relations and functions - mathsbooks.netmathsbooks.net/maths quest 11 methods/by chapter/ch... ·...

6In thisIn this chachapterpter6A Set notation6B Relations and graphs6C Domain and range6D Types of relations

(including functions)6E Function notation6F Special types of function6G Circles6H Functions and modelling

VCEVCEcocovverageerageArea of studyUnit 1 • Functions and graphs

Relations andfunctions

264

M a t h e m a t i c a l M e t h o d s U n i t s 1 a n d 2

Set notation

Set notation is used in mathematics in the same way as symbols which are used torepresent 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 setof things.

5.

The symbol

⊂

means ‘is a subset of’, for example, if

B

⊂

A

, then all of theelements 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,

A

∩

B

is the set of elementscommon to sets

A

and

B

.

8.

The symbol

∪

means ‘union’, for example,

A

∪

B

is the set of all elementsbelonging 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 thatthe set is empty.

C h a p t e r 6 R e l a t i o n s a n d f u n c t i o n s

265

Sets of numbers

Certain letters are ‘reserved’ for important sets that arise frequently in the study ofmathematics.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 asfractions 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 aratio of two whole numbers). These numbers are called

irrational

, for example,

π

,, etc.

Note that

N

⊂

J

⊂

Q

⊂

R

, 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} ∩ Be whether or not: i 8 ∈ A ii B ⊂ A iii 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. e i Yes. 8 ∈ A ii All elements of B belong to A. ii Yes. B ⊂ A iii 3 is an element of C but not A. iii No. C ⊄ A

1WORKEDExample

ab---

3

1

1–2 3–

4 2–3

7–5

33—51

2 340

–1–2 –3

...

...

...

...

R

Q

J

N

Q'

π3

266 M a t h e m a t i c a l M e t h o d s U n i t s 1 a n d 2

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} andD = {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 thata 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 numbersets N, J, Q and R.

a A ∩ B b A ∩ C c A ∩ C ∩ Dd A ∪ B e C ∪ D f A \ Cg 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 \ Hd H \ F e (F ∪ H) \ G.

A B B ∅ C {1, 2} D A ∪ B E A

A B B C C A D A ∪ B E B ∪ C

A B B ∅ C C D A ∩ B E B \ C

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

remember1. {. . .} 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

6AWORKEDExample

1

mmultiple choiceultiple choice


75

C h a p t e r 6 R e l a t i o n s a n d f u n c t i o n s 267

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

The cost of hiring a trailer depends on the number of hours for which it is hired. Wecan say that a relation exists between the number of hours and the cost. The table belowoutlines the relation.

Since the cost depends upon the number of hours, the cost is said to be thedependent variable, while the number of hours is called the independent variable. Theinformation in the table can be represented by a graph, which usually gives a betterindication of how two variables are related. When graphing a relation, the independentvariable is displayed on the horizontal (or x) axis and the dependent variable isdisplayed 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 isthe first element and y is the second element.

This graph clearly shows that the cost increases as the number of hours of hireincreases. The relation appears to be linear. That is, a straight line could be drawn thatpasses 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 hourscan be referred to as a ‘discrete dependent variable.’

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

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

Cost = 10 × number of hours + 20Using 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

Cos

t of

trai

ler

hire

($)

10 2 3 4 5 6 7 8

5040

60708090

100

Number of hours

y

x


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

a 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)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 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)

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

x0 1 2 3 4

14

8

12

16

3

1

2 y

x0–2 –1 1 2

12345

–3–2–1

y = 2x + 1

3

2WORKEDExample


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 2 4 6 81 3 5 7

20

806040

100120140160180

t (min)

R (

beat

s/m

in)2

3

4

3WORKEDExample

remember1. 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


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)}.

1The 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 certaintime, progressively increases from Monday through to Friday. Which graph belowbest 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 ∈ ND y = x − 4, x ∈ J E y = 4 − x, x ∈ {0, 1, 2, 3}

A B C

6B


y

x0 1 2 3 4

1

2

3

4y

x0 1 2 3 4

1

2

3

4y

x0 1 2 3 4

1

2

3

4

y

x0 1 2 3 4

1

2

3

4y

x0 1 2 3 4

1

2

3

4




0 M WT T F

Num

ber

of p

eopl

e

0 M WT T F

Num

ber

of p

eopl

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 d

e The relation which shows the number of student absences per day during term 3 atyour 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 eachis 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 ∈ Re y = 2x + 3, where x ∈ Jf 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 $40commission for each computer she sells. If n is the number of computers she sells perweek 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 WT T FN

umbe

r of

peo

ple

0 M WT T F

Num

ber

of p

eopl

e

y

x0

y

x0

EXCEL Spreadsheet

Plottingrelations

WORKEDExample

2

WORKEDExample

3 t (min) 0 2 4 6 8

T (°C) 80 64 54 48 44


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 therunway, 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 perstudent.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 CalculatorGraphics Calculator tip!tip! Plotting points

1. Press , selectEDIT and 1:Edit andenter the x-coordinatesof the points to beplotted in L1. Enter they-coordinates in L2 asshown below. Deleteany values in the tablethat are not required(such as those from pre-vious calculations).(Note: To clear a com-plete list, scroll to thelist heading and press

followed by.

2. Press [STATPLOT], select 1:Plot 1by pressing andensure the options areselected as shown in thescreen below:

3. Adjust WINDOW set-tings or ZOOM for asuitable view of theplotted points. Press

[FORMAT] andselect GridOn andLabelOn with ,followed by .Press to checkcoordinates.

STAT

CLEARENTER

2nd

ENTER

2nd

ENTERGRAPH

TRACE


Domain and rangeDomain and rangeA relation can be described by either:1. a listed set of ordered pairs2. a graph or3. a rule.

The set of all first elements of a set of ordered pairs is known as the domain andthe 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 ofdependent 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 arelation. 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 notationIf a and b are real numbers and a < b, then the following intervals are defined with anaccompanying number line:

The closed circle indicates that the number is included and the open circle indicatesthat 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

xa b xa b

xa xa

xb xb

xa b xa b


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 20 x–3 50 x5310

4WORKEDExample

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

x100–2

x0 1

5WORKEDExample

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

c d

Weight (kg) 10 15 20 25 30

Cost per kg ($) 3.5 3.2 3.0 2.8 2.7

y

x0

y

x0–4 4

–4

4

6WORKEDExample


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 weight 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 becauseif x < 1, y = .

a

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

When x = 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.

x 1–

1

negative number2 0

1

2 23 34

3 y

x0 1 2 3 4 5

1

–1

2y = x – 1

4

5

7WORKEDExample


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 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)

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

x0 1 2 3 4

2

–2–4

64

81012

y = x2 – 4, x ∈ [0, 4]

3

4

5

6

remember1. 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]

remember

C h a p t e r 6 R e l a t i o n s a n d f u n c t i o n s

277

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)

6CWW

ORKEDORKEDE

Examplexample4

–2 10 50

40–3 90–8

0–1 0 1

30–5 –2 4210–3

WW

ORKEDORKEDEExamplexample

5

312---

1

2-------


210 210

210 210

210

SkillSH

EET 6.1


y

x0 3 7

4

–4

–1

Ch 06 MM 1&2 YR 11 Page 277 Friday, June 29, 2001 11:20 AM


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 intervalnotation.

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


WORKEDExample

6a, b

WORKEDExample

6c, dy

x0

2

–3

y

x0

2y = 2exy

x0

2

–2 2

y

x0 1

y = x – 1y

x0

4 y = 4e–x2

y

x0

–3

y

x0

y = 1–x

y

x0

1

y

x0

–2

WORKEDExample

7

WorkS

HEET 6.1 x 16 x2–

1x---


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-valueand vice versa. For example:

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

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

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

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

1. {(−1, 1), (0, 1), (1, 2)} 2.

Interesting relationsThis 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 = 92 x3 + y3 = 13 sin (x2 + y2) = 14 x2 – y2 = 15 7x2 – 6 xy + 13y2 = 166 x4 = x2– y2

7 x2 + y2 < 258 x2 + y2 > 259 9 < x2 + y2 < 36

10 x sin x + y sin y < 1

3

y

x1–1

–1

1

y

x0

y

x0

y

x0

The graph of y2 (1 – x) = x2(x + 1) produced by GrafEq.


Many-to-many relationsA many-to-many relation exists if there is more than one x-value for any y-value andvice versa. For example:

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

x0

y

x0

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

x0

y

x0

y

x0

1

y

x0x = –1

F

2

y

x0

y = 1

1

2

1

2

8WORKEDExample


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

1. 2.

Vertical line testA function is determined from a graph if a vertical line drawn anywhere on the graphcannot intersect with the curve more than once.

y

x0

y

x0

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

x0

y

x0

9WORKEDExample

remember1. 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

x0

y

x0

remember

Function Not a function


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

6DWORKEDExample

8y

x0

y

x0

y

x0

y

x0

y

x0

y

x0

y

x0

y

x0

y

x0

y

x0

y

x0

y

x0

WORKEDExample

9


y

x0

y

x0


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 eachfunction.

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+

B R and R

C R and R−

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


y

x0

1

–1

y

x0

1

–1

y

x0

1

1

y

x0

1

1

y

x0

1

–1

x 1+


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.

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

Fully defining functionsTo fully define a function one must:1. define the domain, and2. state the rule.That is, if a function f (x) has domain X, the function may be defined as follows:

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

= −2

b Write the rule. b f (x) = x2 − 3Substitute x = −2 into the rule. f (−2) = (−2)2 − 3Simplify. = 4 − 3

= 1

c Write the rule. c f (x) = x2 − 3Substitute x = a into the rule. f (a) = a2 − 3

d Write the rule. d f (x) = x2 − 3Substitute x = 2a into the rule. f (2a) = (2a)2 − 3Simplify the expression if possible. = 22a2 − 3

= 4a2 − 3

123

123

12

123

10WORKEDExample

f :X → Y, f (x) = . . . . . .

Domain Co-domain Rule

C h a p t e r 6 R e l a t i o n s a n d f u n c t i o n s 285Y is not necessarily the range but is a set which contains the range, called the

co-domain. The co-domain gives the set of possible values which y can be. It is usuallyR (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 domf and the range as ran f.

For example, the function defined by {(x, y): y = 2x, x ∈ [0, 3]} can be expressed infunction 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] as the specified domain allows (x = 0 gives y = 0 and x = 3 givesy = 6, which are the minimum and maximum values of y).

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 isdefined. The letters f, g and h are usually used toname 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 maximaldomain.

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]

y

x0 1 2 3

123456

f (x)

Express the following functions in function notation with maximal domain.

a {(x, y): y = x2 − 4} b y = 3x − 4, −2 ≤ x ≤ 5 c y =

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---

1x---

11WORKEDExample


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

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

THINK WRITE

a The domain is given as R. a iii dom f = RThe co-domain is given as R. iii The co-domain is R.Use a graphics calculator to obtain the graph of the function, or sketch it.

iii

From the graph the range is observed to be R.

iii ran f = R

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

The co-domain is given as R. iii The co-domain is R.Use a graphics calculator to obtain the graph of the function, or sketch it.

iii

The range is observed from the graph to be R+.

iii ran g = R+

1

x-------

1

2

3 y

x0 5

5

f (x)

4

1

2

3

x

y

0 1

1g(x)

4

12WORKEDExample

State the i maximal domain and ii the range for the function defined by the rule:

a b .

THINK WRITE

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

a Require x + 1 ≥ 0

Solve this inequation. So x ≥ −1State the maximal domain. ii Maximum dom = [−1, ∞)

y x 1+= y1

x 2+------------=

1

2

3

13WORKEDExample


THINK WRITE

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

ii

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

ii ran = [0, ∞)

b The rule exists for all x, except whenx + 2 = 0.

b x + 2 ≠ 0

Therefore x ≠ −2. x ≠ −2

State the maximal domain. ii Maximum dom = R \ {−2}

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

ii

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

ii ran = R \ {0}

4

x

y

01

1

y = x + 1

5

1

2

3

4

x

y

0–1–2

1

1——x + 2y =

5

remember1. 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.

2. Functions are completely described if the domain and the rule are given.3. Functions are commonly expressed using the notation

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

will remain defined.

f :X → Y, f (x) = . . . . . .


remember


Function notation

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 Express the following functions in notation with maximal domain.

a {(x, y): y = 4x + 1} b

c d

e y = (x + 2)2, where x ∈R+ f y = x2 + 3x, where x ≥ 2

g y = 8 − x, where x ≤ 0 h y = x2 +

5 For each of the following functions, statei the domain ii the co-domain iii the range.

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

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

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

6 State the i maximal domain and ii range for the function defined by the rule:

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 f (x) = 3 − x b f (x) = c y = x3 + 2

d y = 5 − 3x2 e f

6E

SkillSH

EET 6.2

Mathca

d

Function notation

WORKEDExample

10 x 4+

1x--- 1

2---

–12---

–15---

24x

------

SkillSH

EET 6.3

1x---

8 x–

f x )( 10x

------= x,–

WORKEDExample

11 y x 6–=

y1

x 1–-----------= y

2

x 1+----------------=

x

WORKEDExample

12

Mathca

d

Singlefunction grapher

3

x-------

EXCEL

Spreadsheet

Mathca

d

Square root graphs

x2--- 1

x–----------

9 x2–

WORKEDExample

13 5 x

y x 4–= y1

x 3–----------------=


Special types of functionOne-to-one functionsAs we have already seen, one-to-one relations and many-to-one relations are functions. Aone-to-one function has, at most, one y-value for any x-value and vice versa. The graph ofa 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

x0

y

x0

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

x0

3

1

f (x)

3

14WORKEDExample

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 cross the graph only once.

Only b shows a one-to-one function.

y

x0

y

x0

y

x0

15WORKEDExample


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 differentdomains. For example:

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.

y

x0

f (x)y

x0

g(x)

–1 1

y

x0

h(x)

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

THINK WRITE

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

a

State the required domains. For the function to be one-to-one, the domain is (−∞, 2] or and [2, ∞)

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

b

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

y

x0

4

2

y = (x – 2)2

x

y

0

1—x2y =

1 y

x0

4

2

y

x0 2

2

1

x

y

0 x

y

0

2

16WORKEDExample


Hybrid functionsA 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 graphof f (x) is shown at right.

f x( )x 1,+x2,

=for x 0≤for x 0>

y

x0

1

–1

f (x)

17WORKEDExample

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

x01 1 2 3 4 5

1

2

3

–1

f (x)

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

x 1,+5 x,–

=x 0<0 x 2<≤x 2≥

remember1. A function is one-to-one if for each x-value there is only one y-value and vice

versa.2. A one-to-many function may be ‘converted to’ a one-to-one function by

restricting the domain.3. A hybrid function obeys different rules for different subsets of the domain.

remember


Special types of function

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

2 Consider the relations below and state:i which of them are functions ii which of them are one-to-one functions.

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

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) =

a b c d

e f g h

i j k l

a b c

d e f

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

i g(x) = , x ∈ R \ {0} j f (x) = (x + 3)2

6FWORKEDExample

14

1 x2–

WORKEDExample

15y

x0

y

x0

y

x0

y

x0

y

x0

y

x0

y

x0

y

x0

y

x0

y

x0

x

y

0x

y

0

WORKEDExample

16

x

y

0–1

x

y

0 2

y

x0–3 3

–3

x

y

0

(3, 4)y

x0–4

(–2, –2)x

y

0

(–1, 4)

(1, 0)

4 x2–

1x2-----


Use the graph of the relation y2 = x − 1, shown below, to answer the followingquestions.a A one-to-one function can be formed by:

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

5

Consider the following hybrid function:

a The graph which correctly represents this function is:

b The range of this hybrid function is:

6 a Sketch the graph of the function

b State the range of f.


b State the range of g.c Find i g(−1) ii g(0) iii g(1).

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}

A y2 = x − 1 B CD E

A B C

D E

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


x

y

0 1

y x 1–±= y x–= 1–y x 1–= y x 1–=


f x( )x,–

x,

=x 1<x 1≥

x

y

0 21

1

–1x

y

0 21

1

–1 x

y

0 1

1

–1

x

y

0 1

1

–1 x

y

0 1

1

–1

WORKEDExample

17

Mathcad

Hybridfunctions

f x( )1x---,

x 1,+

=x 0<

x 0≥

g x( )x2 1,+2 x,–

=x 0≥x 0<



b State the range of z.c Find i f (−3) ii f (−2) iii 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).

The following screen dumps show how a two-piece hybrid or piecewise defined func-tion 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 >.

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

f x( )x 2,–

x2 4,–

x 2,+

=x 2–<

2 x 2≤ ≤–

x 2>

x

y

0 1

123

–1–1–2

–2

f (x)

x

y

0 1

1

–1

WorkS

HEET 6.2

Graphics CalculatorGraphics Calculator tip!tip! Piecewise defined functions

2nd


A special relationYou 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 getx2 + 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. 12 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?

EXCEL Spreadsheet

Circularrelations

Mathcad

Circularrelations

Coordinate pairs

x y x2 y2 x2 + y2On 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


CirclesA 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 = r 2

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

yr

r

–r

–ry r2 x2–=

y r2 x2––=

y r2 x2–=y r2 x2––=

x

y

r

r–r

y = r2– x2

x

y

–r

r–r

y = – r2– x2

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.

8 x2–

116 4=

y

x0–4 4

–4

42

3

19 3=

y

x0 3

–3

32

3

1

8

y

x0

8

– 8 8

28 8

83

18WORKEDExample


General equation of a circleThe 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 andlower values are entered as separate equations in Y1 and Y2 and then use ZOOM andselect 5:ZSquare to show the graph in true proportion.

y

x0

k + r

k

k – r

h – r h h + r

Range(h, k)

Domain

(x – h)2 + (y – k)2 = r 2

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 = 9THINK WRITEa 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

x0–1 1

4

2

3

x 2 + (y – 3)2 = 1

2

3

451 y

x0–3

–2

–5

1

–6

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

2

3

45

19WORKEDExample

remember1. The general equation of a circle with centre (h, k) and radius r is

(x − h)2 + (y − k)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 r2 x2–=

y r2 x2––=

remember


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 functionor 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 = 49d 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 = 16D (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]

6G

Mathca

d

Circle graphs

EXCEL

Spreadsheet

Circle graphs

y

x0–3 3

–3

3y

x0–1 1

–1

1

y

x0–5 5

–5

5

y

x0–10 10

–10

10

y

x0

6

– 6

– 6 6

y

x0–2 2

–2 2

2 2

2 2

y

x0–3 3

3

y

x0–4 4

–4

WORKEDExample

1814---

y 81 x2–±= y 4 x2–= y 1 x2––=

y 1

9--- x2–= y 1

4--- x2––= y 5 x2–=

y 10 x2–±= x2 y2+ 3, 3 x 0≤ ≤–=


y

x0 2

2

–2

4


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 andrange of each.

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

10 Circular ripples are formed when a water drop hits the surface of a pond. If one rippleis 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 caseb calculate how fast the ripple is moving outwards.(State your answers to one decimal place.)

A B C

D E

A [−3.5, −2.5] B (−4, −2) C RD [2, 4] E [−4, −2]

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 = 9g (x + 5)2 + (y − 4)2 = 36 h (x − )2 + (y + )2 =


y

x0–3–2

1

4

–6

y

x0 3

12

2 4

y

x0–3

12

–4 –2

y

x0–1–2

32 4

y

x0–3

2

1

–3.5 –2.5

WORKEDExample

19

12--- 3

2--- 9

4---


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 thefunction A(r) = π r2.

For a circle to be drawn the radius needs to be a positive number. Hence the domainis (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. aThe 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

250300

150200

50100

350400450

20WORKEDExample

a Express the cost as a hybrid function.b Sketch the graph of the function.

rememberWhen using functions to model situations:1. form an equation involving one variable and sketch a graph2. use the graph to determine domain and range etc.

remember


Functions and modelling

1 The cost of hiring a paper recyclingremovalist is described in thefollowing table:

a State the cost function, $C, interms of the time, t hours, forhiring 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 thentravels 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 placedin 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

6HWORKEDExample

20

12---


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 timberis 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 parklandt 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 perperson. On average, for every $1 rise in admission price, 50 less people attend theconcert. 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.

(x + 4) m

(x – 1) m

N t( ) 1596

t 3+-----------+=


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.

summary

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.

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

• 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

• A function is one-to-one if for each 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 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

(x − h)2 + (y − k)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.

f :X → Y, f (x) = . . . . . .


r2 x2–

r2 x2–


Multiple choice

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

2 Which of the following statements is false?

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

D y =

E y = 2x, x ∈ R+

4 Which one of the relations graphed below is continuous?

5 The interval shown below 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 ∅

A J ⊂ Q B 3.142 ∈ QC π ∈ R D {0, 1, 2, 3} ∈ NE (N ∪ J) = J

A B C

D E

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]

CHAPTERreview

6A

6A

6By

x0 1 2 3 4

2

4

6

8

x2---

6By

x0

y

x0

y

x0

y

x0

y

x0

6Cx–5 –1 410


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

7 The domain of the relation shown below is:

8 A relation has the rule y = x + 3, where x ∈ R+. The range of this relation is:

9 The implied domain of the relation described by the rule is:

10 The range of the function, f (x) = 2 is:

11 The relation shown is:

12 Which of the following is not a relation?

A B

C D

E

A R \ {0, 1} B R \ {1}C R D J \ {1}E R+ ∪ R−

A R+ B R+ \ {3}C [3, ∞) D RE (3, ∞)

A (5, ∞) B R+ C [5, ∞) D (0, 5) E R−

A R B R+ C R− D [0, ∞) E (2, ∞)

A one-to-one B one-to-many C many-to-manyD many-to-one E none of the above

A y = x2 B x2 + y2 = 3 C {(1, 1), (2, 1), (3, 2), (4, 3)}D y = 5 − x E {1, 3, 5, 7, 9}

6Cx20 x20

x20 x20

x20

6Cy

x0

2

4

1

6C

6C y1

x 5–----------------=

6C 4 x–

y

x0

6D

C h a p t e r 6 R e l a t i o n s a n d f u n c t i o n s 30713 Which one of the relations graphed below is not a function?

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

A B y = 2 − 7x C x = 5 D y = 10x2 + 3 E y = −8

15 Which of the functions listed below is not one-to-one?

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

17 The function f : {x: x = 0, 1, 2} → R, where f (x) = x − 4, may be expressed as:

18 If g(x) = 6 − x + x2, then g(−2) is equal to:

A B C

D E

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

A B C

D E

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}

A 6 B 8 C 0 D 12 E 5

6Dy

x0

y

x0

y

x0

y

x0

y

x0

6Dy

x5---=

6D

6Dy

x0

y

x0

y

x0

y

x0

y

x0

6E

6E


19 If f (x) = 3x − 5, then f (2x + 1) is equal to:

20 The hybrid function

is represented by which of the following graphs?

21 The equation of the circle shown is:

The circle with equation (x + 1)2 + (y − 4)2 = 9 applies to questions 22 and 23.

22 The domain is:

23 The range is:

24 A circle has its centre at (4, −2) and a radius of . The equation of the circle is:

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

A 6x − 8 B 6x − 5 C 3x − 5 D 3x − 4 E 6x − 2

A B C

D E

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

A [−10, 8] B [−2, 4] C (−2, 4) D [−3, 3] E [−4, 2]

A [−7, −1] B [−5, 13] C [1, 7] D [−3, 3] E (1, 7)

A (x − 4)2 + (y + 2)2 = 25 B (x − 4)2 + (y + 2)2 = 5 C (x + 4)2 + (y − 2)2 = 5D (x + 4)2 + (y − 2)2 = 25 E 4x2 − 2y2 = 5

Hours (h) Charge (C$)

0 < h ≤ 2 50

2 < h ≤ 4 80

4 < h ≤ 6 100

6E

6Ff x( )

x 1,+x2,

2 x,–

=x 0<0 x 2≤ ≤x 2>

y

x0

1

4

1–1 2

y

x0

1

4

–1 2

y

x0

1

4

2

y

x0

1

4

–1 2

y

x0

1

4

–1 2

6Gy

x0

–2

2

51 3

6G

6G

6G 5

6H

C h a p t e r 6 R e l a t i o n s a n d f u n c t i o n s 309Which of the following graphs best shows the information in the preceding table?

Short answer1 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.

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 Determine which of the following relations are functions.

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

5 Express the following rules in full function notation.

a

b

6 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.

A B C

D E

Time, t (hours) 1 2 3 4 5

No. of cars, n 30 75 180 330 500

a y = 2x2 − 1 b 3x + y = 2c x = y2 + 1 d x2 + y2 = 10e y3 = x f y2 − x2 = 1

c

h0 2 4 6

50

100c

h0 2 4 6

50

100c

h0 2 4 6

50

100

c

h0 2 4 6

50

100c

h0 2 4 6

50

100

6B

12---

6B

6D

6Cx

6Ey

1x---=

y 2 x–( )=

6G


7 Sketch the graph of the function described below.

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

ab (x − 2)2 + (y + 1)2 = 9

9 A chicken farmer delivers chicken manure according to the following fee schedule:Less than half a truckload: $50Half to a full truckload: $75More than 1 but less than 2 truckloads: $100Sketch a graph showing this information.

Analysis1 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 find the maximum area.

2 For the graph below:

a state the domainb state the rangec find the rule for x ∈ (−∞, −2)d find the rule for x ∈ (−2, 0]e find 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

.

6Ff x( )

2 x,–

3,

2x 5,–

=x 1–≤

1 x 3< <–

x 3≥

6Fy 1 x2––=

6H

10 m

y mx m

x m

y

x0 4–2

18

4

(3, 18)

testtest

CHAPTERyyourselfourself

testyyourselfourself

6

f x( )…,

…,

…,

=………

relations and functions - mathsbooks.netmathsbooks.net/maths quest 11 methods/by chapter/ch... ·...

Documents