NJC 03_2016_-_2017_H2_Mathematics_Functions_Notes_(Final)

National Junior College Mathematics Department 2016 Functions Page 1 of 30 National Junior College 2016 – 2017 H2 Mathematics Topic 3: Functions Key Questions to Answer: 1. What is a function? - What kinds of relationships hold between x and f(x)? - What is the difference between a function and a relation? 2. How do you define a function? - What does function notation look like? - When are two functions equal? 3. How do you determine the range of a function? - Is it sufficient to check the endpoints of the domain? Why? 4. What is a one-one function? - What are the other possible type(s) of relations? - What makes a one-one function different from the rest? 5. What is an inverse function? - What is the condition for an inverse function to exist? - Why is this condition necessary? - How do you find the rule of the inverse function? - How is the domain and range of the inverse function related to that of the original function? - What is the graphical relationship between a function and its inverse? 6. What is a composite function? - What is the condition for a composite function to exist? - Why is this condition necessary? - How do you find the rule of a composite function? - What is the domain of a composite function? - How do you find the range of a composite function? - In finding the range, is it sufficient to check the endpoints of the domain? Why? Before we begin the topic on Functions, it is important for students to be familiar with the use of set notation to represent a range of values. §1 Set Notation Recall that a set is a collection of distinct elements. The curly brackets “{...}” represent the p hrase “the set of ...”. When several sets are being discussed simultaneously, they can be denoted using capital letters.        120, 2, 4 0, 2, 4, ... 0A B S S   

National Junior College Mathematics Department 2016 Functions Page 2 of 30 Set operators: Notation Meaning Usage  (element) that belongs to (set)  2 0, 2, 4  union (of two sets)      0 2 0,2 \ (set) take away (set)      0,2 \ 20  intersection (of two sets)      0, 2, 4 0, 2, 0, 2 By convention, particular symbols are reserved for the most important sets of numbers: Symbol Meaning Set of real numbers (see Remark 1 below).  Set of positive real numbers (does not include zero).  Set of negative real numbers (does not include zero).  0 Set of non-negative real numbers. Remarks: 1. A number that can be represented on the number line is a real number. This means that for any real number x, .x  2. By definition, zero is neither positive nor negative. WONDER What are the symbols that represent the set of integers, rational numbers and natural numbers respectively? Answer: , , . 1.1 Representation of Sets of Values The use of set builder notation and/or interval notation is helpful to represent a range of valu