FUNCTIONS Organized by Jabatan Pelajaran Pulau Pinang 2006 ADDITIONAL MATHEMATICS MODULE 1 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
1 CHAPTER 1 : FUNCTIONS Contents Page 1.1 Concept map 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 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
2 CONCEPT MAP Functions Function Notation Composite Functions Inverse Functions Relations Type of relation f: x y Or ……………… Object images ………….,, …………… …………… FUNCTIONS One to one Many to one fg ( x ) = ……………. ……….. Object f ( x ) = y ……………… http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
3 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 . Domain = ……………………………………… Codomain = ……………………………………… Object =…………………… Image =…………………… Range =…………………... 3 2 -2 5 4 3 9 Set P Diagram 1 Set Q -3 1 1 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
4 1.3 Classifying the types of relations State the type of the following relations a) …………………………………………….. b ) …………………………………………. c) …………………………………………….. d) …………………………………………… 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) …………………………… b) …………………………… c) …………………………… x 4 16 2 2 x 36 4 6 A B p q r 1 2 3 A B p q r a b c d A B a b c p q r 3 2 4 9 4 9 Prime -3 -3 x 3 2 4 -2 9 -3 Even x X x X 2 Type of number http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
5 2.2 Express each of the following functions using function notation. a) Function notation f : x …………….. or f ( x ) = …………… b ) Function notation g : x …………….. or g ( x ) = …………… c) Function notation h : x …………….. or h ( x ) = …………… 2.3 a)Find the image for each of the following functions. ( i ) f : x 2x + 9 f (5 ) = ………………………… ( ii ) f : x 2 5x 3 f (-3 ) = ………………………… iii) h : x 5 x +6 find h ( -2 ) ………………………….. b ) Find the object for each of the following functions. i )f : x 2x – 3 , find the object when the image is 5. ii )f : x 3 2x 8 , find the object when the image is 3. i )f : x x 6 – 7 , find the object when the image is -5. A B 2 3 4 4 6 8 A B 2 3 4 4 9 16 A B 1 2 3 5 7 9 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
6 c ) Find the value of x for each of the following function. i ) f ( x ) = 2x + 7 for which f ( x ) = 3 ii ) g ( x ) = 2 1 8 x for which g (x) = 4 iii) h(x) = 2 x 3 for which h ( x ) = x 3.0 Composite Functions 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 find g2 (4) iii) f(x) = 3x+ 2 1 and g ( x ) = 1 1 x .Find fg ( 1 ) a b c fg f: a b g: bc gf : a c f g http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
7 ( 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 - 2 x g ( x ) = x 4 fg ( x ) = ( c) Find the value of x for each of the following composite function. i) f ( x ) = x 3 g ( x ) = 2x + 1 fg ( x ) = 5 ii )f( x ) = 2x + 4 g ( x ) = x -2 fg ( x ) = 2 iii) f ( x ) = 1 - 2 x , g (x ) = x 4 fg ( x ) = -1 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
8 ( d ) Solve i )Given the function f: x 4x + k , g : x x – 2 fg : x mx + 8 Find the value of k and m ii )Given the function f: x 9 – 2x , g : x ax + b and fg: x 1 – 6x Find the value of a and b. iii)Given the function f: x 2x – 1 , g : x 4x and gf : x ax + b Find the value of a and b http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
9 4.0 SPM QUESTIONS SPM 2004 ( paper 1, question no 1 ) 1. Diagram 1 shows the relation between set P and set Q State ( a ) the range of the relation, ( b ) the type of relation. [2 marks] Answer: d e f x y z w Set P Set Q Diagram 1 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
10 SPM 2004 ( paper 1, question no 3) 1. Given the function h ( x ) = x 6 , x 0 and the composite function hg ( x ) = 3x , find ( a ) g ( x ) ( b ) the value of x when gh ( x ) = 5. [ 4 marks] Answer: http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
11 SPM 2005 Question 1. In Diagram 1 , the function h maps x to y and the function g maps y to z. Determine ( a ) gh ( 2 ) Answer: x y z 2 5 8 h g Diagram 1 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
12 5.0 Assessment( 30 minutes) 1) 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 : x2x + 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: a b c q r s P A B Diagram 1 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
13 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: http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
14 6.0 ANSWERS 1.1 Arrow diagram Ordered pair Graph (kel, kb), (Sel, S.Alam), ( Perak,Ipoh) (Tri, 3), (rec, 4), ( Pen,5) 1.2 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 1.3 ( a ) one to one ( b ) One to many( c ) many to one( d ) many to many 2.1 ( a ) function ( b ) Not function ( c ) function 2.2 (a) 2x (b) x2 (c) 2x + 3 2.3 a) ( I ) 19 (ii) -6 (iii ) 3 b) (i) 2 (ii) 2 1 (iii) 3 c) (i) -2 (ii) 2 1 (iii) -3 3.1 a) ( i) 15 (ii ) 102 ( iii) 2 b) (i) 5 – 2x (ii) 20x2 +60x+47 (iii) 1- x 2 c) (i) - 5 1 (ii) 1 (iii) 1 d) (i) k=16,m=4 (ii) a=3,b=4 (iii) a=8,b= - 4 4.0 SPM QUESTIONS SPM 2004( P1,Q1) a) range = x,y b )many to one SPM 2004( P1,Q3) (a) g(x) = 3 2 , x 0 ( b) x = 15 SPM 2005( P1,Q1) 8 Tri rec Pen 4 5 3 A B Kel Sel Per Ipoh SA KB A B KB SA Ipoh Kel Sel Per 3 3 4 5 rec Tri Pen http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
15 5.0 Assessment( 30 minutes) 1) ( a)many to many ( b ) range = p,q,r 2) -2 3) k = 3 4) (i) 8x+11 (ii) -5 5) p = 2 1 , q = 16 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
16 FUNCTIONS ADDITIONAL MATHEMATICS MODULE 2 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
17 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 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
18 CHAPTER 1 : FUNCTIONS 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. i ) f ( x ) = x + 3 ii) f ( x ) = 5 x iii ) f ( x ) = 2 3x 1 Functions Function Notation Composite Functions Inverse Functions Relations Type of relation f: x y Or f(x) = y f ( x ) = y f -1 ( y ) = x Object images Domain Codomain , Range FUNCTIONS Arrow diagram Ordered pairs graph One to one One to many Many to one Many to many fg ( x ) = f [ g(x) ] image Object http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
19 iv ) f ( x ) = 7 – 5x v ) f ( x ) = 4 3 x vi ) f( x ) = 3 5 4x 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 ) = q x p iii) f ( x ) = px + q 1 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
20 c ) Given the function f : x 1 2 x x , x 1 and g(x) =2x -6 , find f -1 g . Answer: d ) Inverse function f is define by f-1 : x 2 1 5 x x , x 2 1 . Find f ( 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 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
21 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 Range :………………………………. b) f : x |2x – 3| for 0 x 4 Range : ………………………………………… f(x) f(x) http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
22 c) f : x | 5 – 2x |for -1 x 4 Range :………………………………. d) f : x |9 – 2x| for 0 x 6 Range :………………………………. e) f : x |2x| – 1 for -1 x 3 Range :………………………………. f ) f : x | 3x | for -2 x 2 Range :………………………………. f(x) http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
23 3.0 SPM QUESTIONS SPM 2004 Question. 1. Given the functions h : x 4x + m and h-1 : x 2kx + 8 5 , where m and k are 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. Determine ( a ) h-1 ( 5 ) Answer: x y z 2 5 8 h g Diagram 1 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
24 SPM 2005 ( Paper 1, Question 2 ) 1.The function w is defined as w ( x ) = 2 x 5 , x 2. ( a ) w-1 ( x ), ( b ) w-1 ( 4 ). [ 3 marks] Answer: SPM 2005 ( Paper 1, Question 3 ) 1.The following information refers to the functions h and g. Find gh-1 ( x ). [ 3 marks] h: x 2x – 3 g: x 4x - 1 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
25 4.0 Assessment ( 30 minutes ) 1.A function f is defined by f: x 6 - 2 1 x Find a ) f ( x ) b ) f-1 ( 5 ) Answer: 2. Inverse function f is defined by f-1 : x 3 5 4x find f ( 2 ). Answer: http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
26 3.Given the function f : x 2x - m and inverse function f -1 : x nx + 3 7 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: http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
27 5.0 ANSWERS 1.1 ( a )( i) x- 3 ( ii) 5x (iii) 3 2x 1 (iv) 5 7 x (v) 3 – 4x (vi) 4 5 3x ( b ) p x q (ii ) xq + p (iii) p x - pq 1 ( c) x x 7 2 2 4 , x 2 7 ( d ) -1 Absolute function 2. . 3 f (x) 8 0 f(x) 5 f(x) x b ) _ 5 | 1 | 2 f(x) x a ) _ 8 | 1 | 2 _ -3 f(x) x f ) _ -1 | -1 | 1 f(x) x e ) _ 1 | -1 f(x) x d ) _ 1 | -1 f(x) x c ) f(x) x b ) _ 1 | -1 f(x) x a ) http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
28 0 f(x) 7 0 f(x) 9 -1 f(x) 5 0 f(x) 6 SPM 2004 ( P1,Q2) K= 8 1 , m=- 2 5 SPM 2005 ( P1,Q2) (a) 2 SPM 2005 ( P1,Q2) ( a ) x 2x 5 , x 0 ( b ) 4 3 SPM 2005 ( P1,Q3) 2x+ 5 Assessment ( 30 minutes ) 1)( a) 12 – 2x ( b ) 8 2) - 4 1 3) n = 2 1 , m=7 4) f(x) x f ) _ 6 | 2 | -2 f(x) x d ) _ 9 | 4 | 5 f(x) x e ) _ 5 | 1 | -1 _ -1 _ 1 | 3 f(x) x c ) _ 7 | 2 | 3 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com
29 The corresponding range of f(x ) = 0 f(x) 7 f(x) a ) _ 5 | 2 | 3 _ 7 | 6 x http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com