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Add Math Form 4 Question

In: Computers and Technology

Submitted By asyrafzaman
Words 2371
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ADDITIONAL
MATHEMATICS
MODULE 1

FUNCTIONS
Organized by
Jabatan Pelajaran Pulau Pinang 2006

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CHAPTER 1 : FUNCTIONS
Contents
1.1 Concept map

Page
2

1.2 Determine domain , codomain , object, image and range of relation

3

1.3 Classifying the types of relations

3

2.1 Recognize functions as a special relation.
2.2 Expressing functions using function notation. 2.3 Determine domain , object , image and range 4-5

3.0 Composite Functions

6 -9

4.0 SPM Questions

9 – 10

5.0 Assessment test

11 – 12

6.0 Answers

13 – 14

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CONCEPT MAP

FUNCTIONS

Relations

Object images ………….,,
……………
……………

Functions

Function
Notation

Type of relation

y
Or
………………

Composite
Functions

Inverse
Functions

f: x

One to one Many to one ………..

fg ( x ) = …………….

Object

f(x)=y
 ………………

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1.1 Functions
Express the relation between the following pairs of sets in the form of arrow diagram, ordered pair and graph.
Arrow diagram
Ordered pair
Graph
a ) Set A =
 Kelantan, Perak ,
Selangor 
Set B =  Shah Alam
, Kota Bharu ,Ipoh 
Relation: ‘ City of the state in Malaysia ‘ b )Set A
=  triangle,rectangle, pentagon 
Set B =  3,4,5 
Relation : ‘ Number of
Sides’

1.2 Determine domain , codomain , object, image and range of relation.
List down the domain , codomain , objects , images and the range of the following relation
.

3

9

2

5

1

4

-2

3

-3

1

Set P

Diagram 1

Set Q

Domain =  ……………………………………… 
Codomain =  ……………………………………… 
Object
=……………………
Image
=……………………
Range
=…………………...

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1.3 Classifying the types of relations
State the type of the following relations
a)

x
2

x

3

4

2

9

16

4

x
X

4

2

-2
-3

36

6

……………………………………………..
c)

x

b)

x

X2

………………………………………….
Type of number

d)
4
9

3
4

2
-3

Prime
Even

-3

9

……………………………………………..

……………………………………………

2.0 Functions
2.1 Recognize functions as a special relation.
2.2 Expressing functions using function notation.
2.3 Determine domain , object , image and range
2.1 Identify each of the following relations is a function or not.

a)

A

B

A

B

b)

p

1

q

2

r

3

A

B

c) p a

a

p

q

b

b

q

r

c

c

r

d

……………………………

……………………………

……………………………

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2.2 Express each of the following functions using function notation.

a)

A
2

b)

B

A

B

4

2

3

6

4

8

c)

B

4

1

5

3

9

2

7

4

Function notation f : x  …………….. or f ( x ) = ……………

A

16

3

9

Function notation g : x  …………….. or g ( x ) = ……………

Function notation h : x  …………….. or h ( x ) = ……………

2.3 a)Find the image for each of the following functions.
( i ) f : x  2x + 9

( ii ) f : x 

f (5 ) =

5x  3
2

x
+6
5 find h ( -2 )

iii) h : x 

f (-3 ) =

…………………………

…………………………

…………………………..

b ) Find the object for each of the following functions. i )f : x  2x – 3 , find the object when the image is 5.

2x  8
, find the
3
object when the image is 3.

ii )f : x 

6
– 7 , find the x object when the image is -5.

i )f : x 

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c ) Find the value of x for each of the following function.
8
2x  1 for which g (x) = 4

ii )

i ) f ( x ) = 2x + 7 for which f ( x ) = 3

g(x)=

iii)

x 3
2
for which h ( x ) = x

h(x) =

3.0 Composite Functions

g

f

a

b

c

fg



f: a  b g: b  c gf : a  c

3.1 ( a ) Find the value for each of the following composite functions

i ) f( x ) = x + 2 and g ( x) = 5x + 3 find fg ( 2 ) =

ii ) g ( x ) = 2 +5x

iii) f(x) = 3x+

find g2(4)
=

1 and g ( x )
2

1
.Find fg ( 1 ) x 1

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( b ) Find the following composite function.

( i ) f( x ) = 2x + 3 g( x ) = 1 – x fg ( x ) =

( ii ) f( x ) = 2x + 3 g( x ) = 2 + 5 x2 gf ( x ) =

( iii ) f ( x ) = 1 g(x)=

x
2

4 x fg ( x ) =

( c) Find the value of x for each of the following composite function.
3
x g ( x ) = 2x + 1

i) f ( x ) =

fg ( x ) = 5

ii )f( x ) = 2x + 4

iii) f ( x ) = 1 -

g ( x ) = x -2

g (x ) =

fg ( x ) = 2

4 x fg ( x ) = -1

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x
,
2

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( d ) Solve i )Given the function f: x  4x + k , g : x x – 2 fg : x  mx + 8

ii )Given the function f: x  9 – 2x , g : x  ax + b and fg: x  1 – 6x

iii)Given the function f: x  2x – 1 , g : x  4x and gf : x  ax + b

Find the value of k and m

Find the value of a and b.

Find the value of a and b

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4.0 SPM QUESTIONS
SPM 2004 ( paper 1, question no 1 )
1. Diagram 1 shows the relation between set P and set Q

 w

d

x

e

y

f

z
Set P

Set Q
Diagram 1

State
( a ) the range of the relation,
( b ) the type of relation.
Answer:

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[2 marks]

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SPM 2004 ( paper 1, question no 3)
1. Given the function h ( x ) =

6
, x  0 and the composite function hg ( x ) = 3x , find x (a)g(x)
( b ) the value of x when gh ( x ) = 5.
Answer:

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[ 4 marks]

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SPM 2005 Question
1. In Diagram 1 , the function h maps x to y and the function g maps y to z. x y

h

g

z

8
5
2

Diagram 1

Determine
( a ) gh ( 2 )
Answer:

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5.0 Assessment( 30 minutes)
1)
P

a

q

b

r

c

s
A
Diagram 1

B

The diagram above shows the relation between set A and set B. State
a) the type of relation
b) the range of relation
Answer:

2) Given that f : x  2x + 7 find the object when image is 3.
Answer:

3) Given that f ( x ) = 10 – kx and f ( 2 ) = 4 ( k constant ) . Find the value of k.
Answer:

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4) Given that f ( x ) = 4x -1 and g ( x ) = 2x + 3 .
Find
i ) fg ( x ) ii ) fg ( - 2 )
Answer:

5 ) Given the function f : x  px + 2 and g : x  qx + 3 . If the composite function fg is such that fg ( x ) = 8x + 8 , find the values of p and q.
Answer:

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6.0 ANSWERS
1.1
Arrow diagram

Ordered pair
KB

Kel



SA

Sel

(kel, kb), (Sel,
S.Alam), (
Perak,Ipoh) 

Ipoh

Per

A

Graph
KB
SA
Ipoh

B

Kel



3

Tri

(Tri, 3), (rec,
4), ( Pen,5) 

3
4

rec

Per

5
4

5

Pen

Sel

3
A

B

Pen

rec

Tri

1.3

Domain =  -3,-2,1,2,3  , codomain =  1,3,4,5,9  , Object =1,2,3,-2,-3
Image = 1,3,4,5,9 , Range =  1,4,9 
( a ) one to one ( b ) One to many( c ) many to one( d ) many to many

2.1

( a ) function

2.2

(a) 2x (b) x2 (c) 2x + 3

2.3

a) ( I ) 19

(ii) -6 (iii ) 3

3.1

a) ( i) 15

(ii ) 102

b) (i) 5 – 2x

(ii) 20x2+60x+47

1.2

( b ) Not function

1
(ii) 1
5
d) (i) k=16,m=4

c) (i) -

4.0 SPM QUESTIONS
SPM 2004( P1,Q1) a)
SPM 2004( P1,Q3)
SPM 2005( P1,Q1)

( c ) function

b) (i) 2 (ii)

1
1
(iii) 3 c) (i) -2 (ii)
(iii) -3
2
2

( iii) 2
(iii) 1-

2 x (iii) 1
(ii) a=3,b=4

(iii) a=8,b= - 4

range =  x,y



(a) g(x) =

2
, x 0
3

b )many to one

( b) x = 15

8

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5.0 Assessment( 30 minutes)
1)

( a)many to many

2)

(i) 8x+11
(ii) -5
1
p = , q = 16
2

5)

p,q,r



k=3

4)



-2

3)

( b ) range =

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ADDITIONAL
MATHEMATICS
MODULE 2

FUNCTIONS

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CHAPTER 1 : FUNCTIONS
Contents

Page

1.0 Inverse Functions ( concept map )

2–4

2.0 Absolute Function

4-6

3.0 SPM Questions

7–8

4.0 Assessment ( 30 minutes )

9 – 10

5.0 Answers

11 – 13

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CHAPTER 1 : FUNCTIONS

FUNCTIONS
Ordered pairs
Arrow diagram
Relations

Functions

graph

Object images Domain
Codomain ,
Range

Function
Notation

Type of relation

f: x

y

Or f(x) = y

One to one Many to one image

Inverse
Functions

Composite
Functions

fg ( x ) = f [ g(x) ]

Object

One to many Many to many

f(x)=y
 f-1( y ) = x

1.0 Inverse Functions
1.1 Determine the object by inverse mapping
1.2 Determine the inverse functions a ) Find the inverse function of each of the following functions. x ii) f ( x ) =
i)f(x)=x+3
5

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iii ) f ( x ) =

3x  1
2

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iv ) f ( x ) = 7 – 5x

v)f(x)=

3 x
4

vi ) f( x ) =

5  4x
3

b ) Find the inverse function of each of the following functions in terms of p and q

i ) f ( x ) = px - q

ii ) f ( x ) =

x p q http://mathsmozac.blogspot.com
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iii) f ( x ) = px +

1 q http://sahatmozac.blogspot.com

c ) Given the function f : x 

x2
, x  1 and g(x) =2x -6 , find f-1 g . x 1

Answer:

d ) Inverse function f is define by f-1 : x 

x 5
1
, x  . Find f ( 2 )
2x 1
2

Answer:

2.0 Absolute Function
1. Sketch the graph of each of the following functions

a)f(x)=x

b)f(x)=x+1

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c) f (x ) = | x |

d)f(x)=|x+1|

e ) f (x) = |x| + 1

f)f(x)=|x|-1

2.Sketch the graph of each the following functions and state the corresponding range.
a) f : x  2x – 3 for 0  x  4

b) f : x  |2x – 3| for 0  x  4

f(x)

f(x)

Range :……………………………….

Range : …………………………………………

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c) f : x  | 5 – 2x |for -1  x  4

d) f : x  |9 – 2x| for 0  x  6

f(x)

Range :……………………………….

Range :……………………………….

e) f : x  |2x| – 1 for -1  x  3

f ) f : x  | 3x | for -2  x  2

Range :……………………………….

Range :……………………………….

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3.0 SPM QUESTIONS
SPM 2004 Question.
1. Given the functions h : x  4x + m and h-1 : x  2kx +

5
, where m and k are
8

constants, find the value of m and of k.
[ 3 marks]
Answer:

SPM 2005 ( Paper 1, Question 1 )
2. In Diagram 1 , the function h maps x to y and the function g maps y to z. x y

h

g

z

8
5
2

Diagram 1

Determine
( a ) h-1 ( 5 )
Answer:

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SPM 2005 ( Paper 1, Question 2 )
1.The function w is defined as w ( x ) =
( a ) w-1 ( x ),
( b ) w-1 ( 4 ).
Answer:

5
, x  2.
2 x

[ 3 marks]

SPM 2005 ( Paper 1, Question 3 )
1.The following information refers to the functions h and g.

h: x g: x

 2x – 3
 4x - 1

Find gh-1 ( x ).

[ 3 marks]

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4.0 Assessment ( 30 minutes )
1.A function f is defined by f: x  6 Find
a)f(x)

1 x 2

b ) f-1 ( 5 )

Answer:

2. Inverse function f is defined by f-1 : x 

5  4x
3

find f ( 2 ).
Answer:

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3.Given the function f : x  2x - m and inverse function f-1 : x  nx +

7
3

Find the value of m and n.
Answer:

4. Sketch the graph of the function f ( x ) = |2x – 5 | for 0  x  6. Hence , state the corresponding range.
Answer:

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5.0 ANSWERS
1.1

2x  1
7x
5  3x
(iv)
(v) 3 – 4x (vi)
3
5
4
xq x 1
(b)
(ii ) xq + p (iii) p p pq
2x  4
7
( c)
, x
7  2x
2
( d ) -1

( a )( i) x- 3 ( ii) 5x (iii)

Absolute function f(x) a)

f(x)

b)

_1
|
-1

x

f(x)

c)

x

f(x)

d)

_1
|
-1

x

f(x)

e)

x

f(x)

f)

_1
|
-1

|
-1

x

x

|
1

_ -1

|
1

|
2 x

2.
a)

f(x)

b)

_8

_

.

f(x)
_5

|
1

|
2 x

-3

 3  f ( x)  8

0  f(x)  5

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c)

f(x)
_7

f(x)

d)

_9

|
2

|
4

|
3 x

0  f(x)  7

0  f(x)  9

f(x)
_5

e)

|
5 x

f(x)
_6

f)

_1
|
-1

|
_ -1 1

|
3x

-1  f(x)  5

|
-2

|
2

0  f(x)  6

SPM 2004 ( P1,Q2)
1
5
K= , m=8
2
SPM 2005 ( P1,Q2)
(a) 2
SPM 2005 ( P1,Q2)
2x  5
3
(a)
, x 0 ( b ) x 4
SPM 2005 ( P1,Q3)
2x+ 5
Assessment ( 30 minutes )
1)( a) 12 – 2x ( b ) 8
1
2) 4
1
3) n = , m=7
2
4)

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x

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a)

f(x)
_7
_5

|
2

|
3

|
6

x

The corresponding range of f(x ) = 0  f(x)  7

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...Question 4 The first problem to be dealt with is if the investment is necessary or not, as there are advantages and disadvantages to the choice. One disadvantage is that there is a risk in the stability of the POS terminal after the OS upgrade, for the reason that the POS application runs stable on the DOS-system at the time. But, after upgrading the OS and porting the POS application, there is a risk of reducing the stability of the POS application due to bugs in the application (after porting). At the same time the investment costs could be avoided if choosing to remain at the old DOS system. One advantage of upgrading the OS is the fact that the PDAs can become interconnected with other stores or the headquarters. This can increase efficiency as employees will be able to look at the inventory of other stores and search for the products which are sold-out at their own store for instance. Another advantage is that the MS DOS system became obsolete and unsupported as of 2003, which means that the POS terminal will not be compatible with the POS software, and in such manner a change in the OS is inevitable. Therefore, we are in favour of investing in a new OS, and the next question is which OS is the best option. When looking in a short-term view, Linux comes out as the best option, as of €0 license cost and only an annual maintenance fee to be paid. As Zara intends to invest in the OS for long-term period, then UNIX is the best choice for a new OS, in the......

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...Week 1 DQ 1 Response 1) Given the enumeration methods; sum rule, product rule, permutations, combinations along with enumeration methods for indistinguishable objects, how can we devise a strategy to solve problems requiring these methods? A basic concept in the branch of the theory of algorithms called enumeration theory, which investigates general properties of classes of objects numbered by arbitrary constructive objects (cf. Constructive object). Most often, natural numbers appear in the role of the constructive objects that serve as numbers of the elements of the classes in question ("Enumeration", 2013). The Sum/Difference Rules refer to the derivative of the sum of two functions is the sum of the derivatives of the two functions ("Basic Derivative Rules", 2013). The product rule is one of several rules used to find the derivative of a function. Specifically, it is used to find the derivative of the product of two functions. It is also called Leibnitz's Law, and it states that for two functions f and g their derivative (in Leibnitz notation, ). The derivative of f times g is not equal to the derivative of f times the derivative of g: .The product rule can be used with multiple functions and is used to derive the power rule. The product rule can also be applied to dot products and cross products of vector functions. The Leibnitz Identity, a generalization of the product rule, can be applied to find higher-order derivatives ("Definition Of Product Rule", 2013). A......

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...short-run cost. I would also like to add that in the short-run, a firm will have fixed capital (it takes time to increase size of factories). In the short-run, the firm can vary the quantity of labor. However, in the short term, a firm is likely to experience diminishing marginal returns. This means as firms employ more workers, there will come a point where extra workers have a declining marginal product. For some producers, the short-run lasts a few days. For others, the short-run can last for decades. Jennifer, Good post and I agree with you. An explicit cost is a business expense that is predictable in occurrence and amount. Company salary and wage expenses, mortgage or rental payments and utilities are among common company explicit costs. Because of their predictability, explicit costs are easier to budget for and account for in financial recordkeeping. Companies use explicit costs in budget preparation. They total the projected amounts of explicit costs based on monthly payments or average costs for an item over time. This provides a basis for understanding the break-even point at which revenue covers explicit costs. A high number of less predictable costs makes budgeting more difficult and leads to greater fluctuation in business income. These costs are never hidden, one has to pay separately. Source: http://smallbusiness.chron.com/explicit-cost-definition-65796.html From the e-Activity, recommend whether the company in question should or should not......

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...included with question papers Ruler graduated in centimetres and Nil millimetres, protractor, compasses, pen, HB pencil, eraser. Tracing paper may be used. Instructions to Candidates_____________________________________________________ Check that you have the correct brain power required to attempt this question paper. Answer ALL the questions. Write your answers in the spaces provided in this question paper. You must NOT phone a friend or ask the audience. Anything you write on the formulae page will gain NO credit. If you need more space to complete your answer to any question, write smaller. Information for Candidates____________________________________________________ The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2). There are 23 questions in this question paper. The total mark for this paper is 110. Calculators must not be used unless the symbol appears Advice to Candidates__________________________________________________________ Show all stages in any calculations – A* questions often require you to explain or prove something. Work steadily through the paper. Do not spend too long on one question. If you cannot answer a question, leave it, attempt the next one and try not to cry. Return at the end to those you have left out. Have a lie down afterwards to help recover. GCSE A* Questions Skill:......

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...Name: Sharon Mohammed Candidate Number: School: Iere High School Class: Form 5s (Science) Project Title: Projectile Motion Additional Mathematics SBA: Teacher: Ms. Lia Chunilal Date: TABLE OF CONTENTS Project Title……………………………………………………………………………pg3 Aim of Project………………………………………………………………………..pg4 Problem Statement……………………………………………………………………pg5 Information about Volleyball…………………………………………………………pg9 Apparatus and Materials………………………………………………………………pg6 Method for Experiment……………………………………………………………….pg7 Solution to Problem…………………………………………………………………..pg8 Verification of Solution……………………………………………………………….pg18 Discussion……………………………………………………………………………..pg19 Conclusion…………………………………………………………………………….pg21 Bibliography...................................................................................................................pg22 PROJECT TITLE: To determine the ideal angle at which Penelope, a national volleyball player, needs to serve the ball in order to obtain the distance that it needs to go over the net but stay in the court y applying the theory of projectile motion. AIM: To determine the ideal angle and distance needed for a volleyball ball to be served over the net within the dimension of the court using projectile motion. PROBLEM STATEMENT: Penelope is a national volleyball player and she wants to know the angle at which she needs to serve the volleyball in order to get the ball over the net but also in the court every time she......

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...completed from data collected. Research question 1. Is the reaction time of an athlete greater than that of a non athlete? How the data was obtained 1. A random athlete was selected. 2. The athlete stands or sits near the edge of a table resting their elbow on the table so that their wrist extends over the side. 3. The ruler is held by a non athlete between the outstretched index finger and thumb of the athlete’s dominant hand, so that the top of the athlete’s thumb is level with the zero centimeter line of the ruler. 4. The non athlete releases the ruler and the athlete catches the ruler between their index finger and the thumb as quick as possible. 5. The distance between the bottom of the ruler and the top of the athlete’s thumb where the ruler has been caught was recorded. 6. Methods 3 – 5 were repeated several times (15 times with 15 different persons). 7. Methods 1 – 6 were repeated 15 different non athletes Raw Data in Centimeter Athlete Right Hand 17, 16, 17, 14, 16, 20, 16, 12, 12, 10, 6, 10, 9, 3, 9 Non- Athlete Right Hand 25, 22, 19, 22, 18, 15, 17, 15, 16, 21, 18, 16, 13, 15, 17 Stem And Leaf Diagrams Athlete Right Hand |0 |3 6 9 9 | |1 |0 0 2 2 4 6 6 6 7 7 | |2 |0 ......

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...4שאלה 4 ניתוח חמשת הכוחות של פורטר עבור יצרני התרכיז: כוח המיקוח של הלקוחות –נמוך: * המבקבקים מחזיקים בבעלותם מערך מכירות וייצור בטריטוריה גיאוגרפית אקסלוסיבית, עם זכויות המובטחות לנצח ע"י הזכייןת סיום ההסכם רק אם המבקבק מחליט. * מספר מועט של מבקבקים גדולים, שחלקם מוחזקים ע"י יצרני התרכיז * התחרות המותגית הפנימית של המשקאות הקלים ב1980 שימרה את זכויות יצרני התרכיז לתת טריטוריות אקסלוסיביות למבקבקים, ובכך נתנו פחות כוח למיקוח לקונים מהמבקבקים.. כי אין להם תחליף. * המבקבקים נעולים בחוזים המאפשרים ליצרני אתרכיז את הזכות לקבוע מחירים ותנאים אחרים של מכירה. * המבקבקים רשאים לנהל עבודה עם מותגים שאינם קולה, של יצרני תרכיז אחרים, לפי ראות עיניהם. * ניתן חופש למבקבקים בכל הקשור ללקיחת החלטה האם להשיק מוצרים חדשים של יצרני התכיז או לא אבל לא יכולים להשיק מתחרים ישירים שלהם. * התחרות על שטח מדף בערוצים הקמעונאים נותן יותר כוח מיקוח לקונים. כוח מיקוח של הספקים –נמוך: * יצרני התרכיז מנהלים מו"מ ישירות מול הספקים של המבקבקים הגדולים – בעיקר של הממתיקים והאירוז – על מנת לעודד אספקה אמינה, משלוח מהיר יותר ומחירים נמוכים יותר. * קוקה קולה ופפסי הם מבין הלקוחות הגדולים ביותר של תעשיות פחית המתכת ושומרים על יחסים עם יותר מספק אחד, מה שנותן לספקים פחות כוח מיקוח כי יש ספקים אלטרנטיביים. פחיות מתכת הם הרוב של האירוז למבקבקים – 60%. בנוסף יש שימוש בבקבוקי פלסטיק (38%) ובקבוקי זכוכית (2%). איום כניסה של מתחרים חדשים –נמוך: * חברות עם ערוצי הפצה מדלת לדלת כמו חברות חטיפים יכולות להחליט לגוון ולהכנס לתעשייה. * "עלויות החלפה" נמוכות ללקוחות שמסכנים מעט...

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