TMJC H2 Chapter 3 Functions Assignment Solutions 2024

Chapter 3 Functions TMJC 2024 Page 1 of 5 H2 Mathematics (9758) Chapter 3 Functions Assignment Solutions 1 2011/9740/II/3 The function f is defined by   1f : ln 2 1 3, , . 2x x x x      (i) Find   1f x and write down the domain and range of 1f  . [4] (ii) Sketch on the same diagram the graphs of  fy x and   1fy x  , giving the equations of any asymptotes and the exact coordinates of any points where the curves cross the x- and y-axes. [4] (iii) Explain why the x-coordinates of the points of intersection of the curves in part (ii) satisfy the equation  ln 2 1 3,x x   and find the values of these x-coordinates, correct to 4 significant figures. [3] Q1 Solution (i) Let  ln 2 1 3y x   . 3 3 2 1 e e 1 2 y y x x        3 1 e 1f 2 x x     1 1 f ff f 1, , 2D R R D             (ii) x y  fy x   1fy x  (0,3) (3,0) 3e 10, 2       3e 1 , 02       1 2y  1 2 x  The functions are still increasing, so do not ‘bend back on itself’. Question asks for ‘exact coordinates’. y x Note: all 3 graphs, ݕ= f(ݔ) , ݕ= f ିଵ(ݔ) and ݕ= ݔ all intersect at the same points (this includes the asymptotes) Reminder to use the same scale for x- and y-axis

Chapter 3 Functions TMJC 2024 Page 2 of 5 (iii) The points of intersection also lies on the line y = x, thus the x-coordinates also satisfy the equation of      f ln 2 1 3 ln 2 1 3x x x x x x         . By GC, x = –0.4847 or x = 5.482 (4 s.f.) There are 2 values of x so make sure the graphs in part (ii) intersect at 2 points. Note that GC does not show the intersection but you can still find the intersection.

Chapter 3 Functions TMJC 2024 Page 3 of 5 2 2017/PJC Promo/Q7 (modified) Functions f and g are defined by   2 f : 1 +2 for x x x    , 1g : for , 5 . 5x x x x    (i) Only one of the composite functions fg and gf exists. Give a definition (including the domain) of the composite that exists, and explain why the other composite does not exist. For the composite function that exist, find its range. [5] (ii) If the domain of f is restricted to x k , k  , state the greatest value of k for which the function 1f exists. For the rest of the question, use the value of k found in part (ii). (iii) Sketch on the same diagram the graphs of  fy x for x k and   1fy x  , showing clearly the relationship between the two graphs. [4] (iv) Find   1f x and state the domain of 1 f  . [3] Q2 Solution (i)  gR \ 0   , fD   Since g fR D , fg exists   1fg = f 5x x      2 1 1 +25x      2 1fg : 1 +2, , 5 5x x x x           fgD = \ 5  fR [2, )  ,  gD = \ 5  Since f gR D , gf does not exist.       g f fg g g fgD D = \ 5 R \ 0 R 2,         (ii) Greatest k = 1 y x (−1, 2) (0, 3) y = f (x) with domain ℝ\{0} y x (−1, 2) (0, 3) y = f (x) y x y = g (x) 10, 5     O O O

Chapter 3 Functions TMJC 2024 Page 4 of 5 (iii) (iv) Let y =  2 1 +2 x , 1x   2 1 2x y   1 2x y   1 2x y   , since 1x 1f ( ) 1 2, , 2x x x x       , 1fD =  [2, ) y x (−1, 2) ݕ= f(ݔ) ݕ= f ିଵ(ݔ) 2, −1) ݕ= ݔ O Reminder to use the same scale for x- and y-axis so that ݕ= f(ݔ) and ݕ= f ିଵ(ݔ) are symmetrical about the line ݕ= ݔ

Chapter 3 Functions TMJC 2024 Page 5 of 5 3 It is given that and that    g g 4x x  for all real values of x. (i) Sketch the graph of y = g (x) for . [4] (ii) Evaluate    g 3 g 7  . [3] Q3 Solution (i) (ii)              2 g 3 g 7 g 1 g 3 6 1 2 3 2 9            2 6 for 0 2,g 2 2 for 2 4,        x xx x x 6 10x   2 4 0 6 8 2 10 x 4 6 y 2