RI Functions C4 Add Prac (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ________________________________________ Additional Practice Questions for Chapter 4: Functions Page 1 of 4 Additional Practice Questions for Chapter 4: Functions 1 The function g is defined by 1g : 1 , where , .1xx x a x   (i) State the value of a and explain why this value has to be excluded from the domain of g. [2] (ii) Find  2g x and  1g, x giving your answers in simplified form. [4] (iii) Find the values of b such that     21gg .bb  [2] 2 The function f is defined by  3 f: l n 8 ,xx  8x . (a) Solve the inequality f( )xx . [3] (b) Find, in a similar form, the inverse function 1f . [3] 3 (a) The function h is defined by 21h: 3 , 2xx for x  . The function g is defined by 1g: , 51 xx x   for x  ,0 . 2 .x (i) Find gh(2). [2] (ii) Find the value of x for which g( ) 1.4.x  [1] (b) The function f is defined by f: , 2 xax xb   for x  ,.xk (i) Give an expression for k and explain why this value of x has to be excluded from the domain of f. [2] The function f is such that 1f( ) f ( )xx  for all x in the domain of f. (ii) Determine the possible values of a and of b. [3] (iii) Find an expression for 1f( 4 ) .  [1] 4 The function f is defined by f: 2 1 4 ,xx x    . (i) Show that the composite function ff exists and give the corresponding definition of ff . State the range of ff . (ii) Solve the equation f( ) 1 0 0x  . (iii) The function g is defined by g: 2 1 4 , .xx x k   Find the greatest value of k for 1g to exist. Using this value of k , find 1g .

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________ ________________________________________ Additional Practice Questions for Chapter 4: Functions Page 2 of 4 5 Functions g and h are defined by 2g: 2 ,xx m x x  where m is a constant,  h: l n 1 2 , 1x xx  . (i) Sketch the graph of h( )yx , stating the exact coordinates of the point of intersection with the axis and the equation of the asymptote. [2] (ii) Find 1h( ) x and write down the domain of 1h . [3] (iii) On the same diagram as in part (i), sketch the graph of 1h( ) .yx  Find the solution of the equation 1h( ) h ( )x x . [3] State the condition for the composite function hg to exist. Find the maximum range of values of m such that hg exists. [3] 6 The functions f , g and h are defined as follows: 2 2 f: 4 1 , 1g : 1 , \{0} h : e , x xx x x k xx x xx        (i) State the smallest value of k such that 1f exists and express 1f in a similar form. [3] (ii) Express each of the following functions in terms of f, g and h as appropriate. (a) ln , 0xx x  (b) 2 11, e xxx  (c) , 3xx x  [4] 7 Functions f and g are defined by 2f : 2 for , where is an unknown constant g: t a n f o r , . 22 xx x a x a xx x x         (i) Find the range of f in terms of a. [2] (ii) Show that the composite function 1fg exists. Find the range of 1fg in terms of a. [3] (iii) It is given that 3a . The domain of f is now restricted to  :1xx . For the new domain, give a definition (including the domain) of -1f. [3] 8 The functions f and g are defined by 3f: 2 1 , 0xx x  , g: l n ( ) ,x xa x a . (i) Give a reason why 1f exists. Hence find 1f( ) x and state the domain of 1f . [3] (ii) Only one of the composite functions fg and gf exists. State the greatest possible value of a such that this composite function exists and using this value of a, give a definition of this composite function. Write down the range of the composite function. Explain why the other composite function does not exist. [5] (iii) By sketching suitable graphs or otherwise, find the set of values of x such that 11ff ( ) f f( )x x  . [2]

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________ ________________________________________ Additional Practice Questions for Chapter 4: Functions Page 3 of 4 9 The functions f and g are defined as follows: 2 5f: , , 1 , 1 g: 2 4 , , 2 . xxx x x xx x x x            It is given that range of f is ( ,1) (1, ).   (i) Show that 1ff  . (ii) Evaluate   51f4 . (iii) Find the range of values of  such that fg exists. For these values of , find the range of fg in terms of . 10 The function f is defined by f : x  (ax + b)2, x   , x  b a where a and b are positive constants. (i) Show that 1f exists. [2] (ii) State the geometrical relationship between the graphs of f and 1f . Given that the graphs of f and 1f intersect, show that 1 4a b . [4] (iii) Show that the composite function ff is defined. [2] (iv) Solve the equation ff(2) = b2. [3] 11 The functions f and g are defined by  2 f: 5 0 2 4 ,xx  ,kxk 2 33g: 81 8 xx xx   , ,x  where k is a positive real number. (a) Without using a calculator, solve the inequality g( ) 2.x  [3] (b) It is given that the range of f is [5 0 , 5 0 ] . (i) Find the value of k. [2] (ii) Without the use of a calculator, solve the inequality gf( ) 0.x  [4] 12 The functions f and 1fg are defined as follows: f : 5,xx x   , 12fg : , , 5xx x x  (i) Find the functions g and fg. (ii) The function h is defined such that its inverse exists and the graphs of f( )yx and 1h( 3 1 )yx  are symmetrical about the line y = x. Find h(x).

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________ ________________________________________ Additional Practice Questions for Chapter 4: Functions Page 4 of 4 13 Functions f and g are defined by  1 2f : 4 2 , , 0 16xx x x    g : 3 1 , xx x  (i) State the range of f. [1] (ii) With the aid of a diagram, show that 1f exists and define 1f in a similar form. [4] (iii) On the same diagram as in part (ii), sketch the graphs of 1f and 1ff , indicating their endpoints. [3] (iv) Explain why the x-coordinates of the point(s) of intersection between the graphs in part (iii) satisfies the equation 2 24 0xx  . [1] (v) State whether the composite function fg exists, justifying your answer. [2] (vi) Not in syllabus (For enrichment) Find the largest possible domain of g in the form [m, n], m, n   , for which the composite function fg exists. 14 The function f and g are defined by 2f( ) e 4 , , g( ) 2, . xxx xx x       (i) Find 1f( ) x and state its domain [3] (ii) Find the exact solution of fg( ) 5x  , giving your answer in its simplest form. [3]