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 values as a set. Range of values Equivalent representation as a set of values Set Builder Notation Interval Notation 23 x    | 2 3xx    ( 2, 3) 23 x    | 2 3xx    [ 2, 3) 23 x    | 2 3xx    [ 2, 3] 2x  |2xx  ( , 2]  3x  |3xx (3, ) In the use of interval notation such as [a, b) or (a, b), note that  ab ,  we use round brackets if we want to exclude the extreme values of the range, and  we use square brackets if we want to include the extreme values of the range.

National Junior College Mathematics Department 2016 Functions Page 3 of 30 UNDERSTAND Can we write  0\ as ( , 0) (0, )   ? Why or why not? Answer: Yes. Both are sets that correspond to the same interval. §2 Special Type of Relations: Functions 2.1 Representations of a Relation A relation is an association between two sets of data (in our case, they are usually numbers). A relation can be represented in the following ways. (i) Numerically (as a table of values) Example: x 1yx 0 1 1 2 2 3 3 4 Note: This form of representation is useful especially when the input s are a set of a finite number of discrete values. (ii) Graphically (using a curve or line) Example: Note: This form of representation is useful when the input is an interval of real numbers. (iii) Diagrammatically (using set diagram) Example: (iv) Algebraically (as an equation) Example: 2yx 22 4xy 2.2 Relations versus Functions Example 2.2.1 x 1yx 0 1 1 2 2 3 3 4 x y y = x + 1 O 1 –1 Y (Output) X (Input) 0 1 2 3 1 2 3 4

National Junior College Mathematics Department 2016 Functions Page 4 of 30 The table above illustrates a relation R (i.e. 1yx ) between the elements of two sets of numbers {0,1,2,3} and {1,2,3,4}, namely x and y. It is obvious that R maps 0 to 1, 1 to 2, 2 to 3, and 3 to 4. Since each input (i.e. x–value) is mapped to exactly one output (i.e. y–value), this relation is a function. Example 2.2.2 The relation depicted in the set diagram above is not a function as the input “0” is mapped to two outputs, namely “1” and “2”. In this topic, our focus is to study functions, which is defined (not so formally) as follows. A function is a relation that has exactly one output for every possible input. A formal definition for function is given in Section 2.3. 2.3 Rule, Domain and Range of a Function We shall introduce the formal definition of a function. Definition 2.3.1 (Function) A relation f: XY is a function if and only if for each element xX , there exists exactly one element yY such that f ( ) .xy In the above definition, the element x is mapped to the element y, and y is said to be the image of x. A function f can be expressed algebraically as f : f ( ), ,x x x X  where f ( )x is the rule of the function f, and the set X is the domain (“inputs”) of the function f. The domain of function f is denoted by fD . Alternatively, a function f can be expressed as f ( ),y x x X or f : , f ( ),X Y x x where the set Y is the codomain of the function f (usually it is the set of real numbers ). Y (Output) X (Input) 0 1 2 3 1 2 3 4

National Junior College Mathematics Department 2016 Functions Page 5 of 30 It is not necessary for all the elements of Y to be the image of some x  X. The range (“outputs”) of the function f is the subset of Y which contains all the possible images of all the elements of X under f. It is usually denoted by fR . Example 2.3.1 Explain why the relations shown in Figures A and B are functions. Also, give two reasons why the relation shown in Figure C is not a function. Solution:  The relation as shown in Figure A satisfies the definition of a function.  The relation as shown in Figure B also satisfies the definition of a function, even though no element of X maps to the element 5 in Y, and two elements of X map to the element 7 in Y.  The relation as shown in Figure C is not a function, as o the element 3 in X maps to two elements in Y, and o the element 4 in X does not map to any element in Y. Example 2.3.2 The function with rule 3f ( )xx defined on domain + can be expressed as 3f ( ) , ,x x x  or 3f : , ,x x x  or 3f : , . xx  Example 2.3.3 Let f be the function with rule 3f ( )xx defined on  f 1,2,3D  . f (1) 1 , f (2) 8 and f (3) 27 . Codomain of f,  1,2,8,24,27Y  . Range of f,  f 1,8,27R  . 2 1 8 27 24 1 2 3 fD Y f x f fXD , domain fR , range f ( )yx Y, codomain Y X 1 2 3 4 3 5 7 9 Figure A Y X Figure B 1 2 3 4 3 5 7 9 Y X Figure C 1 2 3 4 3 5 7 9

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