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# Functions - Functions Advanced Level Pure Mathematics...

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Functions Advanced Level Pure Mathematics Advanced Level Pure Mathematics Calculus I Chapter 1 Functions 1.1 Introduction 2 1.2 Direct Images and Inverse Images 4 1.3 Composition of Functions 5 1.4 Constant Function and Identity Function 6 1.5 Injective, Surjective and Bijective Functions 7 1.6 Some Special Real Functions 12 1.7 Elementary Functions 21 1.8 Revision Exercise 22 Hung Fung Book Calculus and Analytical Geometry I Functions Revision Exercise P.49 ( 1- 15 ) 1.1 Introduction Given a set A which has two elements y x , . We denote { } y x A , = or { } x y A , = . We can write A either equals { } y x , or { } x y , . It is unordered pair . In coordinate system, the x and y coordinates are written in ) , ( y x ( 29 ) , ( x y . It is ordered pair. It is easy to see that two ordered pairs ) , ( 1 1 y x and ) , ( 2 2 y x are equal if and only if 2 1 x x = and 2 1 y y = . Definition A function (or a mapping ) from a set A into a set B , is defined as B A f : (i) A f = 1 Pr Prepared by K. F. Ngai Page 1 1

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Functions Advanced Level Pure Mathematics (ii) f b a b a 2200 ) , ( ), , ( 2 2 1 1 , if 2 1 a a = , then 2 1 b b = . For any A a , ) ( a f is unique. - ) ( a f value of f at a . f 1 Pr is the first projection of the ordered pair of . f - f 2 Pr Image of f - A Domain of f - B Range of f Example Let { } 5 , 4 , 3 , 2 , 1 = A , { } e d c b a B , , , , = The following are functions (mappings) from A to B . { } ) , 4 ( ), , 3 ( ), , 2 ( ), , 1 ( 1 c b c a f = { } ) , 4 ( ), , 3 ( ), , 2 ( ), , 1 ( 2 a b c d f = The following are not functions (mappings) from A to B . { } ) , 3 ( ), , 2 ( ), , 1 ( 1 c b a g = { } ) , 4 ( ), , 3 ( ), , 2 ( ), , 1 ( ), , 1 ( 2 c a a b a g = { } ) , 4 ( ), , 3 ( ), , 2 ( ), , 1 ( 3 h f e d g = Remark A function of real variable is a function whose domain is the set of all real numbers or a subset of R . A real-valued function is a function whose range is the set of all real numbers. Example Let R x , find the domain as long as possible of each of the following functions. (a) 5 2 ) ( - = x x f (b) x x f 1 ) ( = (c) x x f = ) ( Prepared by K. F. Ngai Page 2
Functions Advanced Level Pure Mathematics (d) 3 2 ) ( 2 - - = x x x x f (e) x x f - = 1 ) ( (f) x x x f sin 1 ) ( = (g) x x f tan ) ( = (h) x x f π sin 1 ) ( = (i) ) 1 log( ) ( - = x x f (j) x x f sin ) ( = Example If x x f = ) ( , 0 x , then ) , 0 [ ) , 0 [ : + ∞ + ∞ f Example In the following, which is/are graph(s) of a function(s) of x ? Example For each of the following pairs of functions, are they identical ? If no, explain. (a) x x x f = ) ( , 1 ) ( = x g Prepared by K. F. Ngai Page 3

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Functions Advanced Level Pure Mathematics (b) x x f = ) ( , 2 ) ( x x g = (c) 1 ) ( = x f , x x x g 2 2 cos sin ) ( + = (d) 2 ln ) ( x x f = and x x g ln 2 ) ( = 1.2 Direct Images and Inverse Images Definition Let B A f : be a function from A to B and A X . The direct image ( image ) of X under f [ ] ( 29 X f is defined as [ ] { } X a a f X f = : ) ( Example (1) Let the function B A f : be represented by the following figure. If { } 4 , 3 , 2 = X , then [ ] { } { } e d b f f f X f , , ) 4 ( ), 3 ( ), 2 ( = = . (2) 29 [ C g π 2 , 0 : x i x x g sin cos ) ( + = , where 1 2 - = i . If [ ] π , 0 = X , then [ ] X g is a unit semi-circle above the real axis. Definition Let B A f : be a function from A to B and B Y . The Inverse image [ ] ( 29 Y f 1 - of Y under f is defined as [ ] { } f x f A x x Y f = - ) ( and : 1 Prepared by K. F. Ngai Page 4
Functions Advanced Level Pure Mathematics Example (1) Let the function B A f : be represented by the following figure.

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Functions - Functions Advanced Level Pure Mathematics...

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