Upcoming SlideShare
×

# Functions

1,207 views

Published on

1 Like
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

Views
Total views
1,207
On SlideShare
0
From Embeds
0
Number of Embeds
87
Actions
Shares
0
43
0
Likes
1
Embeds 0
No embeds

No notes for slide

### Functions

1. 1. 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
2. 2. 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
3. 3. 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
4. 4. 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
5. 5. ( b ) Ordered pairs{( Faouzi,Netball),( } ( c ) Graphs Set Q Netball Set P Fauzi
6. 6. 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
7. 7. ( b ) Graphs Set Q 7 5 4 Set P 8 10 14( b ) Ordered pairs {（8, 4 ), (10, 5 ) , (14, 7 ) }
8. 8. 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.
9. 9. 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 }
10. 10. 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 }
11. 11. 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.
12. 12. 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
13. 13. 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
14. 14. 1.2 Functions Learning Outcomes: At the end of this lesson, students should be able to recognise function as a special relation.
15. 15. 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
16. 16. 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
17. 17. 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)
18. 18. 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.
19. 19. 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
20. 20. 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.
21. 21. 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
22. 22. 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
23. 23. 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
24. 24. 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
25. 25. 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
26. 26. 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
27. 27. 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
28. 28. Exercise 1.2.4 ( page 10)1. 2. 3. 4. 5. 6Skill practice 1.2 (page 10) 1. 2. 4. 5. 7 819-1-2009
29. 29. 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
30. 30. 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
31. 31. 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
32. 32. 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
33. 33. 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 :xx 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
34. 34. 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
35. 35. 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