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Chapter 1 FunctionsLearning Outcomes:At the end of this lesson, students should be able torepresent a relation using( a ) arrow diagrams, ( b ) ordered pairs ( c ) graphs
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1.1 RelationsA represents the set of races in MalaysiaB represents the set of festivals in MalaysiaA = { Malay, Chinese, Indian }B = { Hari Raya Aidilfitri, Chinese New Year, Deepavali} A B The elements of set A are associated with the elements of set B as depicted by diagram shown on the left. This association between A and B is called a relation from A to B
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1.1 RelationsA represents the set of races in MalaysiaB represents the set of festivals in MalaysiaA = { Malay, Chinese, Indian }B = { Hari Raya Aidilfitri, Chinese New Year, Deepavali} B A Celebrating Festivals Chinese New Year Malay Chinese Hari Raya Aidilfitri Indian Deepavali
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1.1.1 Representing a Relation If set P is the set of students in Form 4 Newton and set Q is their favourite sports. P={ } Q={ } The relation between set P and set Q is favourite sports. This relation can also be represent by using ( a ) Arrow diagram P Q Favourite Sports
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( b ) Ordered pairs{( Faouzi,Netball),( } ( c ) Graphs Set Q Netball Set P Fauzi
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Given that set P ={8, 10, 14} and set Q={4, 5, 7}. The relationfrom set P to set Q is a factor of. Represent the above relation byusing ( a) an arrow diagram ( b) graphs ( c) ordered pairs ( a ) Arrow diagram factor of Q P 8 4 10 5 14 7
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( b ) Graphs Set Q 7 5 4 Set P 8 10 14( b ) Ordered pairs {（8, 4 ), (10, 5 ) , (14, 7 ) }
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1.1.2 Identifying the Domain, Codomain,Object, Image andRange of a RelationLearning Outcomes: At the end of this lesson, students should be able to ( a ) identify domain, codomain, object, image and range of a relation.
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The arrow diagram below shows a relation one quarter of from set M to set N N M One quarter of 8 2 12 3 20 5 24 6 36 7Domain= { 8, 12, 20, 24, 36 } Object = 8, 12, 20, 24Codomain= { 2, 3, 5, 6, 7 } Image = 2, 3, 5, 6Range = { 2, 3, 5, 6 }
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Given that set R={ 2, 3, 4, 5} and set S={4, 7,9, 16, 25}. The relation is square of from set R to set S N M square of 2 4 3 7 4 9 5 16 25Domain= { 8, 12, 20, 24, 36 } The image of 3 = 9Codomain= { 2, 3, 5, 6, 7 } Range = { 4, 9, 16, 25The object which has 16 as its image = 4 }
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1.1.3 Classifying RelationLearning Outcomes: At the end of this lesson, students should be able to ( a ) classify relations into one to one, many to one, one to many and many to many relation.
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State the type of relations for following arrow diagram(a) Multiple of (b) Examination Kadir 3 9 Nabil PMR 4 12 Siu Lin 5 15 Muthu SPM 6 18 many to one One to one (d) Factor of(c) Factor of 4 2 4 64 3 6 6 4 8 8 24 One to many Many to many
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State the type of relations for following ordered pairs(a) { (3, 6), (3,9), (4,8),(5, 10)}(b) {(Ahmad,Science), (Brian, Science),(Chandran, Mathematics)}(c ) { ( a, 3), (b, 5),(b, 6), (c, 8)｝(d) Set B 8 6 4 2 Set A 3 5 9 12
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1.2 Functions Learning Outcomes: At the end of this lesson, students should be able to recognise function as a special relation.
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1.2 Functions As a Special RelationFunction is a relation in which every element in the domainhas a unique image in the codomain. p Q A Multipler of B Factor of 3 9 2 12 4 4 5 15 3 6 18 6This relation is a function because This relation is not a functionevery object has only one image because object 6 has two image
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A B R S One third of Factor of 3 1 2 2 6 6 3 3 9 4 8This relation is not a function This relation is not a functionbecause not every element in the because object 6 has two imagecodomain has to be related. Set D(d) This relation is not a function 8 because object 5 has two 6 image 4 2 3 5 9 Set C 12
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Exercise 1.1.3 Page 51. ( a), ( b ) , ( c ) , ( d )2. ( a ) , ( b )Skill Practice 1.1 Page 61 ( a) (b) ( c ) 2 ( a) (b) 3 ( a) (b) (c) 4 ( a) (b) (c) (d)
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1.2.2 Expressing Function Using Function Notation Learning Outcomes: At the end of this lesson, students should be able to express functions using function notations.
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A function f from set A to set B is denoted by f : A BThis mean that all the elements in set A are mapped intoset B by function f.The function f which maps x to 2 x 3 is written as : f : x 2 x 3 or f ( x) 2 x 3 A function can be represented by lower-case alphabet such as f , g h and others
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f : x 2 x is read as “function f maps x to 2x”,f ( x) 2x is read as “ 2x is the image of x under the function f”, or f of x is equal to 2x.
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Write the functions below by using function notation. Let the function be g. Square root The notation is: 4 2 g : x ｘ or g ( x) x 9 3 16 4 x g x 4 2 9 3 16 4
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Write the functions below by using function notation.(a) (b) Marks Half of Science 82 2 1 4 2 History 75 8 4 68 English( c) Price of tickets f Child RM3 (d) 1 4 4 Adult RM7 1 RM4 8 15 Senior 1 citizen 15 8
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1.2.3 Determine Domain, codomain, object image andrange of a FunctionThe arrow diagram shows the function f : x 3x 1 x 3x 1 f 1 4 4 1 8 15 1 15 8Domain codomain
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1. Given that f ( x) 3x 1， find the value of f(0), f(3) and f(10) f ( x) 3x 1 x 3x 1 f (0) 3(0) 1 f 1 0 1 3 29 f (3) 3(3) 1 10 8 8 f (10) 3(10) 1 29
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1. Given that f ( x) 7 cos x， find the value of x=0 and 0 x=60 f ( x) 7 cos x x 3x 1 f f (0) 7cos (0) 7(1) 0 7 7 3.5 600f (600 ) 7cos(600 ) 1 7 2 3.5
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2. Given that h( x) 1 3x ， find the value of h(-3) and h(5) h( x) 1 3x x 1-3xh( 3) 1 3( 3) h 10 -3 10 5 14 h(5) 1 3(5) 14 14
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3. Given that f ( x) 3x 4， find the value if (b) f ( x) 10(a) f ( x) 11 f ( x) 10 3 x x )4 f( 11 3x 10 4 3x 11 4 3x 6 3x 15 x 2 x 5 x 3x 4 f 5 11 -2 10
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Composite FunctionsGiven that f ( x) x 3 and g ( x) 2 x 1, find (a) fg fg f [ g ( x)] f [ g ( x)] x g ( x) 3 f(2 x( x1]) g ) 2x 1 3 2x 4
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Composite FunctionsGiven that f ( x) x 3 and g ( x) 2 x 1, find (a) gf gf g[ f ( x)]g [ f ( x)] 2 x 1 (x [ f 3) 2x 3) 1g ( x ( x)] 2x 7
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Composite FunctionsGiven that f ( x) x 3 and g ( x) 2 x 1, find (a) f 2 (b) f 4 f 2 f f f 4 f 2f 2 2 2 f [ f ( x)] f [ f ( x)] 2 f ( x 3) f ( x 6) xx 3 3 xx 6 6 2 4 f x 6 f x 12
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Given that f ( x) 2 x and g ( x) 3 2 x, find(a) fg (2) (b) gf ( 2) fg ( x) f [ g ( x)] gf ( x) g[ f ( x)] f (3 2 x) g ( 2 x) x 2 (3 2x) 3 2 x) (2 x fg ( x) 6 4x gf ( x) 3 4x fg (2) 6 4(2) gf ( 2) 3 4( 2) 2 11
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Determine one of the Functions when the Composite Function and Other Function are Given A function f is defined by . Find the function g f :xx 1 in each of the following(a) fg : x 2x 2 3 (b) gf : x2 3x 5 2 2 g[ f ( x)] x 3x 5 f [ g ( x)] 2x 3 Let y x 1 2g (x ) 1 x 2x 3 x y 1 2 g ( y) ( y 1)2 3( y 1) 5 g ( x) 2x 2 g ( y) y2 2 y 1 3y 3 5 g ( y) y2 y 3 g ( x) x2 x 3
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Exercise 1.3.2 ( Page14) 3-1-09 to 6-1-09(1) Given that f ( x) x 3 and g ( x) 3x 1, find (a) Find the composite functions fg and gf fg 3x 2 gf 3x 8 (b) What are the value of fg(2), gf(-3) and gf(-5) 8 1 7(2) Given that f ( x) 4 x 5, find f 2 16 x 20 The composite function f 2, Hence, find 2 1 (a) f and f 2 ( 2) (b) value of x which f 2 ( x) 9 2 12 11 28 16
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Exercise 3-1-09 to 6-1-09(1) if f : x x 1, find the function g such that 2 fg : x x 2x 4 g ( x) x2 2x 5 2(2) if f : x x 5, find the function g such that gf : x 2x2 9 g ( x) 2 x 1
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