1 FUNCTIONS _ 5 _ 1.0 RELATIONS

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1 1 FUNCTIONS 1.0 RELATIONS Notes : (i) Four types of relations : one-to-one many-to-one one-to-many many-to-many. (ii) Three ways to represent relations : arrowed diagram set of ordered pairs graph. (iii) The arrow diagram representation is of the form below : P :Domain Q : Codomain P Q a b A : Domain B : Codomain A B _ 4 _ 1 _ 5 _ 3 6 a is the object In the above relation b is the imej of a ; a is the object of b. b is the image Set { 4 6 } is the RANGE for the mapping from A to B The relation from A into B above can also be written as set of ordered pairs {(1 4) (3 6) } 4 is the image of 1 ; The Object of 6 is 3. (iv) Set of ordered pairs of a relation takes the form of { (a b) (c d).. } 1. EXAMPLE 1. QUESTION P { 1 3} Q { } P { 4 6} Q { } A relation from P into Q is defined by the set of ordered pairs { (1 4) (1 6) ( 6) (3 7) }. A relation from P into Q is defined by the set of ordered pairs { ( ) (4 ) (6 7) }. State (a) the image of (b) the object of 4 1 (c) the domain { 1 3 } (d) the codomain {46710} (e) the range { } (f) the type of relation many-to-many State (a) the image of 4 (b) the objects of (c) the domain (d) the codomain (e) the range (f) the type of relation Functions 1

2 1.1 FUNCTION NOTATIONS Notes: (i) A function is usually represented using letters in its lower case : f g h.. (ii) Given the function f : x x we usually write it in the form f(x) x + 1 before answering any question. 1. FINDING THE VALUE OF FUNCTIONS [Mastery Exercise] f (a) represent (a) the value of f(x) when x a. (b) the image of f when the object is a EXAMPLE 1. Given the function f : x x find (i) f (3) (ii) f (-4) f(x) x + 1 (i) f ( 3 ) (3 ) (ii) f (- ) (-) Given the function g : x x 3 (i) g(0) (ii) g(4) g (x) x - 3 (i) g ( 0 ) QUESTION 1. Given the function f : x x 3 find (i) f () (ii) f (-1) f(x) x + 3 (i) f(3) ( ) + 3 (ii) f(-1) ( ) + 3. Given the function g : x x 5 (i) g(0) (ii) g() g (x) x - 5 (i) g ( 0 ) (ii) g (4) 4-3 (ii) g () Given the function h : x 3 x 4 find h (). Jawapan : h( x) 6 3x 4 4 x Given the function h : x x 3 find h(3). Jawapan: h( x) 3 x h ( ) 6 3() 4 h ( 3 ) 6 3 Functions

3 1. FINDING THE VALUE OF FUNCTIONS [Reinforcement Exercises] QUESTION 1. Given the function f : x 3x 1 (a) f (0) (b) f (3) 1. f ( x ) 3x + 1 (a) f ( 0 ) 3 ( 0 ) + 1 ANSWER (c) f(-) (d) f(4) (e) f(-3). Given the function g : x x 3x (a) g (0). g(x) x - 3x (a) g (0) (b) g () (c) g (4) (d) g (-3) (e) g (-1) 3. Given the function 6 h : x 3 4x (a) h (0) (b) h (1) (c) h ( 1 ) (d) h(-3) (e) h ( 4 1 ) Functions 3

4 1.3 PROBLEM OF FUNCTIONS INVOLVING SIMPLE EQUATIONS. [1] Note : Questions on functions usually involve solving linear / quadratic equations. EXAMPLE 1. Given that f : x x the value of x if f(x) 5. f (x) x 1 x 1 5 x 6 x 3. Given that f : x 5x 3 the value of x if f(x) -7. f (x) 5 x x x - 10 x - QUESTION 1. Given that f : x 4x 3 the value of x if f(x) 17. f (x) 4 x 3 4 x x x. Given that f : x 3x 7 the value of x if f(x) Given that g : x x the value of y 3. Given that g : x 3x the value of y if if g(y) y - 3 g(x) x + g(y) y + y + y y - y y 5 g(y) y Given that f : x 7x 3 g : x 4x 5 find the value of p if f(p) g(p). 4. Given that f : x x 0 g : x 4x find the value of x if f(x) g(x). f(x) 7x 3 g(x) 4x + 15 f(p) 7p 3 g(p) 4p + 15 f(p) g(p) 7p 3 4p p 18 p 6 Functions 4

5 PROBLEMS OF FUNCTIONS INVOLVING SIMPLE EQUATIONS. [] EXAMPLE 5. Given that f : x x 0 g : x x 7 find the value of x if f(x) g(x) QUESTION 5. Given f : x 3x 9 g : x x 7 the value of x if f(x) 3 g(x). A: f(x) x - 10 g(x) x 7 x 10 - (x 7) x x x 4 x 6 6. Given that f : x 3x 4 the values of x if f (x) x. A: f(x) 3x + 4 x 3x + 4 x - 3x 4 0 (x + 1) (x 4) 0 x -1 or x 4 6. Given that f : x x 8 the values of x if f (x) x. 7. Given that g : x 3x 6 the values of y if g (y) 6. A: g(x) 3x 6 g(y) 3y - 6 3y y 1 0 ( 3) : y 4 0 (y + ) (y ) 0 y - or y 8. Given that g : x 3x 6 the values of p if g (p) 7p. A: g(x) 3x 6 g(p) 3p - 6 3p - 6 7p 3p 7p (3p+) (p 3) 0 p atau p Given that g : x x 5 the values of y if g (y) Given that g : x x 3 the values of p if g(p) - 5p. Functions 5

6 PROBLEM OF FUNCTIONS INVOLVING SIMPLE EQUATIONS [Reinforcement Exercises] QUESTION 1. Given the functions f : x x 5 g : x x (a) the value of x if f(x) 7 (b) the value of y if g(y) y - 3 (c) the value of p if f(p) g(p) (d) the value of x if f(x) g(x) ANSWER 1. f(x) x 5 g(x) x + (a) f(x) 7 x 5 7 x x. Given the functions f : x 3x 4 g : x x (a) the value of p if f(p) -5 (b) the value of k if g(k) 3k - 4 (c) the value of x if f(x) g(x) (d) the values of y jika f(y) y. f (x) g(x) (a) f (p) -5 3p Given the function f : x 3x (a) evaluate f (-). (b) solve the equation f(x) 0. (c) find the values of p if f(p) 3p f (x) (a) f (-) f f(-) f( ) f(-) 3(-) 1 4. Given the functions f : x x 4. f (x) g(x) g : x 1 4x (a) the image of 3 that is mapped by f. (b) the values of x if f(x) x (c) the values of y if f(y) g(y). Functions 6

7 1.4 FINDING THE VALUE OF COMPOSITE FUNCTIONS [Mastery Exercises] Notes : To find the value of fg(a) given the functions f g we can first find fg(x) then substitute x a. We can also follow the EXAMPLES below. EXAMPLE 1 : Given that f : x 3x 4 g : x x find fg(3). f(x) 3x - 4 g(x) x g(3) (3) 6 fg(3) f [ g(3) ] f ( 6) 3 (6) EXAMPLE : Given that gf(4). f : x 3 x f(x) 3 x g(x) x. f(4) 3 (4) gf(4) g (-5) (-5) 5 g : x x EXERCISES 1. Given that f : x x g : x 3x find f g(1).. Given that f : x x 9 g : x 1 3x find gf(3). 3. Given the functions f : x x 3 4. Given the functions f : x 3x 7 g : x 4x (a) f g(1) (b) gf(1) g : x 4 x (a) f g(0) (b) gf(0) Functions 7

8 FINDING THE VALUE OF COMPOSITE FUNCTIONS [Reinforcement Exercises] 1. Given that f : x x 3 g : x 4x find fg().. Given that f : x x 5 g : x 5x gf(7). 3. Given that f : x x 4 g : x x 4. Given that f : x 3x 4 g : x 5 x find find (a) fg(1) (b) gf(1) (a) fg(0) (b) gf(0) 5 Given the functions f : x 3x 6 Given that f : x 3x g : x x g : x x (a) fg(-) (b) gf(-) (a) fg(-1) (b) gf(-1) 7. Given the functions f : x x 8. Given the functions f : x x g : x 3x x g : x 1 4 x 3x (a) fg(1) (b) gf(1) (a) fg(-1) (b) gf(-1) Functions 8

9 1.5 FINDING THE VALUE OF COMPOSITE FUNCTIONS f g Notes : f (a) ff (a) f [ f (a) ] EXAMPLE 1 : Given that f : x 3x f (). EXAMPLE : Given that g : x 3 4x evaluate gg(1). f(x) 3x - f() 3() 4 f () f [ f() ] f ( 4) 3 (4) 10 g(x) 3 4x g(1) 3 4(1) -1 gg(1) g [g(1)] g ( -1) 3 4 (-1) EXERCISES 1. Given that f : x x 3 f ().. Given that g : x x 5 evaluate gg(3). 3. Given that g : x x evaluate g (3). 4. Given that f : x 3x 4 ff(0). 5. Given that g : x 5x g (1). 6. Given that f : x 3x ff(1). 7. Given that g : x 4x evaluate gg(-). 8. Diberi f : x x 4x 3. Nilaikan f (0). Functions 9

10 1.6 BASIC EXERCISES BEFORE FINDING THE COMPOSITE FUNCTIONS [ 1 ] EXAMPLE 1 : Given f : x 3x f(x) 3x + 1 thus (a) f() 3() (b) f(a) 3 a + 1 (c) f(p) 3 p + 1 (d) f(k) 3 (k) + 1 6k + 1 (e) f(x) 3 (x) + 1 6x + 1 (f) f(x ) 3 x + 1 EXAMPLE : Given g : x 5 4x. g(x) 5 4x thus (a) g() 5 4() (b) g(a) 5 4a (c) g(p) 5 4p (d) g(3k) 5 4(3k) 5 1 k (e) g(x ) 5 4x (f) g (3+x) 5 4 (3+x) 5 1 8x - 7 8x EXERCISES 1. Given f : x x 3 f(x) x + 3 Given g : x 4x. g(x) thus (a) f() () + 3 thus (a) g () (b) f(a) (b) g(a) (c) f(p) (c) g(s) (d) f(k) (d) g(3x) (e) f(x ) (e) g(x ) (f) f(-x ) (f) g (3+x) (g) f( - 3x) (g) g( 4x) Functions 10

11 BASIC EXERCISES BEFORE FINDING THE COMPOSITE FUNCTIONS [ ] 1. Given f : x 4 x Given g : x x. f(x) 4 x g(x) thus (a) f(3) 4 (3) (b) f(-x) (c) f(+x) thus (a) g(-1) (b) g(x) (c) g(x-) (d) f(3 -x) (d) g(-3x) (e) f(x ) (e) g(x ) (f) f(-x +) (f) g (1-x) 3. Given f : x 1 x f(x) thus (a) f(10) 4 Given g : x ( x ). g(x) thus (a) g(-1) (b) f(3x) (b) g(x) (c) f( - x) (c) g(x-1) (d) f(4+x) (d) g(-x) (e) f(x- 3) (e) g(x ) (f) f(x ) (f) g (1+x) Functions 11

12 1.7 FINDING COMPOSITE FUNCTIONS EXAMPLE 1 : Given that f : x 3x 4 g : x x find fg(x). f(x) 3x - 4 g(x) x fg(x) f [ g(x) ] f ( x) 3 (x) - 4 6x 4 OR fg(x) f [ g(x) ] 3 [g(x)] (x) 4 6 x - 4 EXAMPLE : Given that f : x 3 x composite function gf. Answer: f(x) 3 x g(x) x. OR thus gf(x) g[f(x)] g (3 x) (3- x) gf(x) g[f(x)] [f(x)] (3- x) gf : x (3 x) g : x x the EXERCISES 1. Given that f : x x 3 g : x 4x find fg(x).. Given that f : x x 5 g : x 5x the composite function gf. 3. Given the functions f : x x 4 4. Given that f : x 3x 4 g : x 5 x g : x x find (a) fg(x) (b) gf(x) (b) f g (b) gf Functions 1

13 REINFORCEMENT EXERCISES FOR FINDING COMPOSITE FUNCTIONS 1. Given the functions f : x 3x. Given that f : x 3x g : x x g : x x the composite functions (a) fg(x) (b) gf(x) (a) fg (b) gf 3. Given the functions f : x x 4 g : x 1 3x (a) fg(x) (b) gf(x) 4. Given that f : x 3 x g : x 4 x 3 find the composite functions (a) f g (b) gf 5. Given the functions f : x x g : x 5 x (a) fg(x) (b) gf(x) 6. Given that f : x 5x g : x 1 x find the composite function gubahan (a) fg (b) gf Functions 13

14 REINFORCEMENT EXERCISES FOR FINDING COMPOSITE FUNCTIONS [ f g ] 1. Given that f : x 3x f (x).. Given that f : x 3x the function f. f(x) 3x + f (x) f [f(x)] f (3x + ) or 3f(x) + 3(3x+) + 3. Given that g : x 1 3x gg(x) 4. Given that g : x 4 x 3 the function g. 5. Given that f : x x ff(x). 6. Given that f : x 5x the function f. 7. Given that f : x 3 4x f (x). 8. Given that f : x 5x the function f Given that g : x x g (x). 10. Given that h : x the function f. x Functions 14

15 1.8 INVERSE FUNCTIONS Notes : 1. Students must be skillful in changing the subject of a formula!. If f (x) y then x f 1 (y) OR if f 1 (x) y then x f(y) EXAMPLE 1. Given f(x) x 3 then f 1 ( x 3 ) x. Given that f(x) x 3 f 1 (x). METHOD 1 f 1 ( y ) x when y x 3 f 1 (y) f 1 (x) y 3 x 3 y + 3 x x y 3 Given f(x) x 3 then f(y) y 3 f 1 ( y 3 ) y METHOD f 1 ( x ) y when x y 3 f 1 (x) Students are advised to master only one of the methods! x 3 x + 3 y y x 3 DRILLING PRACTICES ON CHANGING SUBJECT OF A FORMULA [ 1 ] 1. If y 3x then x 3x y 3x y + x y 3. If y 4x then x 4x y 4x x 3. If y 3 6x then x 4. If y 4 5x then x 5. If y 3 + 5x then x 6. If y 10x 4 then x Functions 15

16 DRILLING PRACTICES ON CHANGING SUBJECT OF A FORMULA [ ] 1. If y x 5 then x 4. If y 3x then x 5 x 5 4 y x 5 4y x x 3. If y 5 + 6x then x 4. If y 4 1 x then x 5. If y x then x 6. If y 6x 9 then x 7. If y 1 3x then x 8. If y 4 x then x 9. If y x 4x then x 10. If y x then x 3 x Functions 16

17 DETERMINING THE INVERSE FUNCTION FROM A FUNCTION OF THE TYPE ONE-TO-ONE CORRESPONDENCE EXAMPLE 1 : Given that f(x) 4x 6 f 1 (x). EXAMPLE : Given that f : x x x x find f 1 (x). Given f(x) 4x 6 so f (y) 4y 6 then f 1 ( 4y 6 ) y. f 1 ( x ) y when x 4y 6 f 1 (x) x 6 4 x + 6 4y y x 6 4 Given f(x) x x so f(y) y 1 y then f 1 y 1 ( ) y y f 1 ( x ) y when x y 1 y x( - y) y + 1 x xy y + 1 x 1 y + xy x 1 y( + x) y x x f 1 ( x ) x x x -. EXERCISES 1. Given that f : x 4 + 8x f 1.. Given that g : x 10 x g Given that f : x 4 3x f Given that g : x x g 1. Functions 17

18 MASTERY EXERCISES ON FINDING THE INVERSE OF A FUNCTION 1. Given that f : x 10 8x f 1.. Given that g : x 3 1 x g Given that f : x 5 + x f Given that g : x 3 x g Given that f(x) x 5 f 1 (x). 6. Given that g(x) 4 3x g 1 (x) Given that f : x 6x - 15 f Given that g : x x g Given that f(x) x 4x f 1 (x). 10. Given that g(x) x 4 x g 1 (x). Functions 18

19 FINDING THE VALUE OF f 1 (a) WHEN GIVEN f(x) Example : Given that f : x x + 1 the value of f 1 (7). METHOD 1 [ Finding f 1 first ] f(x) x + 1 f(y) y + 1 f 1 (y + 1) y f 1 (x) y when x y + 1 x 1 y f 1 x (x) f 1 7 (7) 3 y x Let METHOD [ Without finding f 1 (x) ] f(x) x + 1 f 1 (7) k f(k) 7 k k 6 k 3 f 1 (7) 3 ( If you are not asked to find f 1 ( x ) but only want to have the value of f 1 ( a ) then use Method ) EXERCISES 1. Given that f : x x + the value of f 1 (3).. Given that g : x 7 x the value of g 1 (). 3. Given that f : x 7 3x the value of f 1 (-5). 4. Given that g : x 5 + x the value of g 1 (5). 5. Given that f : x 3x - the value of f 1 (10). 6. Given that g : x 7 x the value of g 1 (). 7. Given that f : x 4 + 6x the value of f 1 (-). 8. Given that g : x 1 x the value of g 1 (4). 9. Given that f : x f 1 ( -1 ). 4 the value of x Diberi g : x 3 x cari nilai g -1 (-1). Functions 19

20 Finding the Other Related Component Function when given a Composite Function One of its Component Function TYPE 1 ( Easier Type ) TYPE ( More Challenging Type ) Given the functions f fg the function g. OR Given the functions g gf the function f. EXAMPLE : 1. Given the functions f : x x 3 Given the functions f gf the function g. OR Given the functions g fg the function f.. Given the functions f : x x 5 fg : x 6x the function g. f(x) x + 3 fg(x) 6x 1 Find g(x) from fg(x) 6x 1 f [g(x)] 6x 1 g(x) + 3 6x 1 g(x) 6x 4 g(x) 3x - g f : x 10x 5 the function g. f(x) x 5 gf(x) 10x 5 Find g(x) from gf(x) 10x 5 g [ f(x) ] 10x 5 g ( x 5) 10x 5 g ( y 5) 10y 5 g(x) 10y 5 when x y 5 x 5 So : g(x) 10 ( ) 5 g(x) 5x 5x x + 5 y y x 5 EXERCISES 1. Given the functions f : x x. Given the functions f : x x fg : x 4 6x the function g. g f : x 5 6x the function g. Functions 0

21 REINFORCEMENT EXERCISES on finding Component Function from Composite Function ( TYPE 1 ) 1. Given the functions f : x x 3 fg : x x + 3 the function g.. Given the functions g: x x + 3 gf : x x the function f. 3. Given the functions f : x 3x + 4 fg : x 6x + 1 the function g. 4. Given the functions g : x x - 1 gf : x 6x + 7 the function f. 5. Given the functions f : x x fg : x x the function g. 6. Given the functions p : x x pq : x 4 x the function q. 7. Given the functions g : x + 4x gf : x 6 + 4x the function f. 8. Given the functions f : x x + 7 fg : x 7 4x the function g. 9. Given the functions h : x 1 x hg : x 1 3x the function g. 10. Given the functions h : x 3x + 1 hf : x 7 9x the function f. Functions 1

22 REINFORCEMENT EXERCISES on finding Component Function from Composite Function ( TYPE ) 1. Given the functions f : x x - 3 gf: x x the function g.. Given the functions g: x x + 3 fg : x x + 3 the function f. 3. Given the functions f : x 3x + 4 gf : x 6x + 7 the function g. 4. Given the functions g : x x - 1 fg : x 6x + 1 the function f. 5. Given the functions f : x x gf : x 4 x the function g. 6. Given the functions g : x x fg : x x the function f. 7. Given the functions f : x 3x gf : x 1-3x the function g. 8. Given the functions f : x + 4x gf : x x + 16x the function g. Functions

23 QUESTIONS BASED ON SPM FORMAT 1. P { 1 } 3} Q { } Base on the information above a relation from P into Q is defined by the set of ordered pairs { (1 4) (1 6) ( 6) ( 8) }. State (a) the images of 1 (b) the object of 4 (c) the domain (d) the codomain (e) the range (f) the type of relation 3. Given the functions f : x x g : x x 3 (a) f 1 (5) (b) gf (x).. a b c A B The above diagram shows the relation between set A set B State (a) the images of b (b) the objects of p (c) the domain (d) the codomain (e) the range (f) the type of relation 4. Given the functions g : x 4x h : x x 3 (a) g 1 (3) (b) hg (x). p q r s 5. Given the function f : x 3x h : x x x (a) f 1 (3) (b) hf (x) (c) f (x). 6. Given the functions f : x x h : x x (a) f 1 ( 1) (b) hf (x) (c) f h(x). Functions 3

24 7. Given that f : x 4x m find the values of m n. 3 f : x nx 4 8. Given that f : x x g : x 4x fg : x ax b the values of a b. 9. Given that f : x x 3 g : x a bx [ m 3 ; n 41 ] gf : x 6x 36x 56 the values of a b. 10. Given that g : x m 3x find the values of m k. [ a 8 ; b 1 ] 4 g : x kx 3 [ a ; b 6 ] 11. Given that g(x) mx + n g (x) 16x 5 find the values of m n. [ k 61 ; m 4 ] 3 1. Given the inverse function 1 x f ( x) (a) the value of f(4) (b) the value of k if f 1 (k) k 3. [ m 4 n 5 ; m 4 n 3 5 ] 11 1 [ (a) ; (b) k ] Functions 4

25 QUESTIONS BASED ON SPM FORMAT 1. (SPM 06 P Q) Given the function f : x 3x x g : x 1 5 (a) f 1 (x) (b) f 1 g(x) (c) h(x) such that hg(x) x Given the function f : x x x g : x 3 (a) f 1 (x) (b) f 1 g(x) (c) h(x) such that hg(x) 6x 3. (a) x (b) x 15 (c) 10x (SPM 06 P1 Q) Diagram shows the function m x h: x x x 0 where m is a constant. 8 x Find the value of m. h Diagram m x x 1 (a) x 1 (b) 1 x 1 6 (c) 18x Diagram 3 shows the function x where p is a constant. 7 x Find the value of p. g Diagram 3 p 3x g: x x p 3x x 5 m 4 p 4 Functions 5