Document Sample

My Additional Mathematics Module Form 4 (Version 2007) Topic: 1 FUNCTIONS by NgKL (M.Ed.,B.Sc.Hons.,Dip.Ed.,Dip.Edu.Mgt.,Cert.NPQH) Kajang High School, Kajang, Selangor IMPORTANT POINTS 1.1 CONCEPTS OF RELATIONS 1. The relation between two sets, P and Q, is the pairing of elements in set P with elements in set Q. P 2 3 4 2. Q 4 9 16 25 (a) (b) (c) (d) (e) 3. Set P = domain Set Q = codomain Each elements in set P = object Each elements in set Q = image The images 4, 9, 16 in set Q = range Relations can be represented by; (a) Arrow diagrams: X 2 Y 8 (b) Ordered pairs: (2, 8), (3, 27), (4, 64) 3 4 27 64 (c) Graphs: 64 27 8 2 3 4 4. Four types of relations: (a) One-to-one relation X 2 Y 8 (b) One-to-many relation X 18 Y 2 3 3 27 12 4 (c) Many-to-one relation X 2 3 4 Y 8 9 (d) Many-to-many relation X Y Exercise 1.1(a): 1. Complete the table below of the following relations. X (a) 2 4 6 Y 4 10 14 (b) {(3, 5), (5, 9), (7, 13)} (c) z y x 2 Answers: Domain (a) 8 14 Codomain Object Image Range (b) (c) Exercise 1.1(b): 1. Based on the relations given, identify the images or objects. (a) X Y 7 10 8 3 6 (b) {(3, 5), (3, -1), (4, 9)} (c) 6 4 2 3 Answers: Object of 6 (a) Object of 5 (b) Object of 2 (c) 6 9 Object of 8 Image of 3 Image of 7 Object of 9 Image of 3 Image of 4 Object of 6 Image of 6 Image of 9 2. State the type of relations for each of the following. (a) X a b c Y Answer: p q r (b) {(3, s), (4, t), (5, s)} Answer: (c) 6 4 2 Answer: p q s (d) X Y Answer: (e) X Y m Answer: 1 n 2 p 1.2 CONCEPTS OF FUNCTIONS 1. A function = special relation where every object in the domain has only one image in the range. One-to-one relation and many-to-one relation are functions. One-to-many relation and many-to-many relation are not functions. A function can be expressed by function notation, in which the function is represented by the symbol f and the object by the symbol x. f : x y is read as ‘function f maps x to y’, or, f(x) = y which is read as ‘y is the image of x under function f’ Example: f : :x 3x …. fuction f maps x to 3x. f(x) = 3x …. 3x is the image of x under function f. 2. 3. 4. Exercise 1.2(a): 1. State whether the following relation are functions and give reasons. (a) X Y Answer: (b) {(1, 3), (3, 5), (5, 7)} Answer: (c) {(a, p), (a, q), (b, r)} Answer: (d) 7 5 3 a b c Answer: (e) X Y Answer: m a n b p 2. Write the following relations in function notation. f (a) 3 2 1 x x2 9 4 1 Answer: f (b) x 1 1 -1 2 15 4x2 - 3 Answer: Exercise 1.2(b) 1. Determine the image of the object for each of the following functions. (a) f(x) = 4x + 5, x = 1, 5 (b) f(x) = x2 + 2, x = 0, -3 (c) f(x) = 4 – x, x = 1, 6 (d) f(x) = 2 x 3 , x = 2, 7 Exercise 1.2(c) 1. A function is defined as f : x 3x + 6, find (a) the object if the image is 18. (b) the value of x if f(x) = 2x. 2. A function is defined as f : x x2 + 6x, find (a) the object if the image is 7. (b) the object that maps to itself. 3. Given a function g : x 3 x2 , x 2, find (a) the value of x if g(x) = 6 (b) the value of x if g(x) = x – 4. 4. Given a function h : x 2x 5 3 , find (a) the value of x if h(x) = 3. (b) the object if the image is 5. Exercise 1.2(d) 1. Given f(x) = px + q, f(0) = -5 and f(3) = 7. Find the values of p and q. 2. A function is defined as f: x = 2x2 – mx + n. If f(1) = 4 and f(2) =7, determine the values of m and n. 3. Given f: x = 4 mx n ,x n , f(2) = 2 and f(5) = 4. Find m and n. m 4. The arrow diagram shows the function f :x determine the values of p and q x x p xq x p xq , x q, 2 -3 4 5 1.3 COMPOSITE FUNCTIONS A B C f x f(x) g gf(x) gf(x) If f is the function which maps set A onto set B and g is the function that maps set B onto set C, then set A can be mapped directly to set C by a composite function, represented by gf(x). Exercise 1.3(a) 1. Given f:x → 2x + 5 and g:x → 3x – 4. Find (a) fg(x) (b) gf(x) (c) fg(3) 2. Given f:x → 5x + 2 and g:x → x2 – 1. Find (a) f2(x) (b) f2g(x) (c) fg(2) 3. Given f:x → (a) fg(x) 3 , x 2 and g:x → x + 4. Find x2 (b) gf(x) (c) f2(x) 4. Given f:x → │2x + 3│ and g:x → 2x – 3. Find (a) fg(x) (b) gf(x) (c) fg(─1) Exercise 1.3(b) 1. Given f(x) = 2x + 5 and fg(x) = 8x – 5, find g(x). 2. Given f(x) = x + 3 and fg(x) = x2 + 2x + 1, find g(x). 2. Given f(x) = 2 2 , x -2 and and fg(x) = , x ─ ½, find g(x). x2 2x 1 Exercise 1.3(c) 1. Given g(x) = 3x and fg(x) = 15x – 9, find f(x). 2. Given f(x) = x - 3 and gf(x) = x2 - 4x + 5, find g(x). 3. Given f(x) = 2 4 , x 0 and and fg(x) = 5 , x 0, find g(x). x x 1.4 INVERSE FUNCTIONS X f Y x y f—1 If f is the function which maps elements in set X onto the elements in set Y, then when the elements in set Y are mapped onto the elements in set X, the function is called an inverse function of f. The notation for the inverse function of f is written by f—1 When f(x) = y, then f—1 (y) = x. When f—1f = x, then ff—1 = x. Not all functions have inverse functions. The inverse function exists if and only if the function is a one-to-one relation. Exercise 1.4(a) 1. Find the value of each object by inverse mapping. f (a) 3 a x x2 + 2 11 6 f (b) x 2x 2 – 3 1 2 b 5 15 f (c) c 4 x 5 – 4x 3 -9 f (d) x x 4 3 d e 5 9 f (e) x x3 x 5 -3 5 3 f 2. Determine the inverse function of the following functions. (a) f : x → 2x + 4 (b) f : x → 2x 5 6 (c) g : x → 2x 3 , x -4 x4 3. (a) Find the inverse function, h-1(x) of the function h : x → What is the value for h-1(2)? 3 2 , x 0. x (b) Find the inverse function, f-1(x) of the function f : x → What is the value for f -1(3)? 3 , x - 2. x2 (a) Find the inverse function, g-1(x) of the function g : x → 7x - 3. What is the value for g-1(5)? 4. (a) Given the inverse function, f f(x). – 1 : x → 3x + 2, determine the function (b) Given the inverse function, g function g(x). – 1 : x → 2 , x -5, determine the x 5 (c) Given the inverse function, f f(x). –1 :x→ 3 4x , determine the function 2 (d) Given the inverse function, h function h(x). –1 :x→ 5x , x - ½, determine the 2x 3 5. (a) Given function, f(x) = 7x + 2, determine f reason whether the inverse function exists. – 1 (x) and state and give (b) Given function, g(x) = x2 + 9, determine g reason whether the inverse function exists. – 1 (x) and state and give (c) Given function, f(x) = x3 - 16, determine f reason whether the inverse function exists. – 1 (x) and state and give Problem Solving 1. Given the function g : (x) → (a) the value of m. (b) g –1(x). 3 , x m, find x4 2. Given f : x → 2x2 – 5 and g : x → x + 3, find (a) fg(x). (b) the value of gf(-1). 3. Given the function h(x) = 3x + 5, find the value of x (a) if h2(x) = h(-x)’ (b) when x is mapped onto itself. 4. Given the function g : x → 3x + 2, find the function f(x) if (a) fg : x → 2x2 + 5 (b) gf : x → 2x – 3 5. Given f(x) = px + q and f2(x) = 4x – 16. Find (a) the values of p an q. (b) the value of ff–1(2). 6. Given that f(x) = │5x - 8│, find (a) f(2). (b) the values of the objects that have the image 7. Past Years SPM Papers P = {1, 2, 3} Q = {2, 4, 6, 8, 10} 1. Based on the the above information, the relation between P and Q is defined by the set of ordered pairs {(1, 2), (1, 4), (2, 6), (2, 8)}. State (a) the image of 1. (b) the object of 2. [2 marks] SPM2003/Paper 1 2. Given that g : x → 5x + 1 and h : x → x2 – 2x + 3, find (a) g–1(3), (b) hg(x). [4 marks] SPM2003/Paper 1 3. Diagram 1 shows the relation between set P and set Q. . w d x e y f . z State (a) the range of the relation, (b) the type of the relation. [2 marks] SPM2004/Paper 1 4. Given the functions h : x → 4x + m and h–1 : x → 2kx + are constant, find the value of m and of k . 5 , where m and k 8 [3 marks] SPM2004/Paper 1 5. Given the function h(x) = 6 , x 0 and the composite function hg(x) = 3x, x find (a) g(x), (b) the value of x when gh(x) = 5. [4 marks] SPM2004/Paper 1 6. In Diagram 1, the function h maps x to y and the function g maps to z. x h 5 2 Determine (a) h–1 (5), (b) gh(2). y g z 8 DIAGRAM 1 [2 marks] SPM2005/Paper 1 7. The function w is defined as w(x) = (a) w–1(x), (b) w–1(4). 5 , x 2. Find 2x [3 marks] SPM2005/Paper 1 8. The following information refers to the functions h and g. h : x → 2x – 3 g : x →4x - 1 Find gh–1 (x). [3 marks] SPM2005/Paper 1

DOCUMENT INFO

Shared By:

Categories:

Stats:

views: | 6304 |

posted: | 11/28/2009 |

language: | English |

pages: | 30 |

OTHER DOCS BY nklye

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.