a negative number, there is no solution and therefore no x-intercepts.An inspection of the equation of the graph would also have revealed this.

x-intercept: y = 0

0

1

23

2=

−+

( )x1

23

2( )x −= −

( )x − =

−2 2 1

3

( )x − = ±−

2 13

y

x0

x = b

y = c

b

c

y = + ca

(x − b)2

192

6 Find the y-intercept. y-intercept: x = 0

y =−

+1

0 23

2( )

= +14

3

= 134

7 To sketch the graph:(a) draw the set of axes and label them(b) use dotted lines to draw asymptotes(c) mark the x- and y-intercepts(d) treating the asymptotes as the new set of axes, draw

the basic truncus curve(e) make sure it intersects the axes in the right places.

y

x

246

−2 10 2 3 4−1

x = 2

y = 3

13 )(0,4

The square root functionThe square root function is given by• y x= .This can be written as the power function • y x=

12 .

The function is defined for • x ≥ 0; that is, the domain is R+ ∪ {0}, or [0, ∞).As can be seen from the graph, the range of the square root function is also • R+ ∪ {0}, or [0, ∞).Throughout this section we will refer to the graph of • y x= as ‘the basic square root curve’.

y

x0

y = x√

Let us now investigate the effects of various transformations on the basic square root curve.•

Consider the function • y a x b c y a x b c= − + = − +or ( ) .12

dilationThe value • a is a dilation factor; it dilates the graph from the x-axis. The domain is still [0, ∞).

ReflectionIf • a is negative, the graph of a basic square root curve is reflected in the x-axis. The range becomes (-∞, 0]. The domain is still [0, ∞).If • x is replaced with -x, the graph is reflected in the y-axis. For example, the graphs with

equations y x= and y x= − are reflected across the y-axis.The domain becomes (• -∞, 0] and the range is [0, ∞).

y

x10 2

y = x√

y

x−2 10 2−1

y = −x√




The value • h translates the graph horizontally. If b > 0, the graph is translated to the right, and if b < 0, the graph is translated to the left. The graph with the equation • y x= −1 results when the basic curve is translated one unit to the right. This translated graph has domain [1, ∞) and range [0, ∞). If the basic curve is translated three units to the left, it becomes • y x= + 3 and has domain [-3, ∞) and range [0, ∞).

y

x

12

−2−3 10 2 3−1

y = x + 3√

3√

The domain of a square root function after a translation is given by [• b, ∞).

Vertical translationThe value • c translates the graph vertically. If c > 0, the graph is translated vertically up, and if c < 0, the graph is translated vertically down.

If • y x= is translated three units vertically up, the graph obtained is y x= + 3, with domain [0, ∞) and range [3, ∞).

−1

y

x

12345

−2 10 2 3 4−1

y = x + 3√

If the basic curve is translated two units down, it becomes • y x= − 2, with domain [0, ∞) and range [-2, ∞).The range of the square root function is [• c, ∞) for a > 0.

The graph of• y a x b c y a x b c= − + = − +or ( )12 shows the combination of these

transformations.

y = x + b + c

(b, c)(−b, c)

a√y = b − x + ca√

y

x

WoRkEd ExAMPlE 12

Sketch the graph of y x== −− ++3 1 2, clearly marking intercepts and the end points.

Think WRiTE/dRAW

1 Write the equation. y x= − +3 1 2,

2 Write the coordinates of the end point. End point: (1, 2)

194

3 State the shape of the graph. Shape: .

4 Inspection of the equation reveals that there is no y-intercept.

There is no x-intercept.

5 Inspection of the equation reveals that there is no y-intercept.

There is no y-intercept.

6 To help sketch the graph, determine the coordinates of a second point.Let x = 3.

x y= = − +3 3 3 1 2:

= × +3 2 2

Point: ( , )3 3 2 2+

7 Sketch the graph by plotting the end point, showing the second point, and drawing the curve so that it starts at the end point and passes through the second point.

−1

y

x

1234567

−2 10 2

(1, 2)

3 4−1

y = 3 x − 1 + 2√

(3, 3 2 + 2)√

The graph of 1. yx

= 1 is called a hyperbola.

• The graph of ya

x bc=

−+ is the graph of the basic hyperbola, dilated by the factor

of a in the y-direction, translated b units horizontally (to the right if b > 0 or to the left if b < 0) and c units vertically (up if c > 0 or down if c < 0). If a < 0, the graph is reflected in the x-axis. The equations of the asymptotes are: x = b and y = c. The domain of the function is R \ {b} and its range is R \ {c}.

The graph of 2. yx

= 12

is called a truncus.

• The graph of ya

x bc=

−+

( )2 is the basic truncus curve, dilated by a factor of a in the

y-direction and translated b units along the x-axis (to the right if b > 0 or to the left if b < 0) and c units along the y-axis (up if c > 0 or down if c < 0). If a is negative, the graph is reflected in the x-axis. The vertical asymptote is x = b. The horizontal asymptote is y = c. The domain is R \ {b}.The range is y > c if a > 0, or y < c if a < 0.

The graph of the function 3. y a x b c= − + is the graph of y x= , dilated by the factor of a in the y-direction and translated b units along the x-axis and c units along the y-axis.• If a < 0, the basic graph is reflected in the x-axis.• Theendpointofthegraphis(b, c).• Thedomainisx ≥ b.• Therangeisy ≥ c for a > 0, or y ≤ c for a < 0.• If y a b x c= − + , the domain is x ≤ b; the graph of y a x= is reflected in the

y-axis.

REMEMBER



Power functions (hyperbola, truncus and square root function) 1 State the dilation factor and the vertical and horizontal translations or reflections for each of

the following. Write a short statement about the effects each has on the basic graph of that function.

a yx

=−

+35

1 b yx

= +21

2

c yx

=−

−14

2 d y x= +2 4

e yx

=−1

3 2( ) f y x= −− 4 5

2 WE 10, 11, 12 Sketch a graph for each of the following functions, clearly showing x- and y-intercepts and any asymptotes.

a f xx

( ) = +31

2 b f x x( ) = − −3 1

c f xx

( ) =−

+21

5 d f xx

( )( )

= −−

22

1 2

3 Assuming the dilation factor for each of the following is 1 , write the equation of the graph.y

x

123456789

10 2 3 4−1

5–2

3–2

x = 2

y = 3

Function notationConsider 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.If x = 1, then y = f (1) If x = 2, then y = f (2) = 2 × 1 = 2 × 2 = 2 = 4, and so on.

Evaluating functionsFor 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.

ExERCiSE

4E

4F

196

WoRkEd ExAMPlE 13

If f (x) = x2 - 3, find:a f (1) b f (-2) c f (a) d f (2a).

Think WRiTE/diSPlAy

Method 1: Technology-free

a 1 Write the rule. a f (x) = x2 - 3

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

3 Simplify. = 1 - 3= -2

b 1 Write the rule. b f (x) = x2 - 3

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

3 Simplify. = 4 - 3= 1

c 1 Write the rule. c f (x) = x2 - 3

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

d 1 Write the rule. d f (x) = x2 - 3

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

3 Simplify the expression if possible. = 22a2 - 3= 4a2 - 3

Method 2: Technology-enabled

1

2 Write the answers. abcd

f (1) = -2 f (-2) = 1 f (a) = a2 - 3

f (2a) = 4a2 - 3

Fully defining functionsTo fully define a function:1. define the domain, and2. state the rule.


Use a CAS calculator. Define the function f(x) = x2 - 3 on the Main screen. To do this, tap:• Action• Command• DefineComplete the entry line as:Define f (x) = x2 - 3 and press E.Complete the entry line as:f({1, -2, a, 2a})Then press E.


That is, if a function f (x) has domain X, the function may be defined as follows:

f :X → Y, f (x) = . . . . . .

Domain Co-domain Rule

Y is not necessarily the range but is a set that contains the range, called the co-domain. The co-domain gives the set of possible values that contains y. It is usually R (the set of Real numbers). The actual values that y can be — the range — is deter mined by the rule. When using function notation the domain can be abbreviated as dom f and the range as ran f.

For example, the function defined by {(x, y): y = 2x, x ∈ [0, 3]} can be expressed in function notation as f : [0, 3] → R, f (x) = 2x.

For this function we can write dom f = [0, 3]. The co-domain = R.Also, ran f = [0, 6] (x = 0 gives y = 0 and x = 3 gives y = 6, which are the minimum and maximum values of y).

R

– 2

3

–5.1

11–3

23

0.6

0etc.

etc.7–8–

11— 3–

–10

Domain

R

2 36

2–3

24

0 etc.

1.2

Range

f : domain co-domain , f(x) = rule

Co-domain

[0, 3] [0, 6]

The graph of this function is shown at right.The maximal domain of a function is the largest possible set of values

of x for which the rule is defined. The letters f, g and h are usually used to name a function, that is, f (x), g(x) and h(x).Note: If a function is referred to by its rule only, then the domain is assumed to be the maximal domain.

Express the following functions in function notation with maximal domain.a {(x, y): y = x2 - 4}b y = 3x - 4, -2 ≤ x ≤ 5c y

x== 1

Think WRiTE

a The rule has meaning for all values of x (it is a quadratic), so the domain of the function is R.

a f : R → R, f (x) = x2 - 4

b The rule has meaning for all values of x in the given domain [-2, 5].

b f : [-2, 5] → R, f (x) = 3x - 4

c The rule has meaning for all values of x except 0. c f : R \ {0} → R, f (x) = 1x

WoRkEd ExAMPlE 14

y

x0 1 2 3

123456

f(x)

198

WoRkEd ExAMPlE 15

State i the domain, i i the co-domain and i i i the range for each of the following functions.

a f : R → R, f (x) = 5 - x b g : R+ → R, g(x) == 1

xThink WRiTE/dRAW

a 1 The domain is given as R. a i dom f = R

2 The co-domain is given as R. i i The co-domain is R.

3 Use a CAS calculator to obtain the graph of the function, or sketch it.

y

x0 5

5

f(x)

4 From the graph the range is observed to be R. i i i ran f = R

b 1 The domain is given as R+. b i dom g = R+

2 The co-domain is given as R. i i The co-domain is R.

3 Use a CAS calculator to obtain the graph of the function, or sketch it.

x

y

0 1

1g(x)

4 The range is observed from the graph to be R+. i i i ran g = R+

WoRkEd ExAMPlE 16 eBookpluseBookplus

Tutorialint-0289

Worked example 16

State the i maximal domain and i i the range for the function defi ned by the rule:

a y x== ++ 1 b yx

==++1

2.

Think WRiTE/dRAW

a 1 The rule has meaning for all x if x + 1 ≥ 0 (that is, contents of are positive).

a Require x + 1 ≥ 0

2 Solve this inequation. So x ≥ -1

3 State the maximal domain. i Maximal domain = [-1, ∞)

x

y

0(−1, 0)

1

y = x + 14 Use a CAS calculator, or other technology, to

obtain the graph of the function, or sketch it by plotting selected points.



5 The range is observed from the graph to be [0, ∞).

i i Range = [0, ∞)

b 1 The rule exists for all x, except when x + 2 = 0. b x + 2 ≠ 0

2 Therefore x ≠ -2. x ≠ -2

3 State the maximal domain. i Maximal domain = R \ {-2}

4 Use a CAS calculator to obtain the graph of the function, or sketch it by plotting selected points.

x

y

0−1−2

1

1——x + 2y =

5 The range is observed from the graph to be R \ {0}.

i i Range = R \ {0}

f1. (x) = . . . is used to describe ‘a function of x’. To evaluate the function, for example when x = 2, fi nd 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.2. Functions are commonly expressed using the notation3.

f :X → Y, f (x) = . . . . . .


dom 4. f is an abbreviation for the domain of f (x).ran 5. f is an abbreviation for the range of f (x).The maximal domain of a function is the largest domain for which the function will 6. remain defi ned.

REMEMBER

Function notation1 WE13

a If f (x) = 3x + 1, fi nd: i f (0), ii f (2), iii f (-2) and iv f (5).

b If g(x) = +x 4, fi nd: i g(0), ii g(-3), iii g(5) and iv g(-4),

c If g(x) = 4 - 1x

, fi nd: i g(1), ii g 1

2

, iii g

−

12

, and iv g−

15

,

d If f (x) = (x + 3)2, fi nd: i f (0), ii f (-2), iii f (1) and iv f (a).

e If h (x) = 24x

, fi nd: i h (2), ii h (4), iii h (-6) and iv h (12).

ExERCiSE

4F

eBookpluseBookplus

Digital docsSkillSHEET 4.2

SubstitutionSkillSHEET 4.3

Transposition of equations

200

2 Find the value (or values) of x for which each function has the value given.a f (x) = 3x - 4, f (x) = 5 b g(x) = x2 - 2, g(x) = 7

c f (x) = 1x

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

e g(x) = x2 + 3x, g(x) = 4 f f (x) = 8 − x , f (x) = 3

3 Given that f xx

x( ) ,= −10 find:

a f (2) b f (-5)c f (2x) d f (x2)e f (x + 3) f f (x - 1)

4 WE14 Express the following functions in function notation with maximal domain.

a {(x, y): y = 4x + 1} b y x= − 6

c yx

=−1

1d y

x=

+2

1

e y = (x + 2)2, where x ∈R+ f y = x2 + 3x, where x ≥ 2g y = 8 - x, where x ≤ 0 h y = x2 + x

5 WE15 For each of the following functions, state:i The domain ii the co-domain iii the range.

a f : {0, 1, 2, 3} → Z, f (x) = 3x - 7 b g: (0, 10] → R, g(x) = 3

x

c f : {2, 4, 6, 8, 10} → N, f (x) = x2 d f : (-∞, 0) → R, f (x) =

1− x

e g: R+ → R, g(x) = x2 - 2 f h: [-3, 3] → R, h(x) = 9 2− x

6 WE16 State the i maximal domain and ii range for the function defined by the rule:a f (x) = 3 - x b f (x) = 5 x

c y = x3 + 2 d y = 5 - 3x2

e y x= − 4 f yx

=−

1

3

Special types of function (including hybrid functions)one-to-one functionsAs 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:

y

x0

y

x0

A function which is not one-to-one; this passes the A one-to-one function; this passes the vertical line test but not the horizontal line test. vertical line test and the horizontal line test.

eBookpluseBookplus

Digital docSpreadsheet 126

Square root graphs

4GeBookpluseBookplus

eLessoneles-0077

Hybrid functions



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/dRAW

1 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.

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

c y

x0

3

1

f(x)

It is a one-to-one function.3 Write a statement to answer the question. The functions are one-to-one for b and c .

WoRkEd ExAMPlE 17

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

x0

b y

x0

c y

x0

Think WRiTE

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

Only b is a one-to-one function.

WoRkEd ExAMPlE 18

Restriction of functionsRestrictions can be placed on a function through its domain. If we have one relation, for example f (x) = x2, we can create several different functions by defining different domains. For example:

y

x0

f(x) y

x0

g(x)

−1 1

y

x0

h(x)

f : R → R, f (x) = x2 g : [-1, 1] → R, g(x) = x2 h: R+ → R, h(x) = x2

The restriction imposed on the function f to produce the function h has created a one-to-one function.

202

For each function graphed below state two restricted, maximal (largest possible) domains which make the function one-to-one.a y

x0

4

2

y = (x − 2)2 b

x

y

0

1—x2y =

Think WRiTE/dRAW

a 1 One-to-one functions will be formed if the curve is split into two through the vertical line x = 2.

a y

x0

4

2

y

x0 22 State the required domains. For the function to be one-to-one, the

domain is (-∞, 2] or [2, ∞).

b 1 One-to-one functions will be formed if the curve is split into two through the line x = 0.

b

x

y

0

x

y

0

2 State the required domains. For the function to be one-to-one, the domain is (-∞, 0) or (0, ∞).

WoRkEd ExAMPlE 19

hybrid functionsA hybrid, mixed, or piecewise defi ned function is a function which has different rules for different subsets of the domain. For example:

f xx x

x x( )

,

,=

+ ≤>

1 0

02

for

for 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.


Tutorial

Worked example 20

a Sketch the graph of f x

x x

x x

x x

( )

,

,

,

==<<

++ ≤≤ <<−− ≥≥

0

1 0 2

5 2 b State the range of f.

Think WRiTE/diSPlAy

Method 1: Technology-free

a Calculate and plot points as shown. a If x = -1, y = x = -1.

y

x0

1

−1

f(x)


int-0986


1 Sketch the graph of y = x for the domain (-∞, 0).

If x = 0, y = x = 0.

2 On the same axes sketch the graph of y = x + 1 for the domain [0, 2).

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

3 On the same axes sketch the graph of y = 5 - x for the domain [2, ∞).

If x = 2, y = x + 1 = 3.If x = 2, y = 5 - x = 3.If x = 5, y = 5 - x = 0.

y

x01 1 2 3 4 5

1

2

3

−1

f(x)

b The range is made up of (or is the union of) two sections, (-∞, 0) with (-∞, 3].

b ran f = (-∞, 3]

Method 2: Technology-enabled1

2

Use a CAS calculator. On the Graph & Tab screen complete the function entry lines as:y1 = x | x < 0y2 = x + 1 | 0 ≤ x < 2y3 = 5 - x | x ≥ 2Then press E.

Draw the graph of f (x) on the Graph & Tab screen by tapping:•!

204

A function is one-to-one if for each 1. x-value there is only one y-value and vice versa.A one-to-many function may be ‘converted to’ a one-to-one function by restricting the 2. domain.A hybrid function obeys different rules for different subsets of the domain.3.

REMEMBER

Special types of function (including hybrid functions) 1 WE17 Which of the following functions are one-to-one? Use a CAS calculator, or other tech-

nology, to obtain the graph of the function where appropriate. 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 - 1g y = x2, x ≤ 0 h g(x) = −1 2x

2 WE18 Consider the relations below and state: i which of them are functionsi i which of them are one-to-one functions.a y

x0

b y

x0

c y

x0

d y

x0

e y

x0

f y

x0

g y

x0

h y

x0

i y

x0

j y

x0

k

x

y

0

l

x

y

0

3 WE19 For each function below state two restricted, maximal domains which make the function one-to-one.

a

x

y

0−1

b

x

y

0 2

c y

x0−3 3

−3

ExERCiSE

4G



d

x

y

0

(3, 4)e y

x0−4

(−2, −2)

f

x

y

0

(−1, 4)

(1, 0)

g f (x) = 1 - x2 h g(x) = −4 2x , x ∈ [-2, 2]

i g(x) = 12x

, x ∈ R \ {0} j f (x) = (x + 3)2

4 MC Use the graph of the relation y2 = x - 1, shown below, to answer the following questions.a A one-to-one function can be formed by: A restricting the domain to R+

b restricting the domain to [1, ∞) c restricting the domain to (1, ∞) d restricting the range to [0, ∞) e restricting the range to R \ {0}

b A rule that describes a one-to-one function derived from the relation y2 = x - 1 is:

A y2 = x - 1 b y x= ± −1

c y x= −− 1 d y x= − 1

e y x= −1

5 MC Consider the following hybrid function:

f xx x

x x( )

,

,=

<≥

− 1

1

a The graph that correctly represents this function is: A

x

y

0 21

1

−1

b

x

y

0 21

1

−1

c

x

y

0 1

1

−1

d

x

y

0 1

1

−1

e

x

y

0 1

1

−1

b The range of this hybrid function is: A R b R \ {-1} c (-1, ∞)

d [0, ∞) e R+

6 WE20 a Sketch the graph of the following function:

f x xx

x x( )

,

,=

<

+ ≥

10

1 0

b State the range of f.

x

y

0 1

206

7 a Sketch the graph of the function g xx x

x x( )

,

,=

+ ≥− <

2 1 0

2 0b State the range of g.c Find

i g(-1) i i g(0) i i i g(1).

8 a Sketch the graph of the function f x

x x

x x

x x

( )

,

,

,

=− <− ≤ ≤

+ >

−

−

2 2

4 2 2

2 2

2

b State the range of z.c Find i f (-3) i i f (-2) i i i f (1) iv f (2) v f (5).

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

10 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.

11 a Sketch the graph of the function f : R → z, f (x) = (x - 3)2.b By restricting the domain of f, form two one-to-one functions that have the same rule as f

(use the largest possible domains).

12 a Sketch the graph of the function g : R → R, g(x) = x2 + 2x + 1.b By restricting the domain of g, form two one-to-one functions that have the same rule as

g (use maximal domains).

inverse relations and functionsA relation is a set of ordered pairs that can be graphed or described by a rule. The inverse of a set of ordered pairs is obtained simply by interchanging the x and y ele ments. So, the inverse of {(1, 5), (2, 6), (3, 7)} is {(5, 1), (6, 2), (7, 3)}. If these points are plotted on a set of axes, it can be seen that when each original point is refl ected across the line y = x, the inverse points are obtained.

Similarly, if the graph of a function is given, then its inverse function can be sketched by refl ecting the original function across the line y = x.

WoRkEd ExAMPlE 21

Sketch the graph of the following and then sketch the inverse.a {(3, -6), (4, -4), (5, -2), (6, 0)}b

x

y c y = x2 for x ≥ 0

x

y

0 1

123

−1−1−2

−2

f(x)

x

y

0 1

1

−1

eBookpluseBookplus

Digital docWorkSHEET 4.2

4h



Think WRiTE/dRAW

a 1 Plot the points on a set of axes. a

54321

–1–2–3–4–5–6

1–2–1–3–4–5 2 3 4 50–6 x6

y6

y = x2 Interchange the x- and y-values and plot them.

Alternatively, reflect the original points across the line y = x.So, (3, -6) → (-6, 3) (4, -4) → (-4, 4) (5, -2) → (-2, 5) (6, 0) → (0, 6)

b 1 Re-draw the given graph. b y = x

x

y

2 On the same set of axes, plot the line y = x.

3 Sketch a reflection of the original graph across the line y = x.

c Sketch the original function. The graph is a parabola with turning point (0, 0). The domain is restricted.

c

x

y

1

10

y = xy = x2

An inverse relation is obtained by interchanging the x- and y-values.The graph of a function and its inverse are reflections of each other across the line y = x.

REMEMBER

inverse relations and functions 1 WE21a Sketch the graph of the following and then sketch the inverse.

a (1, 7), (2, 5), (3, 3)} b (1, 3), (2, 6), (3, 9)} c (-2, 11), (0, 6), (2, 1)}

2 WE21b Sketch the graph of the following and then sketch the inverse. (Assume each set of axes has the same scale for x and y.)

a

x

y b

x

y c

x

y

ExERCiSE

4h

208

d

x

y e

x

y f

x

y

3 WE21c Sketch the graph of the following and then sketch the inverse.

a y = 4x b y = x2 + 3, x ≥ 0

c y = 1

2x + 1 d y = x3 + 4

CirclesA circle is a many-to-many relation.

The rule that defi nes a circle with its centre at (0, 0) and of radius r is

x2 + y2 = r 2

The graph of this circle is shown at right.The vertical-line test clearly verifi es that the circle graph is not a

function.Solving the equation for y we have y2 = r 2 - x2, so y r x= −2 2

or y r x= −− 2 2 .These two relations represent two semicircles that together make a complete circle. y r x= −2 2 is the ‘upper semicircle’ (above the x-axis).

y r x= −− 2 2 is the ‘lower semicircle’ (below the x-axis).

x

y

r

r−r

y = r2− x2

x

y

−r

r−r

y = − r2 − x2

WoRkEd ExAMPlE 22

Sketch the graphs of the following relations.a x2 + y2 = 16b x2 + y2 = 9, 0 ≤ x ≤ 3

c y x== −−8 2

Think dRAW

a 1 This relation is a circle of centre (0, 0) and radius = 16 = 4.

a y

x0−4 4

−4

42 On a set of axes mark x- and y-intercepts

of -4 and 4.

3 Draw the circle.

4i

eBookpluseBookplus

Digital docInvestigation

A special relation

x

yr

r

−r

−r



b 1 This relation is part of a circle of centre (0, 0) and radius = 9 = 3.

b y

x0 3

−3

3

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

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

c 1 This relation is an ‘upper semicircle’ (as y > 0) of centre (0, 0) and radius = 8.

c y

x0

8

− 8 8

2 On a set of axes mark the x-intercepts of − 8 and 8 and y-intercepts of 8 .

3 Draw a semicircle above the x-axis.

General equation of a circleThe general equation of a circle with centre (h, k) and radius r is (x - h)2 + (y - k)2 = r 2.

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

y

x0

k + r

k

k − r

h − r h h + r

Range(h, k)

Domain

(x − h)2 + (y − k)2 = r2

WoRkEd ExAMPlE 23

Sketch the graphs of the following circles. State the domain and range of each.a x2 + (y - 3)2 = 1b (x + 3)2 + (y + 2)2 = 9

Think WRiTE/dRAW

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

x0−1 1

4

2

3

x 2 + (y − 3)2 = 12 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.

3 Draw a circle which passes through these four points.

4 State the domain. Domain is [-1, 1].

5 State the range. Range is [2, 4].

210

b 1 This circle has centre (-3, -2) and radius 3. b

x

y1

−1

−2

−3

−4

−5

−1−2−3−4−5−6 10

2 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.

3 Draw a circle which passes through these four points.

4 State the domain. Domain is [-6, 0].

5 State the range. Range is [-5, 1].

Note: When using a CAS calculator to plot circle graphs, ensure that the upper and lower values are entered as separate equations on the Graphs & Geometry page; for example,

f1(x) = −− 16 2x and f2(x) = −16 2x .

The general equation of a circle with centre (1. h, k) and radius r is (x - h)2 + (y - k)2 = r2.An ‘upper semicircle’ with centre (0, 0) and radius 2. r is y = −r x2 2 .

A ‘lower semicircle’ with centre (0, 0) and radius 3. r is y = −− r x2 2 .

REMEMBER

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

a y

x0−3 3

−3

3b y

x0−1 1

−1

1

c y

x0−5 5

−5

5

d y

x0−10 10

−10

10

e y

x0

6

− 6

− 6 6

f y

x0−2 2

−2 2

2 2

2 2

g y

x0−3 3

3

h y

x0−4 4

−4

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

3 WE 22 Sketch the graphs of the following relations.a x2 + y2 = 4 b x2 + y2 = 16 c x2 + y2 = 49

d x2 + y2 = 7 e x2 + y2 = 12 f x2 + y2 = 14

ExERCiSE

4i

eBookpluseBookplus

Digital docSpreadsheet 007Circle graphs



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

a y = ± −81 2x b y = 4 2− x c y = − −1 2x

d y = 1

92− x e y = − −1

42x f y = 5 2− x

g y = ± −10 2x h x2 + y2 = 3, − 3 ≤ x ≤ 0

5 MC Consider the circle below.

a The equation of the circle is:A x2 + (y - 2)2 = 4 b (x - 2)2 + y2 = 16c (x + 2)2 + y2 = 16 d (x - 2)2 + y2 = 4e (x + 2)2 + y2 = 4

b The range of the relation is:A R b [-2, 2] c [0, 4]d [2, 4] e [-2, 1]

6 MC Consider the equation (x + 3)2 + (y - 1)2 = 1. a The graph which represents this relation is:

A y

x0−3−2

1

4

−6

b y

x0 3

12

2 4

c y

x0−3

12

−4 −2

d y

x0−1−2

32 4

e y

x0−3

2

1

−3.5 −2.5

b The domain of the relation is:A [-3.5, -2.5] b (-4, -2)c R d [2, 4]

e [-4, -2] 7 WE 23 Sketch the graph of the following circles. State the domain and range of each.

a x2 + (y + 2)2 = 1 b x2 + (y - 2)2 = 4c (x - 4)2 + y2 = 9 d (x - 2)2 + (y + 1)2 = 16e (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 - 1

2)2 + (y + 3

2)2 = 9

4

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.

y

x0 2

2

−2

4

212

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 fi nd the radius (in cm) of the ripple in each caseb calculate how fast the ripple is moving outwards.(State your answers to 1 decimal place.)

Functions and modellingWhen 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) = p 2.

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


Tutorialint-0291

Worked example 24

The table describes hire rates for a removal van. a Express the cost as a hybrid function.b Sketch the graph of the function.

4J

Hours of hire (h) Cost ($C)

Up to 3 200

Over 3 up to 5 300

Over 5 up to 8 450



Think WRiTE/dRAW

a 1 The cost is $200 if 0 < h ≤ 3. a

2 The cost is $300 if 3 < h ≤ 5.

3 The cost is $450 if 5 < h ≤ 8.

4 State the cost function C(h). C h

h

h

h

( )

,

,

,

=< ≤< ≤< ≤

200 0 3

300 3 5

450 5 8

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

b C ($)

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

250300

150200

50100

350400450

When using functions to model situations:form an equation involving one variable and sketch a graph1. use the graph to determine domain and range etc.2.

REMEMBER

Functions and modelling 1 WE 24 The cost of hiring a paper recycling removalist is described

in the fol lowing table:

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 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.

Distance d (km)Up to 50 km

50 to 100 km

100 to200 km

200 to 700 km

Over 700 km

Cost $C 0.40 0.60 0.80 1.70 2.00

a State the cost function in terms of the distance.b Sketch the graph of the function.

ExERCiSE

4J

214

3 A car travels at a constant speed of 60 km/h for 11

2 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 i i 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 $21 600 and $58 000 is $6264 plus 30 cents for every dollar earned over $21 600.a Write a rule for the tax payable, $T, for a taxable income, $x, where

21 601 ≤ x ≤ 58 000.b Sketch a graph of this function.c Calculate the tax paid on an income of $32 000.

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

(x + 4) m

(x − 1) m

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 func tion N tt

( ) = ++

1596

3 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 nb sketch the graph of T versus nc find the admission price which will give the maximum takings.

216

SuMMARy

Set notation

{. . .} refers to a set.•∈• 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.Z• 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.

a b

(a, b]

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.•

Power functions (hyperbola, truncus and square root function)

The graph of 1. yx

= 1 is called a hyperbola.

The graph of • ya

x bc=

−+ is the graph of the basic hyperbola, dilated by the factor

of a in the y-direction, translated b units horizontally (to the right if b > 0 or to the left if b < 0) and c units vertically (up if c > 0 or down if c < 0). If a < 0, the graph is reflected in the x-axis. The equations of the asymptotes are: x = b and y = c. The domain of the function is R \ {b} and its range is R \ {c}.



The graph of2. yx

= 12

is called a truncus.

The graph of • ya

x bc=

−+

( )2 is the basic truncus curve, dilated by a factor of a in the

y-direction and translated b units along the x-axis (to the right if b > 0 or to the left if b < 0) and c units along the y-axis (up if c > 0 or down if c < 0). If a is negative, the graph is reflected in the x-axis. The vertical asymptote is x = b. The horizontal asymptote is y = c. The domain is R \ {b}.The range is y > c if a > 0, or y < c if a < 0.

The graph of the function 3. y a x b c= − + is the graph of y x= , dilated by the factor of a in the y-direction and translated b units along the x-axis and c units along the y-axis.

If • a < 0, the basic graph is reflected in the x-axis.The end point of the graph is (• b, c).The domain is • x ≥ b.The range is • y ≥ c for a > 0, or y ≤ c for a < 0.

If • y a b x c= − + , the domain is x ≤ b; the graph of y a x= is reflected in the y-axis.

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.•Functions are commonly expressed using the notation•

f :X → Y, f (x) = . . . . . .


dom • f is an abbreviation for the domain of f ( x).ran • f is an abbreviation for the range of f ( x).The maximal domain of a function is the largest domain for which the function will remain defined.•

Special types of function (including hybrid functions)

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

Inverse relations and functions

An inverse relation is obtained by interchanging the • x- and y-values of the original relation.The graph of a function and its inverse are reflections of each other across the line • y = x.

circles

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

(x - h)2 + (y - k)2 = r2

An ‘upper semicircle’ with centre (0, 0) and radius• r is y r x= −2 2 .

A ‘lower semicircle’ with centre (0, 0) and radius • r is y r x= −− 2 2 .

Functions and modelling

When using functions to model situations:•form an equation involving one variable and sketch a graph1. use the graph to determine domain and range etc.2.

218

ChAPTER REviEW

ShoRT AnSWER

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

Time, t (hours) 1 2 3 4 5

No. of cars, n 30 75 180 330 500

a Plot these points on a graph.b Explain why the dots cannot be joined.c Estimate the number of cars in the park

21

2 hours after the car park opens.

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

b State the domain and range of this relation.

3 State the implied domains of the following functions.

a y x= b yx

= 5c y x= −12

4 If g x x( ) ,= + 2 where x ≥ 0, then find:a g ( x2)b the domain and range of g ( x).

5 Determine which of the following relations are functions.a y = 2x2 - 1 b 3x + y = 2c x = y2 + 1 d x2 + y2 = 10e y3 = x f y2 - x2 = 1

6 Express the following rules in full function notation.

a yx

= 1b y x= −( )2

7 Sketch the graph of the function described below.

f x

x x

x

x x

( )

,

,

,

=− ≤

< <− ≥

−

−

2 1

3 1 3

2 5 3

8 Sketch the graph of each of the following, stating the domain and range.

a y x= −− 1 2

b (x - 2)2 + (y + 1)2 = 9

9 a Sketch the graph of the relation x2 + y2 = 100.b From this relation form two one-to-one

functions (with maximal domains) and state the domain and range of each.

10 A chicken farmer delivers chicken manure according to the following fee schedule:

Less than half a truckload: $50 Half to a full truckload: $75 More than 1 but less than 2 truckloads: $100 Sketch a graph showing this informations.

11 Sketch a graph for the following equation

yx

=+

+23

1

12 State the dilation factor, reflections and translations that have occurred to each of the following equations.

a f x x( ) = −− 2 4

b f xx

( ) = +52

2

c f xx

( ) =−

+− 3

27

MulTiPlE ChoiCE

1 If A = {-2, -1, 0, 1, 2, 3} and B = {-2, 0, 2, 4, 6} then A ∪ B is:A {-2, -1, 0, 1, 2, 3, 4, 6} b {-2, 0, 2}c {-1, 1, 3, 4, 6} d {-1, 1, 3}e ∅

2 Which of the following statements is false?A Z ⊂ Q b 3.142 ∈ Qc p ∈ R d {0, 1, 2, 3} ∈ Ne (N ∪ Z) = Z

3 The rule describing the relation shown is:A y = 2xb y = 2x, x ∈ {1, 2, 3, 4}c y = 2x, x ∈ N

d yx=2

e y = 2x, x ∈ R+

4 Which one of the following graphed relations is continuous?A y

x0

y

x0 1 2 3 4

2

4

6

8



b y

x0

c y

x0

d y

x0

e y

x0

5 The interval shown below is:

x−5 −1 410

A [-5, -1] ∪ [0, 4]b [-5, -1) ∪ [0, 4]c (-5, -1) ∪ (1, 4]d (-5, -1) ∪ (1, 4)e [-5, -1) ∪ (1, 4]

6 The set R+ \ {2} is correctly represented on which number line below?A

x20

bx20

cx20

dx20

ex20

7 The domain of the relation shown below is:y

x0

2

4

1

A R \ {0, 1} b R \ {1}c R d Z \ {1}e R+ ∪ R-

8 A relation has the rule y = x + 3, where x ∈ R+. The range of this relation is:A R+ b R+ \ {3}c [3, ∞) d Re (3, ∞)

9 The implied domain of the relation described by the

rule yx

=−

1

5 is:

A (5, ∞) b R+ c [5, ∞)d (0, 5) e R-

10 The range of the function, f x x( ) = −2 4 is:A R b R+ c R-

d [0, ∞) e (2, ∞)

11 The relation shown is:y

x0

A one-to-one b one-to-manyc many-to-many d many-to-onee none of the above

12 Which of the following is not a relation?A y = x2

b x2 + y2 = 3c {(1, 1), (2, 1), (3, 2), (4, 3)}d y = 5 - xe {1, 3, 5, 7, 9}

13 Which one of the following graphed relations is not a function?A y

x0

b y

x0

220

c y

x0

d y

x0

e y

x0

14 Which of the following rules does not describe a function?

A yx=5

b y = 2 - 7xc x = 5d y = 10x2 + 3e y = -8

15 Which of the functions listed below is not one-to-one?A {(10, 10), (11, 12), (12, 13)}b {(5, 8), (6, 10), (7, 8), (8, 9)}c {(x, y): y = 4x}d {(x, y): y = 5 - 2x}e f ( x) = 2 - x3

16 Which of the graphs below represents a one-to-one function?A y

x0

b y

x0

c y

x0

d y

x0

e y

x0

17 The function with the domain -5 ≤ x ≤ 5, range -8 ≤ y ≤ 17 and rule y = x2 - 8 can be written in function notation as:A f :R → R where y = x2 - 8b f :R → R where f (x) = x2 - 8c f :[-5, 5] → R where f (x) = x2 - 8d f :R → [-8, 17] where f (x) = x2 - 8e f :[-8, 17] → R where f (x) = x2 - 8

18 The function f : {x: x = 0, 1, 2} → R, where f ( x) = x - 4, may be expressed as:A {(0, -4), (1, -3), (2, -2)}b {0, 1, 2}c {(0, 4), (1, 3), (2, 2)}d {(-1, -5), (1, -3), (2, -2)}e {-4, -3, -2}

19 If g ( x) = 6 - x + x2, then g (-2) is equal to:A 6 b 8 c 0d 12 e 5

20 If f ( x) = 3x - 5, then f (2x + 1) is equal to:A 6x - 8 b 6x - 5 c 3x - 5d 3x - 4 e 6x - 2

21 The graph below is to be restricted to a one-to-one function. A possible restricted domain could be:

y

x0–3 –1 3

9

A [-3, 1) b (1, 3) c (1, ∞)d (3, ∞) e (-3, ∞)

22 The hybrid function

f x

x x

x x

x x

( )

,

,

,

=+ <

≤ ≤− >

1 0

0 2

2 2

2

is represented by which of the following graphs?A y

x0

1

4

1−1 2

b y

x0

1

4

−1 2



c y

x0

1

4

2

d y

x0

1

4

−1 2

e y

x0

1

4

−1 2

23 The inverse of the graph shown below is:

x

y

A

x

y

b

x

y

c

x

y

d

x

y

e

x

y

24 The graph of the circle relation (x - 2)2 + (y + 1)2 = 4 is:A

x

y

2

1

−1

−2

−4

−3

−1 1 2 3 4 50

b

x

y

1

−1

−2

−3

−1 1 2 3 40

c

x

y

4321

−1−2−3−4−5

1−2−1−3 2 3 4 5 6 70

222

d

x

y

56

4321

−1−2−3−4

1−2−1−3−4−5−6−7 2 30

e

x

y

3

4

2

1

−1

−2

−1−2−3−4−5 10

25 The equation of the circle shown is:y

x0

−2

2

51 3

A (x + 3)2 + y2 = 4 b (x - 3)2 + y2 = 2c (x + 3)2 + y2 = 2 d (x - 3)2 + y2 = 4e x2 + (y - 3)2 = 4

The circle with equation (x + 1)2 + (y - 4)2 = 9 applies to questions 26 and 27.

26 The domain is:A [-10, 8] b [-2, 4] c (-2, 4)d [-3, 3] e [-4, 2]

27 The range isA [-7, -1] b [-5, 13] c [1, 7]d [-3, 3] e [1, 7]

28 A circle has its centre at (4, -2) and a radius of 5. The equation of the circle is:A (x - 4)2 + (y + 2)2 = 25b (x - 4)2 + (y + 2)2 = 5c (x + 4)2 + (y - 2)2 = 5d (x + 4)2 + (y - 2)2 = 25e 4x2 - 2y2 = 5

29 The graph which best represents the function f :[-2, 2] → R where f (x) = 4 2− x is:A

x

y2

−2

−2 20

b

x

y2

−2

−2 20

c

x

y2

−2

−2 20

d

x

y4

−4

−4 40

e

x

y4

−4

−4 40



30 The table of maths tutoring fees charged by a Year 11 student is as follows:

Hours (h) Charge (C$)

0 < h ≤ 2 50

2 < h ≤ 4 80

4 < h ≤ 6 100

Which of the following graphs best shows the information in the preceding table?A c

h0 2 4 6

50

100b c

h0 2 4 6

50

100

c c

h0 2 4 6

50

100

d c

h0 2 4 6

50

100

e c

h0 2 4 6

50

100

31 The graph of the function f is obtained from the graph of the function y x= by a refl ection in the y-axis followed by a dilation of two units from the x-axis.

The rule for f is:

A f x x( ) = −2 b f x x( ) = − 2

c f x x( ) .= − 0 5 d f x x( )f x( )f x = − 0 5.0 5.

e f x x( ) = −2

[VcAA 2003]

32 The function f : [a → ∞) with rule f (x) = 2(x - 3)2 + 1 will have an inverse function if:A a ≤ -3 b a ≥ -3 c a ≥ 1d a ≤ 3 e a ≥ 3

[VcAA 2005]

33 Part of the graph of the function with rule

ya

x bc=

++

( )2 is shown below.

y

x

12

10 2

The values of a, b and c respectively are:

a b c

A 2 -1 0

b -2 -1 2

c 2 1 1

d 2 -2 1

e -2 1 2

[VcAA 2005]

ExTEndEd RESPonSE

1 Consider the diagram shown at right.a Find an expression for the area, A, in terms of x and y.b Find an expression for the perimeter, P.c If the perimeter is 72 cm, express A as a function of x.d What is the domain of A( x)?e Sketch the graph of this function.f Hence fi nd the maximum area.

10 m

y mx m

x m

224

2 For the graph at right:a state the domainb state the rangec fi nd the rule for x ∈ (-∞, -2)d fi nd the rule for x ∈ (-2, 0]e fi nd the rule for x ∈ [0, 3], given it is of the form y = ax2

f determine the rule when x ≥ 3g describe the relation using hybrid function notation

of the form f x( ) . .

.

=

… …… …… …

3 A function f is defined as follows: f :[-2, a] → R, where f (x) = (x - 1)2 - 4.a Find f (-2), f (-1), f (0), f (1), f (3).b If f (a) = 12, fi nd the value of a.c Sketch the function f, labelling the graph appropriately.d From the graph or otherwise, state the: i domain of f (x) ii range of f (x).

4 The perimeter for a new rectangular penguin enclosure is to have a maximum side length of 8 m. The width is to be twice the length (x).a Draw a diagram of the enclosure and label the sides.b Defi ne a rule which gives the perimeter P, of the new enclosure.c What is the largest value that x can be?d State the domain and range.e Write in function notation the rule for the perimeter.f Defi ne the function for the area of the enclosure, A(x).g If the maximum area allowed is 18 m2, fi nd the dimensions of the enclosure.

5 Thomas is looking to connect to a mobile phone service. He has to decide on one of two plans from Busytone Communications. The details are as follows.

Plan Flag fallCost per

minute (cents)

A 20 30

B 50 20

a Write a function A, for the cost of making a call from plan A.b Write a function B for the cost of making a call from plan B.c What is the cost of a 2-minute call from: i plan A? ii plan B?d What is the length of one call costing $5 from: ii plan A? ii plan B?e How long would a call be for the call costs to be the same for both plans?f If Thomas frequently makes calls lasting more than 4 minutes, which plan should he connect with?

y

x0 4−2

18

4

(3, 18)

eBookpluseBookplus

Digital docTest YourselfChapter 4



eBookpluseBookplus ACTiviTiES

chapter openerDigital doc

10 Quick Questions: Warm up with ten quick •questions on relations, functions and transformations (page 170)

4c domain and rangeTutorial

WE 7 • int-0287: Watch how to sketch relations and state their domain and range using interval notation (page 180)

Digital docs

SkillSHEET 4.1: Practise stating the domain and •range of relations (page 182)WorkSHEET 4.1: Use set notation and interval •notation, recognise appropriate specifi c number fi elds, recognise whether graphs are discrete or continuous and state the domain and range of relations (page 183)Investigation: Investigate some interesting relations •using a graphing program to sketch them (page 183)

4d Types of relations (including functions)Tutorial

WE 8 • int-0288: Watch a tutorial on identifying the type of relation given a graph (page 184)

4e Power functions (hyperbola, truncus andsquare root function)

Interactivity

Domain and range • int-0263: Consolidate your understanding of domain and range for relations and functions (page 187)

4F Function notationDigital docs

SkillSHEET 4.2: Practise substituting values into •functions (page 199)SkillSHEET 4.3: Practise transposing equations •(page 199)Spreadsheet 126: Investigate the graph of a square •root function (page 200)

Tutorial

WE 16 • int-0289: Watch a tutorial on stating maximal domain and range for functions with defi ned rules (page 198)

4G Special types of function (including hybrid functions)

eLesson

Hybrid functions eles-0077: Watch an eLesson to •learn more about hybrid functions (page 200)

Tutorial

WE 20 • int- : Watch a tutorial on sketching the graph of a hybrid function and stating its range (page 202)

Digital doc

WorkSHEET 4.2: Recognise types of relations and •functions, determine maximal domain and range, identify codomains and sketch graphs of relations (page 206)

4I circlesDigital docs

Investigation: Investigate the graph of • x2 + y2 = 25 by constructing a table of values and plotting the graph (page 208)Spreadsheet 007: Investigate graphs of circle •relations (page 210)

4J Functions and modellingTutorial

WE 24 • int-0291: Watch a tutorial on expressing the hire costs of a removal van as a hybrid function and sketching the graph of the function (page 212)

chapter reviewDigital doc

Test Yourself: Take the end-of-chapter test to test •your progress (page 224)

To access eBookPLUS activities, log on to

www.jacplus.com.au

0986

4a set notation 4b relations and graphs 4c 4d 4e relations ...mathsbooks.net/jacplus books/11...

Documents