ACJC 2025 Functions Lecture Notes

1 2 FUNCTIONS SYLLABUS ▪ Concept of function, domain and range ▪ Finding inverse functions and composite functions ▪ Conditions for the existence of inverse functions and composite functions ▪ Domain restriction to obtain an inverse function ▪ Relationship between graphs of a one-to-one function and its inverse

ACJC 2025/26 H2 Mathematics (9758) 2 CONTENTS 0 Basic Concepts on Set Language and Notation ................................ 2 1 Concept of Function ......................................................................... 3 1.1 Relations.................................................................................. 3 1.2 Definition of a function; Domain of a function ...................... 4 1.3 Range of a function ................................................................. 6 2 Types of Functions ........................................................................... 7 2.1 Some Basic Algebraic Functions ............................................ 7 2.2 Even and Odd Functions ......................................................... 7 2.3 Periodic Function .................................................................... 8 2.4 One-one Function .................................................................. 10 3 Inverse Function ............................................................................. 11 3.1 Existence of an Inverse Function .......................................... 11 3.2 Definition of an Inverse Function ......................................... 12 3.3 Domain and Range of an Inverse Function ........................... 12 3.4 Graph of an Inverse Function................................................ 14 4 Composite Function........................................................................ 16 4.1 Existence of a Composite Function ...................................... 16 4.2 Definition of a Composite Function...................................... 18 4.3 Domain and Range of a Composite Function ....................... 19 Annex A: Some Basic Algebraic Functions ............................................ 26 Annex B: Graphs of Inverse Trigonometric Functions............................ 28 Annex C: Practice Questions on Functions ............................................. 30

2 Functions 3 LECTURE 1 Lesson Outline • Basic concepts on set language and notation • Concept of function including mapping, definition, domain and range of a function • Types of functions including basic algebraic, even and odd functions 0 RECAP: SET LANGUAGE AND NOTATION A set is a collection of objects such as numbers, letters, people etc. The objects in a set are called elements of the set. Conventions and Notations • Uppercase letters such as A, B, C etc. are often used to denote a set. • Lowercase letters such as a, b, c etc. are often used to denote the elements of a set. • We write aA to mean that the element a belongs to the set A, or a is an element of A. • We write bB to mean that the element b does not belong to the set B, or b is not an element of B. The set of all elements that are not in B is denoted by B. • We write AB if set A is a subset of set B i.e. every element of A is an element of B.

ACJC 2025/26 H2 Mathematics (9758) 4 Notations for important sets of numbers  the set of natural numbers =  1, 2,3,...  the set of integers =  ... 3, 2, 1,0,1, 2,3, ...− − −  the set of rational numbers. For example, an element in  may be 1 2 , 5 4− , 0 or 9 8 .  the set of real numbers. For example, an element in  may be 1, , e, 1 2 , 0.1212... or ln 9.  the set of complex numbers. For example, an element in  may be 1, , e, 1 2 , 1− , 32+− . Other Common Set Notations Interval notation  ,x a b { : }x a x b    ),x a b { : }x a x b   ( ,x a b { : }x a x b   ( ),x a b { : }x a x b   Superscript x + { : 0}xx x − { : 0}xx Union (of sets)  0x + { : 0}xx  0x − { : 0}xx Difference (of sets)  \0x  { : 0 or 0}x x x    \ 0,1x  { : 0,1 }xx Note that     .

2 Functions 5 1 CONCEPT OF FUNCTION In Mathematics, we attempt to explain physical phenomenon by relating a physical quantity x to another physical quantity y. Ordered pairs 11( , )xy , 22( , ), ..., ( , ) nnx y x y form a relation. This relationship can be represented in the form of a formula, graph or table. The concept of function is essentially a mathematical expression that serves to describe this relationship. 1.1 Relations A relation or mapping is any rule that associates two sets of items. There are many examples of relations in real life: Diagrams illustrating the various types of relationship between elements of sets. A relation can be one-to-one, one-to-many, many-to-one, or many-to-many. Here are some examples of mathematical relations. In each example, the items on the left form an input, and the items on the right form an output. One-to-one relation Identification Numbers Names T001 T002 T003 T004 T005 Jane John Joel Andy Aaron Identity Jenny Lynn Ann Parentage Mothers Jane John Joel Andy Aaron Children Many-to-one relation Subject Combinations GP Math Physics History Jane John Subjects Students One-to-many relation 3 5 7 9 11 −1 0 1 2 3 0 1 0 45 90 135 180 Equations  Solutions

ACJC 2025/26 H2 Mathematics (9758) 6 1.2 Definition and Domain of Function A relation that is one-to-one or many-to-one is called a function. The set of inputs is called the domain of the function. More formally, a function f is a rule or relation which assigns each and every element x in the domain, fD , to a unique image, f(x). In other words, for each fxD , there corresponds one and only one element f(x). The set of images is called the range, fR , of the function. Notes 1 If an element a is mapped to an element b, we write f( )ab= . 2 When defining a function , both the rule and domain must be stated. Note that the functions f : ln , 0x x x  and g : ln , 1x x x  are different functions as they have different _________________ (although the ___________________ is the same). 3 There are several different but equivalent ways of writing a function. For example, the function which maps x to 25x + , where the domain is the set of positive real numbers, can be written in any of the following ways: f ( ) 2 5xx=+ , x + f ( ) 2 5, 0x x x= +  f : 2 5xx + , x + f : 2 5, 0x x x + 4 A function can be represented visually by a graph, a table or mappings between sets. 5 The “vertical line test” can be used as a quick check to see if a graph represents a function. If the line xk= , for all fkD , cuts the graph of f ( )yx= exactly once, then the graph represents a function. f(x) x f

2 Functions 7 Example 1 Relations f, g, h and m are defined by the following diagrams. Identify which of them are functions. (a) (b) (c) (d) ■ Example 2 Which of the following are graphs of functions? (a) (b) Solution No, this is not a function. The vertical line xk= (see graph) cuts the graph of f ( )yx= twice. This means one value of x is mapped to two values of y, hence f is not a function. Yes, this is a function. The vertical line xk= (see graph), for any real value of k, cuts the graph of g( )yx= exactly once . This means each value of x in the domain is mapped to only one value of y, hence g is a function. ■ –1 1 –2 2 –3 f 1 4 p q r g a b p q r h a b c x m(x) 1 −1 1 1 4 −2 4 2 9 −3 9 3 y x a −a y x

8 Self-Practice Exercise 1 Which of the following are graphs of functio