FUNCTIONS

RELATIONS (Week 1)

Types of relationsOne-to-one One-to-many

Many-to-one Many-to-many

Three ways to represent relationsArrowed diagram Set of ordered pairs Graph

Example QuestionA = { 1, 2, 3}B = {2, 4, 6, 7, 10}

A relation from A into B is defined by the set of ordered pairs

{ (1, 4), (1, 6), (2, 6), (3, 7) }State

a. The image 1b. The object of 4c. He domaind. The codomaine. The rangef. The type of relation

A = { 2, 4, 6}B = {2, 3, 6, 7, 10}

A relation from A into B is defined by the set of ordered pairs

{ (2, 2), (4, 2), (6, 7) }State

a. The image 1b. The object of 4c. He domaind. The codomaine. The rangef. The type of relation

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FUNCTION NOTATIONS

Questions

1. Given the function f : x 2x1, find

a. f (3)

b. f (-4)

2. Given the function f : x 2x3 ,find

a. f (2)

b. f (-1)

3. Given the function g : x x2 3, find

a. g(0)

b. g(4)

4. Given the function g : x x2 5 , find

a. g(0)

b. g(2)

5. Given the function f : x 3x 1, find

(a) f (0)

(b) f (3)

(c) f(-2)

(d) f(4)

(e) f(-3)

6. Given the function g : x x2 3x , find

(a) g (0)

(b) g (2)

(c) g (4)

(d) g (-3)

(e) g (-1)

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7. Given the function h : x→6

3−4 x

find

(a) h (0)

(b) h (1)

(c) h (1/2 )

(d) h(-3)

(e) h (1/4 )

8. Given that f : x 2x1, find the value of x if f(x) = 5.

9. Given that f : x 5x 3, find the value of x if f(x) = -7.

10. Given that g : x x 2 , find the value of y if g(y) = 2y – 3

11. Given that g : x 3x 2 , find the value of y if g(y) = 2y + 4.

12. Given that f : x 7x3 , g : x 4x 15 , find the value of p if f(p) = g(p).

13. Given that f : x 2x10 , g : x 4x 2 , find the value of x if f(x) = g(x).

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Exercise

1. Given that f : x 2x10 , g : x x 7 , find the value of x if f(x) = 2 g(x). [ans: x = 6]

2. Given f : x 3x9 , g : x x 7 , find the value of x if f(x) = 3 g(x). [ans: x = 5]

3. Given that f : x 3x 4 , find the values of x if f (x) = x2. [ans: p = -1 or p = 4]

4. Given that g : x 3x2 6 , find the values of if g (y) = 6 . [ans: p = -2 or p = 2]

5. Given that g : x 2x2 5 , find the values of y if g (y) = 45 . [ans: p = -5 or p = 5]

6. Given that g : x 3x2 6 , find the values of p if g (p) = 7p. [ans: p = -2/3 or p = 3]

7. Given that g : x 2x2 3 , find the values of p if g(p) = - 5p . [ans: p = -3 or p = 1/2]

8. Given the functions f : x 2x 5 and g : x x 2 , find

(a) the value of x if f(x) = 7. [ans: x = 6]

(b) the value of y if g(y) = 2y – 3. [ans: y = 5]

(c) the value of p if f(p) = g(p). [ans: p = 7]

(d) the value of x if f(x) = – g(x). [ans: x = 1]

9. Given the functions f : x 3x 4 and g : x 2x , find

(a) the value of p if f(p) = -5. [ans: p = -3]

(b) the value of k if g(k) = 3k – 4. [ans: k = 4]

(c) the value of x if 2f(x) = g(x). [ans: x = -2]

(d) the values of y jika f(y) = y2. [ans: y = -1 or y = 4]

10. Given the functions f : x x2 and g : x 12 4x , find

(a) the image of 3 that is mapped by f . [ image = 9]

(b) the values of x if f(x) = x, [ x = 0 or x = 1 ]

(c) the values of y if f(y) = g(y). [ y = -6 or y = 2 ]

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COMPOSITE FUNCTION (Week 2)

EXAMPLE1. Given that f : x 3x 4 and g : x 2x ,

find fg(x), gf(x), f 2 (x), g2(x).2. Given that f : x 32x and g : x x2, find

fg(x), gf(x) , f 2(x), g2(x).

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Questions

1. Given that f : x 2x1 and g : x 3x , find f g (x) .

2. Given that f : x 2x9 andg : x 13x , find g f (3).

3. Given the functions f : x x3 andg : x 4x 1 , find(a) f g (x) (b) g f (x)

4. Given the functions f : x 3x7 andg : x 4 2x , find(a) f g (1) (b) g f (2)

5. Given that f : x 2x 3 and g : x 4x ,find f g (x) .

6. Given that f : x 2x 5 and g : x 5x , find g f (7) .

7. Given that f : x x4 and g : x 2x 1,find(a) f g (1) (b) g f (1)

8. Given that f : x 3x 4 andg : x 5 2x , find(a) f g (0) (b) g f (0)

9. Given the functions f : x 3x2 1 and 10. Given that f : x 3x and g : x 2 x2 , find

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g : x 2 x , find(a) f g (x) (b) g f (x)

(a) f g (-2) (b) g f (-2)

11. Given the functions f : x x 2 andg : x 2 3x x2 , find(a) f g (x) (b) g f (x)

12. Given the functions f : x 2 x andg : x 1 4 x 3x2 , find(a) f g (-1) (b) g f (-1)

13. Given that f : x 2x 3 , find f 2(x) . 14. Given that g : x x 5, evaluate g g (3)

15. Given that g : x 2x 1 , evaluate g 2(x). 16. Given that f : x 3x 4 , find f f (x) .

17. Given that g : x 2 5x , find g 2(1). 18. Given that f : x 3x2 1, find f f (x) .

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Exercise

1. Given that f : x 3x 4 and g : x 2x , find f g (x).

2. Given that f : x 32x and g : x x2, find the composite function gf.

3. Given that f : x 2x3 and g : x 4x , find f g (x) .

4. Given that f : x 2x5 and g : x 5x , find the composite function gf .

5. Given the functions f : x x4 and g : x 2x 1 , find

(a) f g (x)

(b) g f (x)

6. Given that f : x 3x 4 and g : x 5 2x , find

(a) f g

(b) g f

7. Given the functions f : x 3x 2 and g : x 2 2 x , find

(a) f g (x)

(b) g f (x)

8. Given that f : x 3x and g : x 2 x2, find the composite functions

(a) f g

(b) g f

9. Given the functions f : x x 4 and g : x 1 3x , find

(a) f g (x)

(b) g f (x)

10. Given that f : x 3 x and g : x 4 x 3, find the composite functions

(a) f g

(b) g f

11. Given the functions f : x x 2 and g : x 5 2x , find

(a) f g (x)

(b) g f (x)

12. Given that f : x 2 5x and g : x 1 x2, find the composite function

(a) f g

(b) g f

13. Given that f : x 34x , find f 2(x).

14. Given that f : x 5x , find the function f 2.

15. Given that f : x 2 5x , find the function f 2.

16. Given that g : x 4 x 3 , find the function g 2.

17. Given that g : x 1 3x , find gg(x)

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INVERSE FUNCTIONS (Week 3)

Example:

1. Given that f (x) = 4x – 6 , find f –1(x).2. Given that f : x →

2 x+12−x

,

Find f –1(x).

QuestionsGiven that f : x 4 + 8x , find f –1. Given that g : x 10 – 2x , find g–1.

Given that f : x 4 – 3x , find f –1. Given that g : x 15 + 6x , find g –1 .

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Given that f : x 10 – 8x , find f –1 .Given that g : x→ 3−1

2x, find g–1 .

Given that f : x 5 + 2x , find f –1 . Given that g : x 3 – x , find g–1 .

Given that f ( x )=2 x−54

, find f –1 (x). Given that f ( x )=3 x−26

, find g–1 (x).

Given that f : x 6x - 15 , find f –1.Given that g : x→ 3−3

4x , find g–1.

Given that f ( x )= x+24 x−2

, find f –1 (x). Given that g ( x )= 2 x4−x

, find g–1 (x).

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Exercise

1. Given that f : x 10 – 8x , find f –1.

2. Given that f : x 5 + 2x , find f –1 .

3. Given that g : x 3 – x , find g–1 .

4. Given that f : x 6x - 15 , find f –1 .

5. Given that g : x→ 3−12

x , find g–1.

6. Given that f (x)=2 x−54

, find f –1 (x).

7. Given that g : x→3 x−2

6, find g–1.

8. Given that g : x→ 3−34

x, find g–1 .

9. Given that f (x)= x+24 x−2

, find f –1 (x).

10. Given that g : x→2 x

4−x , find g–1 .

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Finding component function from composite function (Week 4)

Given the functions f : x 2x – 3 andf g : x2x + 3, find the function g .

Given the functions g: x x + 3 andg f : x 2x , find the function f .

Given the functions f : x 3x + 4 andf g : x6x + 1, find the function g .

Given the functions g : x 2x - 1 andg f : x6x + 7 , find the function f .

Given the functions f : x 2 – x andf g : x2 – 2x , find the function g,.

Given the functions g : x 2x andg f : x4 – 2x , find the function q.

Given the functions g : x 2 + 4x andg f : x 6 + 4x2, find the function f .

Given the functions f : x 2x + 7 andf g : x 7 – 4x , find the function g .

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Given the functions f : x 2x - 3 andg f : x2x , find the function g,.

Given the functions g : x x + 3 andf g : x 2x + 3 , find the function f.

Given the functions f : x 3x + 4 andg f : x 6x + 7, find the function g,.

Given the functions g : x 2x - 1 andf g : x 6x + 1 , find the function f.

Given the functions f : x 2 – x andg f : x4 – 2x , find the function g,.

Given the functions g : x 2x andf g : x2 – 2x , find the function f.

Given the functions f : x 3x andg f : x 1 - 3x , find the function g,.

Given the functions f : x 2 + 4x andg f : x5 + 16x + 16x2, find the function g,.

SPM Standard Questions 13

1. P = { 1, 2, 3} Q = {2, 4, 6, 8, 10}

Based on the information above, a relation

from P into Q is defined by the set of ordered

pairs

{(1, 4), (1, 6), (2, 6), (2, 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.

2.

The above diagram shows the relation

between set A and 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

3. Given the functions f : x 2x 1 andg : x x2 3 , find(a) f –1 (5) ,(b) g f (x) .

4. Given the functions g : x 4x 1 andh : x x2 3 , find(a) g –1 (3) ,(b) h g (x) .

5. Given the function f : x 2 3x andh : x x2 x 2 , find(a) f –1 (3) ,(b) h f (x) ,(c) f 2 (x) .

6. Given the functions f : x 2x 1 andh : x 2 x2 , find(a) f –1( –1) ,(b) h f (x) ,(c) f h(x).

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7. Given that f : x4x m and f−1: x → nx+ 3

4, find the values of m and n.[ans: m = –3 ; n = 1/4 ]

8. Given that f : x2x 1, g : x4x andfg : xax b , find the values of a and b [ans: a = 8 ; b = –1 ]

9. Given that f : xx 3 , g : xa bx2 and gf : x6x 2 36x 56 , find the values of a and b.[ans: a = 2 ; b = 6 ]

10. Given that g : xm 3x and

g−1 : x→ 2 kx−43, find the values of m

and k.[ans: k = 1/6 ; m = 4 ]

11. Given that g(x) = mx + n and g 2 (x) = 16x – 25 , find the values of m and n.

[ans: m = 4 , n = –5 ; m = –4 , n = 25/3]

12. Given the inverse function f−1 (x )=2 x−3

2, find(a) the value of f (4),(b) the value of k if f –1 (2k) = –k – 3 .[ans: (a) 11/2 ; (b) k = -1/2 ]

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