DHS Functions (9758) Topical Revision

3. Functions 1 3. Functions 1 PJC/2008Promo/8 The functions f and g are defined by 2f : (2 ),x x x→− x , 1g : 2 , , 0x x x x→ +   . (i) Find ( ) 1g x− and state its domain. [2] (ii) Determine whether the composite function gf exists. [2] (iii) Solve the equation fg(x) = 1, give your answer to 3 significance figures. [3] 2 SAJC/2008Promo/8 The function f and g are defined by f : x e x− , x  + g : x 3x2 + 2, x  − (a) Determine, with reason, whether the inverse for f exists. If 1f− exists, define 1f− in a similar form and state its range. On the same axes, sketch the graphs of f, 1f− and 1ff− . [5] (b) Determine, stating reason, whether fg exists. If the function exists, give its domain, rule and range. [4] 3 NYJC/2013Promo/7 The function f is defined by 2 1f : , , 1 2x x x x x→ −    . (i) Show, by differentiation, that f is strictly increasing. [2] (ii) State the range of f. [1] (iii) Solve the equation ( ) ( ) 1ff xx −= , giving your answer to two decimal places. [2] The function g is defined by πg: 1 sin , , 0 2x x x x→ +    . (iv) 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. [3] (v) For the composite function which exists, state its range. [1]

3. Functions 2 4 DHS/2009Promo/7 The functions f and g are defined by f : tan 1, 0 , 2x x x +   1g : , 0.xxx x +  (i) Find the derivative of h( ),x where 1h( ) f ( )x x= . Hence show that h is a one - one function. [3] (ii) Find an expression for 1h ( )x− . [1] (iii) Show that the composite function gf exists. Define gf in similar form and state the range of gf. [4] 5 JJC/2012Promo/5 The function f is defined by 2f: 1 xx x + − , for x , 1x . (i) Find 2f ( )x and 2012f ( ) x . [3] The function g is defined by g : cosxx , for 02 x  . (ii) Explain why the composite function fg exists. [2] (iii) Define fg, giving its domain. [2] (iv) Find the range of fg. [1] 6 JJC/2010Promo/6 An inverse function is defined by 12f ( ) ln( 1), , 1.x x x x− = −   (i) Find f(x) and state the domain of f. [3] (ii) Explain why 1ff− exists and find 1ff− in a similar form. [2] (iii) Sketch the graph of 1ff ( )yx −= and state the range of 1ff .− [2] 7 HCI/2008Promo/14 [Part ii removed. Out of syllabus] The functions f and g are defined as follows: 2f : 3 2x x x −− , , x x k , 4g : , 0 4 xx e x −  . State the largest value of k such that 1f− exists, and find 1f− in a similar form. [4] (i) Show that the composite function gf does not exist. [1] (iii) Find the set of values of x such that g–1g(x + 1) = g g–1(x + 1). [3]

3. Functions 3 8 NJC/2010Promo/5 The functions f and g are defined by 1f : , 2 1, 1xx x −  −+ 2g : 4 , 2 ,x x x x − where  is a real constant. (i) Find ( ) 1g x− in terms of  , stating the domain of 1g− . [3] (ii) Determine the set of values of  for which the composite function gf exist . [2] (iii) Given that 1 =− , find the range of gf. [2] 9 NJC/2013Promo/10 The function f is defined as follows: ( ) 2f : ln 1xx + , x . (i) Without the use of a calculator, find the set of values of x for which the graph of f ( )yx= is concave upwards. [4] (ii) Sketch the graph of f ( )yx= . [2] (iii) The function f has an inverse if its domain is restricted to .xk State the set of all possible values of k . [1] The function g is defined by ( ) 2g : ln 1 , 1. + x x x (iv) Find g -1 (x) and state the exact domain of g -1 . [3] 10 RI/2013Promo/14 The function f is defined as follows : 2 , 0 2, f ( ) (2 ) , 2 4.4 −  = −  xx x xx x (i) Sketch the graph of f and show that the inverse function of f exists. [3] (ii) Sketch the graph of 1f− on the same diagram as the graph of f, showing clearly their relationship. State the range of values of x for which 1f ( ) f ( )xx −= . [3] (iii) Solve ( ) 1f3 x− = . [2] (iv) Find the exact value of 3 2 f ( ) d . xx Hence, or otherwise, find the exact value of 2 1 3 4 f ( ) d− − xx . [4]

3. Functions 4 11 DHS/2013MYE/10 The functions f and g are defined as follows: 2 3 , 2 0,f: 4 , 0 2. x xx xx  − −     g : , 1 0.x x x −   (i) Sketch the graph of ( )fyx= and show that 1f− exists. [3] (ii) Find 1f− in a similar form. [4] The function h is a restriction of f to 2 0.x−   (iii) Show that the composite function hg exists. [1] (iv) Solve ( ) ( )gh hgxx= , showing your working clearly. [2] 12 MJC/2015Promo/9 The function f is defined by 2f : 2 1, 1 1.x x x x − − −   (i) Define 1f − in a similar form and sketch the graphs of ( )fyx= and 1f ( )yx −= on a single diagram, showing clearly the relationship between the graphs. [6] The function g is defined by ( ) 2 9 3 , 0 3, g: 3 , 3 6, xx x xx −   −   and that ( ) ( )g g 6xx=+ for all real values of x. (ii) Sketch the graph of ( )gyx= for 28 x−   . [3] (iii) Give a reason why the composite function gf exists and hence state its range. [2]

3. Functions 5 13 PJC/2015Promo/7 Functions f and g are defined by 2f : , , 1, 1 g : 1 2 , . x x x x x x x −  − − (i) Only one of the composite functions fg and gf exists. Give a definition, domain and range of the composite that exists, and explain why the other composite does not exist. [4] A function h is said to be self-inverse if 1h( ) h ( )xx −= for all x in the domain of h. (ii) Show that gf is self-inverse. [3] 14 HCI/2015Promo/5 [Part iii removed. Out of syllabus] The function f and g are defined by  f : 2cos , 2 π, 2πx x x − , 21g : , , 1. 1 xx x x x − − (i) Give a reason why f does not have an inverse. [1] (ii) The function f has an inverse if its domain is restricted to π .2 xb Find the greatest value of b. Define 1f− in similar form. [3]

3. Functions 6 15 VJC/2013MYE/12 The functions f is defined by 2f : 2 3, for . x x x x k+ −  (i) Explain why f −1 does not exist when 1k = . [2] (ii) State the largest value of k such that f −1 exists. [1] With the value of k found in part (ii), (iii) define f −1 in a similar form, [4] (iv) sketch the graphs of y = f(x) and y = f −1(x) on the same diagram, [2] (v) write down the equation of the line in which the graph of y = f (x) must be reflected in order to obtain the graph of y = f −1(x)