RI Functions C4 Lect Notes

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ______________ Chapter 4: Functions Page 1 of 36 Chapter 4: Functions SYLLABUS INCLUDES  concepts of function, domain and range;  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. PRE-REQUISITES  Basic exponential, logarithmic and trigonometric expressions  Algebraic manipulations involv ing exponential, logarithmic and trigonometric expressions.  Solving inequalities CONTENT 1 Relations and Functions 1.1 Relations 1.2 Functions 1.3 Representation of a Function 1.4 Existence of a Function 1.5 Graphical Method to determine Range of a Function 1.6 Even and Odd Functions 1.7 One-One Functions 2 Inverse Functions 2.1 Condition for the Existence of an Inverse Function 2.2 Definition and Range of an Inverse Function 2.3 Graphical Relationship betw een a Function and its Inverse 3 Composite Functions 3.1 Condition for the Existence of a Composite Function 3.2 Definition of a Composite Function 3.3 Range of a Composite Function 3.4 Composition of a Function and its Inverse 4 Trigonometric and Inverse Trigonometric Function s 4.1 Sine and Inverse Sine function 4.2 Cosine and Inverse Cosine function 4.3 Tangent and Inverse Tangent function 5 Solving Inequalities Graphically Appendix: Graphing Calculator Task

Raffles Institution H2 Mathematics 2025 Year 5 ________________________________________________________________________________________ ______________ Chapter 4: Functions Page 2 of 36 INTRODUCTION One of the most important themes in calculus is the analysis of relationships between physical and mathematical quantities. These re lationships can be represented in terms of formulae, graphs or numerical data. In this chapter, we will develop the concept of a function, which is a platform for almost all mathematical and physical relationships. 1 Relations and Functions 1.1 Relations A relation from a given set A to a given set B is a rule for connecting elements in A (inputs) to elements in B (outputs). Set A is known as the domain and set B the codomain of the relation. Consider {0, 1, 2}A and {0, 1, 2, 3}B . Let’s say that an element x in A is related (or mapped) to an element y in B if and only if x is less than y. This relation connects the elements in set A to the elements in set B (e.g. 1 is related to 2 and 3 since 1 is less than 2 and 3). We can illustrate this relation from set A to set B by using an arrow diagram as shown in Fig 1.1. There are countless examples of relations in our daily life. Can you formulate some of them? Note that there are many different types of relations. For example, we can have a one to one relation as shown in Fig. 1.2, a many to one relation as shown in Fig. 1.3 and a one to many relation as shown in Fig. 1.4. The domain of the relation shown in Fig. 1.1 is {0, 1, 2}. Can you identify the domains for the examples in Fig 1.2, 1.3 and 1.4? 0 1 0 2 3 B Fig 1.1  0 1 A 2  1 2 3 Individuals Thumbprints A B C  3A 3B Students Classes Fig 1.2 A B C D E F G Fig 1.3  A B Students Subjects Fig 1.4 H1 GP H2 History H2 Physics H2 Chemistry H2 Art

Raffles Institution H2 Mathematics 2025 Year 5 ________________________________________________________________________________________ ______________ Chapter 4: Functions Page 3 of 36 1.2 Functions A function f: XY is a relation which maps each element in the set X, to one and only one element in the codomain Y. The set X is called the domain of f, denoted by fD , and the set Y is called the codomain of f. Based on the definition of a function, which of the relations shown in Fig. 1.1 – 1.4 are functions? Notes: (1) We usually use lower case letters such as f or g to name a function. (2) If the element aX is mapped to the element bY , then we write f( )ab . b is called the image of a under f . The range of f is the set of images of X under f. i.e. Range of f  f( ):xxX . In this case, 1, 2, 3X  is the domain and 1, 2, 8, 24, 27Y is the codomain. Range of f1 , 8 , 2 7 . * Range of f  Codomain of f. (3) Two functions are equal if and only if they have the same rule and domain. 1.3 Representation of a Function At ‘A’ Levels, most functions considered are relations between real numbers where the rule can be stated algebraically. It is not necessary to state the codomain in this context, and it suffices to represent a function by defining its domain and rule. In this case, we take range of f as the codomain of f. Eg. we write (1) 2f: xx x  , where x   , (2) 2f( )xxx , where x   . 1.4 Existence of a Function A relation f is more often shown as a graph on which the points ( x, f( )x ) are plotted. The equation of the resulting curve can be written in the form f( )yx . Graphically, to decide whether a relation is a function, any vertical line f,Dxk k drawn should cut the graph of f at one and only one point. 1 2 2 1 8 27 24 Y X 3 

Raffles Institution H2 Mathematics 2025 Year 5 ________________________________________________________________________________________ ______________ Chapter 4: Functions Page 4 of 36 1.5 Graphical Method to dete rmine Range of a Function Consider a function f with domain denoted by fD . The range of f, denoted by fR , is the set of images f( ):xxX . One of the more commonly used methods for finding the range of a function is by sketching its graph with equation fyx , where fDx . When sketching graphs, one should take note of characteristics such as symmetry, inte rsections with the axes, turning points and asymptotes. A graphing calculator is a useful tool in sketching graphs. However, it has certain limitations. For example, it cannot draw vertical asymptotes, and it also does not know what critical features of a graph to display. It is thus usef ul to be acquainted with the equations of basic functions and understand the properties of their graphs. Example 1 (a) Sketch the graphs of the following functions and state its range: (i) 2f: 8 xxx  , x  , 11 0x , (ii) g: l nxx , x   , (iii) h: 1 e xx  , x  , 2x , (iv) 2 ,, 0 ,k( ) 1, , 0. xx xx xxx      (b) A curve C has equation 2 2yx , 0x   . By considering the graph of C, show that C does not represent a function. Solution : (a)(i) 2f: 8 xxx  , x  , 11 0x , f (1) = 7 , f (4) = 16 , f (10) = 20 fR1 6 , 2 0 y = f(x) (8, 0) O y x (10, 20) (1,7) (4,16)

Raffles Institution H2 Mathematics 2025 Year 5 ________________________________________________________________________________________ ______________ Chapter 4: Functions Page 5 of 36 GC Keyst