4.Functions Solutions 2023 (RVHS)

River Valley High School, Mathematics Department 2023 Functions T1 1 x y Solutions Functions 1 2 2 2xxye              2 2 2 2 ln 2 1 1 ln 2 1 1 ln 2 1 1 ln 2 x x y xy xy xy                Since x < 0,  1 1 ln 2xy    Df–1 = fR =  1,   1f : 1 1 ln 2 , 1x x x       1 1,gg fR D R            -1gf2, 1,1 1 1 ln 3, 0        2 (i) Since any horizontal line y = k, k  R will cut the graph of f at most once, hence f is one-one. Thus, f -1 exists. (ii) Let xxy 1 42 1 2 2  yyx But x > 0, 42 1 2 2  yyx When g( ) 1x  23(2) 1 21g    gD 2, Least 2k 

River Valley High School, Mathematics Department 2023 Functions T2 Hence, f -1 : x  42 1 2 2  xx , x  (- , ) (iii) [B1] – correct graph of f -1 with y = x [B1] - intercepts (iv) Df : (0,) and Rg : [-1 , 1]. Since Rg  Df , therefore fg does not exist. (v) Largest Dg : (0 ,) fg(x) = f(sin x) = xx sin 1sin  Hence, fg : x  xx sin 1sin  , x  (0 ,). 3a  1f ( ) 2 tan 2x x x  ' 2 4f ( ) 1 14 x x   2 2 2 2 30 4 1 1 4 4 11 14 14 414 14 x x x x         Since 2 414 14 x   , 2 410 14 x     , 'f ( ) 0x  So f ( )x decreases when x increases. 1 x y 1

River Valley High School, Mathematics Department 2023 Functions T3 f 3 2 3 2,2 3 2 3R     Or f ( 1.23,1.23)R  For ff to exist, ffRD 3 2 3 2 ,2 3 2 3 fR     or f ( 1.23,1.23)R  33,22fD  or g ( 0.866,0.866)R  ffRD ff does not exist. Let  12 tan 2y x x  . Then  12 tan 2x y y  b) 2h( ) 2xx     24 2 2gh 2 5 1 4x x x x          22g 2 2 3 4xx          2 g 3 4xx   Alternatively, 2h( ) 2xx         42 22 2 2 222 gh 2 5 2 4 4 2 5 2 6 2 13 x x x x x x xx                 2g 6 13x x x    4 (i)    D , and R 2,fg     Since    R 2, D ,gf       function fg exists.         2 41 3, fg x f g x x fg x x x         (ii) Since    R 3, D 0,fg g      function gfg exists        3 4 7, 0h x gfg x g fg x x x x       

River Valley High School, Mathematics Department 2023 Functions T4 (iii)    1h x g g x , x > 0    1g fg x g g x           2 2 7, 7 70 1 1 4 1 7 1 1 292 1 2 1 1 29 sin2 x x x xx xx x x ce x                   5 (i) g( ) f ( 1)xx The transformation is a translation of +1 unit in the direction of the x-axis Alternatively, 1 11g( ) 2 (2 ) f ( )22 xxxx    The transformation is a scaling parallel to the y-axis by factor 1 2 (ii) A horizontal line y = k (k > 9 4 ) cuts the graph at 2 points, h is not one-