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 −   Rule Domain h : sin , x x x  Recall: Notation Example ( )ba, is equivalent to  :x a x b   . ( )10,8− is equivalent to  : 10 8xx −   . ( ),−  is equivalent to  x . ( ba, is equivalent to  :x a x b   . ( 10,8− is equivalent to  : 10 8xx −   . ( ,8− is equivalent to  :8xx .  )ba, is equivalent to  :x a x b   .  )10,8− is equivalent to  : 10 8xx −   .  )10,− is equivalent to  : 10xx − .  ba, is equivalent to  :x a x b   .  10,8− is equivalent to  : 10 8xx −   . Note: • A function is not completely defined if the domain is not indicated. • There should be exactly one result (output) originating from each xX (input). • All functions are relations since a function is a special type of relation. However, a relation may not be a function.

Chapter 3 Functions TMJC 2024 Page 4 of 27 1.4 Test for Function (Vertical line test) (a) Given a relation YX →:f where ,XY  , i f for each aX , the vertical line xa= cuts the graph of f at exactly one point, then f is a function. (b) When a relation is not a function, we can show this by just giving a counter-example, i.e. find the equation of a particular vertical line that cuts the graph more than once or does not cut the graph at all for some xX . Example 1 Determine which of the following relations are functions: (a) 2g( )xx= for x ax = Any vertical line ax = , a , cuts the graph of g at exactly one point.  g is a function (b) h( )xx= for 0x x = 1 The line 1x= cuts the graph of h at 2 points, (1, −1) and (1, 1).  h is not a function. x y O x y O

Chapter 3 Functions TMJC 2024 Page 5 of 27 1.5 Maximal Domain The maximal domain is the largest possible domain for which a function can be defined. Example 2 State the maximal domain for each of the following functions to exist. (a) )1ln(:f −xx  ( )1ln −x is defined if 1 0 1xx−    Maximal domain for function f to exist = ( )1, (b) xx −+ 32:g  x−3 is defined if 303 − xx Maximal domain for function g to exist = ( 3,− (c) xx 1:h  x 1 is defined if 0x Maximal domain for function h to exist =  \0 1.6 Restricting the Domain of a Function Consider the following functions and graphs obtained using GC: 2f : , x x x  2g : , 3 2x x x −   0 ,:h 2 xxx  Q: Are functions f, g and h considered the same functions? Ans: No, they are not the same function. Though the three functions have the same rule, their domains are not the same. Two functions are the same when BOTH the rule and domain are the same. In fact, in this case, since gD and hD are subsets of fD , we say that the functions g and h are restrictions of f. x y O x y O (−3,9) (2,4) x y O (0,0) (3,2) 1x= 0x= 0y=

Chapter 3 Functions TMJC 2024 Page 6 of 27 y x O 1.7 Determine the range of a function The range of a function, f, can be determined by sketching the graph of f ( )yx= to find the set of values y can take. Example 3 Determine the range of each of the following functions. (a) f : 1, x x x ++ (b) g : sin , x x x  (c) q : (3 )(1 ), 2 2x x x x+ − −   (d) 1r : 1 , \{0}xx x− (a) f : 1, x x x ++ ( ) ( ) f f D 0, R 1, = = (b) g : sin , x x x  ( )   g g D, R 1,1 = −  =− (c) q : (3 )(1 ), 2 2x x x x+ − −       q q D 2, 2 R 5, 4 =− =− (d) 1r : 1 , \{0}xx x−    D \ 0 R \ 1 r r = = Note: (1) The range of a function depends on both the rule and the domain. Hence, it is important to only consider the part of the graph corresponding to the required domain. (2) Range must be stated in set notation. (3) Indicate end-points with ‘open’ (if exclusive) or ‘closed’(if inclusive) circle to remind yourselves whether to include the values when writing down the range. y x O (0,1) y x O (2,−5) (−2, 3) (−1, 4) y x O y = 1 1 1− 0x=

Chapter 3 Functions TMJC 2024 Page 7 of 27 Q: Can we substitute the endpoints of the domain into the function to find range? Ans: Generally, no. See Example 3(c). However, if the function is increasing / decreasing on that domain, then we can just substitute the endpoints of the domain into the function to find range (see Example 3(a)). Using GC to sketch the graph and find the range of a function with the given domain Consider Example 3(c) q : (3 )(1 ), where 2 2x x x x+ − −   . Sketch the graph and find the range of q. INSTRUCTIONS TO USE KEYS SCREEN ON GC Step 1: Enter and plot the function (3 )(1 )y x x= + − for the restricted domain [–2, 2] using piecewise function. Go to ! and select m , option B: piecewise. In this case, select 1 piece. Inequality symbols “  ” and “ 