RI Functions C4 Tut (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ___________________________________________________________________________ _________________ Tutorial 4: Functions Page 1 of 4 Tutorial 4: Functions Section A (Basic Questions) 1 Find the range of the following functions: (a) 2f: 2 3 , , 1 5xx x x x     , (b) g: ( 5 ) ( 8 ) , , 3 5xx x x x x      , (c) 2h: t a n , , , 63 2xx x xx     [(a) [4,12] (b) [151,66.5] (c) 1,3 , 3        ] 2 For each of the following functions, sk etch the graph and determine if the inverse function exists. If it does, find it in a similar form. (a) 2f( ) 4 3 , , 1xx x x x    , (b) 2 3g( ) ,xx x  , (c)  1h( ) ln 2 1 , , 2xx x x    . [(a) 1f( ) 2 1 , 0xx x    (c)  1 1h( ) e 1 , 2 xxx   ] 3 The function f is defined by 2f: 2 ( 1 ) , , xx x x k   . Determine the largest value of k for which the function 1f exists. [largest value of k = 1] Section B (Discussion Questions) 1 The function f is defined by 2 3 for 2 ,f( ) 1 for < 2 . xxx xx    Sketch the graph of f and find its range. Hence show that f is one-one and find 1f . 1 1 for 1 f( ) 3 for 1 xx x xx        2 Functions f and g are defined as follows: f: , ,xx x  g : e , . xxx  . Show that the composite function gf exists and find its range. [ (0,1] ]

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _________________ Tutorial 4: Functions Page 2 of 4 3 The functions f and g are defined as follows:  2 f: 7 1xx a  , x  , 3x g: 1 e xxb  , 0x , where a and b are real constants. (i) State the smallest value of a such that 1f exists. For the rest of this question, 4a . (ii) Explain if 1f exists, and if it does, find   1f x and state its domain. (iii) Find the largest value of b such that the composite function fg exists. For this value of b, find  fg x , state its domain, and find its range. [(i) 3a (ii)  1 1f4 7 xx  , 6x (iii) 2b ;    2 fg 7 1 e 1 xx   ;  fgD0 , ;  fgR6 , 2 7 ] 4 The functions f , g and h are defined as follows: 3f: ( 3 ) , 3xx x  , g: ln( 1 ), 1xx x   , 1h: , 0xx x  . (i) Find 1ff and 1ff , stating clearly their rules and domains. Hence state the set of values of x for which 11f f () f f ()x x  . (ii) The functions 1k and 2k are defined as follows: 1 1k: l n ( 1 ) , 0 ,xx x 2k: , 0 .xx x  Express 1k and 2k as appropriate compositions of the functions f, g and/or h. [(i) [3, )x ] 5 The function f is defined by , x  2. (i) Find the range of f. [1] (ii) State, giving a reason, whether ff exists. [2] The function g is defined by , x  1. (iii) Sketch the graphs of , and on a single diagram, showing clearly their geometrical relationships. [4] (iv) If 1g( ) g ( )  , find the values of the constants p and q such that . [2] (v) Find an expression for . [3] [(i) (iv) 1p  , 2q  (v)  1g1 3x x   ]  2 f: 3 1xx   g: fx x g( )yx 1g( )yx  1gg ()yx  2 0pq  1g( ) x  fR, 3 

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _________________ Tutorial 4: Functions Page 3 of 4 6 Functions f and g are defined by f: | 1 | ,xx x   , 2g: l n ( ) , , 0xx x x  . (i) Sketch the graphs of f and g, and state the range of each function. (ii) Explain why fg exists, defi ne fg and state its range. (iii) Using a graphical approa ch, solve the inequality fg( ) f ( ) 0xx . [(i) 0  ,  (ii) 0  (iii) 5.36 0.703 or 0.464 0 or 0 0.314xx x        ] 7 The functions f and g are defined as follows: 1f: , 1x x ,0 o r 1xx , 1g: 1 ,x x ,0 o r 1xx . It is given that the range of g is  \0 , 1 . (i) Prove that the composit e function fg exists. [2] (ii) Express fg in a similar form and hence de scribe the relationship between f and g. [3] (iii) Evaluate   2011 1994 1 2fg . [3] [(ii) fg : x  x, ,0 o r 1xx , f and g are inverse functions (iii) −1] 8 (a) The function f is given by .f : 1 , for 0 ,xx x x  (i) Find 1f( ) x and state the domain of 1f. [3] (ii) Show that if ff ( )x x then 23 41 0 .4xxx   Hence find the value of x for which ff ( ) .x x Explain why this value of x satisfies the equation 1f( ) f ( ) .x x [5] (b) The function g, with domain the set of non-negative integers, is given by 1 2 1f o r 0 , g( ) 2 g( ) for even, 1 g( 1) for odd. n nn nn n     (i) Find g(4), g(7) and g(12). [3] (ii) Does g have an inverse? Justify your answer. [2] [(a)(i) 1 12 ff( ) ( 1 ) , D [ 1 ,)xx     (ii) 1 2 (3 5)x  (b)(i) 6,8,9 ]

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _________________ Tutorial 4: Functions Page 4 of 4 9 The functions f and g are defined as follows.    2 f 2 1 , , 12 , 0 2,33g 1 3 , 2. xx x xx x xx            (i) Show that 1f does not exist. [1] (ii) If the domain of f is restricted to [, )k  such that 1f exists, state the least value of k and define 1f in a similar form. [3] Use the new domain of f found in part (ii) for the following parts. (iii) Find the range of the composite function gf. [2] (iv) Find the value of x such that gf 1x  . [1] [(ii) 2, 2 + (x  1)2, x  1 (iii) (, 1 ] (iv) 6] 10 It is given that  2 2,0 2f( ) 24 ,2 4 xxx xx      and that f( ) f( 4 )xx for all real values of x. (i) State the value of f( 2 1 ) . [1] (ii) Sketch the graph of f( )y x for 66 .x [2] The function g is defined by g( ) 4, 4 20.xx x   (iii) Find fg in a similar form as f. [4]     2 42, 4 8 fg( ) 22 1 44 , 8 0 xxx xx         i iii