Chp 4 Tutorial 4 APQ solutions (RVHS)

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2022 FUNCTIONS - Additional Practice Questions’ Solutions 1. [EJC/2019/JC1 Promo/Q7] The function f is defined by f : , for , ,bx bx x xax b a  where a and b are non-zero constants. (i) Find   1f x and state the domain of 1f . [3] (ii) Hence show that   2f xx , and write down  f n x where n is an odd number. [2] The function g is defined by g : 2 e , for , 0.xx x x    If 2a and 1b for function f, (iii) Explain why the composite function gf does not exist. [1] (iv) Find an expression for  fg x and state the domain and exact range of fg . [3] EJC JC1 Promo 9758/2019/01/Q7 (Solutions) Suggested solution (i) f : , for , ,bx bx x xax b a  Let  bxy y ax b bxax b    axy bx by byx ay b      1f : , for , , bx bx x xax b a   (ii) Since 1f ( ) f ( )xx  ,     21f f fx x x        32f f f fx x x  , therefore for    ffn bxxx ax b  for n odd, (iii) If 2a and 1b for function f, 1f : , for , ,2 1 2 xx x x x  f 1R\ 2    D 0,g  As fRD g , gf does not exist. (iv)    f2f eg xx    2e 2 2 e 1 2e ,03 2e x x x x x         fg gD D [0, )   Range of fg 32R = , 53  

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2022 2. [MI/2019/J1 Promo/01/Q5] The function f is defined by 2 1f : for , 2 4x x x x   . (i) Find   1f x and write down the domain of 1f . [3] (ii)On the same diagram, sketch the graphs of  fyx ,   1fyx  and   1ffyx  , stating the equations of any asymptotes and showing the relationships between the graphs clearly. [4] M1 JC1 Promo 9758/2019/01/Q5 (Solutions) 5(i)       2 2 2 1 1 1 = 4 14 14 114 rejected 2 or 4 1f 4 Domain of f Range of f 0, y x x y x y x x x yy x x                 5(ii) 3. [TMJC/2019/JC2 Prelim/01/Q7] The function f is defined by x y O x y O

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2022   for 1, where is any positive odd integer, f for 1, where is any positive even integer .2 n x n x n n x nx n x n n         (i) Show that  f 1.5 0.5 and find  f 2.5 . [2] (ii) Sketch the graph of  fyx for 15 x . [2] (iii) Does f have an inverse for 15 x ? Justify your answer. [2] (iv) The function g is defined by 21g : , , 1 1 xx x x x    . For 23 x , find an expression for  gf x and hence, or otherwise, find   1 2gf 3    . [4] TMJC JC2 Prelim 9758/2019/01/Q7 (Solutions) For 12 x ,