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