TMJC Term 4 Revision Chapter 3 Functions Solutions

Term 4 Revision Topical Quick Check: Functions TMJC 2024 JC1 H2 Mathematics (9758) Term 4 Revision Topical Quick Check Chapter 3 Functions Revision Guide Chapter 3 Page 3-4 6. One-one functions (b) Horizontal Line Test (MUST SKETCH GRAPH) The function f is one-one if any horizontal line yb= ( b ) cuts the graph of f at most once. • If not one -one: find a counter -example (sketch a horizontal line yk= that cuts the graph twice (providing the value of k) or find f,Dab  such that ( ) ( )ff ab= ). INVERSE FUNCTIONS 7. The inverse of a function f is denoted by 1f− . (Do not confuse inverse function ( ) 1f− with reciprocal graph ( ) 1 f x   ) 8. 1f− exists  f is one-one. 9. -1 ffDR = and -1 ffRD = 10. The graph of 1f− can be obta ined by reflecting the graph of f about the line yx= (provided that f has an inverse). 11. ( ) ( ) 11ff f f x x x−− == ; but 1ff− and 1ff− have different domains. (a) -1 -1 fff fD D R== (b) -1 fffDD = 12. If  occurs in the process of finding inverse, use the domain of f to determine whether to choose (positive square root) or − (negative square root). COMPOSITE FUNCTIONS 13. If f and g are two functions such that fgR D, then the composite function g of f (i.e. gf) exists and is defined by ( ) ( )( )gf g fxx= for all fD.x 14. For gf to exist, fgR D . 15. Domain of composite function gf, gf fD. D= 16. Range of composite function gf, gf gR. R (a) Method 1: Sketch both functions and use f f gf fgD R R⎯⎯ → ⎯⎯ → [**Using fR as restricted domain of g] (b) Method 2: Find the composite function and sketch the graph, keeping in mind of the domain of the first function, i.e., fD since gf fD D .=

Term 4 Revision Topical Quick Check: Functions TMJC 2024 Let’s Try Now 1 YIJC Promo 9758/2022/Q9 (modified) The function f is defined by 2f : 4 5xx x −− , for x , 2x . (a) Find 1f ( )x− and state the domain of 1f− . [3] (b) On the same diagram, sketch the graphs of f and 1f− . [3] (c) Find the exact solution of the equation 1f ( ) f ( )xx −= . [3] The function g is defined by 2g: 1x x− , for , 1x x . (d) Explain why the composite function fg exists and find the range of fg. [3]

Term 4 Revision Topical Quick Check: Functions TMJC 2024 1 YIJC Promo 9758/2022/Q9 (modified) The function f is defined by 2f : 4 5xx x −− , for x , 2x . (a) Find 1f ( )x− and state the domain of 1f− . [3] Q1 Solution (a) Let 2 45y x x= − − . Then 2 2 ( 2) 9 9 ( 2) 2 9 29 yx yx x y xy = − − + = − −= + =  + Since ( fD ,2=−  , 29xy= − + . Hence, 1f ( ) 2 9xx− = − + )1fD 9,− = −  (b) On the same diagram, sketch the graphs of f and 1f− . [3] Q1 Solution (b) yx=

Term 4 Revision Topical Quick Check: Functions TMJC 2024 (c) Find the exact solution of the equation 1f ( ) f ( )xx −= . [3] Q1 S