TMJC H2 Chapter 3 Functions Learning Package 2024

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Chapter 3 Functions TMJC 2024 Page 1 of 27 H2 Mathematics (9758) Chapter 3 Functions Core Concept Notes Success Criteria: Surface Learning Deep Learning Transfer Learning  Identify the rule of a function  Find the domain of a function  Find the range of a function  Define a function using functions notation  Identify whether a relation is a function  Check if a function is one-one Inverse Functions  Check the condition for the existence of inverse functions Composite Functions  Check the condition for the existence of composite functions Periodic Functions  Identify the period of a periodic function Inverse Functions  Find the rule of an inverse function  Find the domain of an inverse function  Find the range of an inverse function Composite Functions  Find the rule of a composite function  Find the domain of a composite function  Find the range of a composite function using its rule and domain Periodic Functions  Find the value of a periodic function f(x) given x Inverse Functions  Restrict a domain to obtain an inverse function  Explain the relationship between a function and its inverse Composite Functions  Find the range of a composite function using the mapping method Periodic Functions  Sketch the graph of a periodic function

Chapter 3 Functions TMJC 2024 Page 2 of 27 §1 Functions 1.1 Relations Let X and Y be two sets, where the elements in X and Y may or may not be the same. A relation is a rule that maps some elements of X to some elements of Y. There are many ways to relate 2 sets of elements, e.g. X and Y. X Y Fig (a) 1 2 −1 0 1 1 2 3 X 2 4 6 Y Fig (b) 0 1 X 0 −1 1 Y Fig (c) 2 −2 −3 X Y Fig (d)

Chapter 3 Functions TMJC 2024 Page 3 of 27 1.2 Functions, Domain and Range 1. A function YX →:f is a relation where f maps each and every element xX to exactly one element yY . (In other words, every input has exactly one output.) 2. Set X is known as the domain of f , and denoted by fD . (Set of input) 3. The element y is called the image of x under f. This is denoted by ( )f xy= . 4. The range of f (denoted by fR ) is the set of all images of xX . (Set of output) Q: In Figure (a)-(d), which of the relations are functions? (b) and (d) 1.3 Defining a Function To define a function, both the rule and the domain of the function must be clearly stated. Examples: Rule Domain f : 1, 0x x x − 2 Rule Domain g : , 1 1x x x −  