Functions - Functions Advanced Level Pure Mathematics...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of 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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/26/2011 for the course COMPUTER S 1003 taught by Professor Angelosstavrou during the Spring '11 term at King Saud University.

Page1 / 23

Functions - Functions Advanced Level Pure Mathematics...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online