composite functions Part Marks Level Calc. Content Answer U1 OC2 (a) 2 C CN A4 3 3 x 2000P2Q3 (b) 2 C CN A4 x (b) 1 A/B CN A x 5 x

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1 composite functions [SQA] 1. f(x) =3 xandg(x) = 3,x =0. x (a)findp(x)wherep(x) =f(g(x)). (b)ifq(x) = 3,x =3,findp(q(x))initssimplestform. 3 3 x Part Marks Level Calc. Content Answer U1 OC (a) C CN A4 3 3 x 000PQ3 (b) C CN A4 x (b) 1 A/B CN A4 1 ic: interpretcompositefunc. pd: process 3 ic: interpretcompositefunc. 4 pd: process 5 pd: process 1 f ( ) 3 x statedorimpliedby 3 3 x 3 p ( ) 3 3 x statedorimpliedby x 5 x [SQA]. Part Marks Level Calc. Content Answer U1 OC (a) C NC A P1 Q19 (b) A/B NC A4

2 3. Onasuitablesetofrealnumbers,functionsfandgaredefinedbyf(x) = 1 x + andg(x) = 1 x. Findf ( g(x) ) initssimplestform. 3 Part Marks Level Calc. Content Answer U1 OC 3 C NC A4 199P1Q6 [SQA] 4. f(x) =x 1,g(x) =3 xandh(x) = 1 4 (5 x). (a)findaformulafork(x)wherek(x) =f ( g(x) ). (b)findaformulaforh ( k(x) ). (c)whatistheconnectionbetweenthefunctionshandk? 1 Part Marks Level Calc. Content Answer U1 OC (a) C NC A P1 Q13 (b) C NC A4 (c) 1 A/B NC CGD

3 5. Afunctionfisdefinedonthesetofrealnumbersbyf(x) = x,x =1. 1 x Find,initssimplestform,anexpressionforf ( f(x) ). 3 Part Marks Level Calc. Content Answer U1 OC 1 C NC A4 1994P1Q19 A/B NC A4 [SQA] 6. Thefunctions f andg,definedonsuitabledomains,aregivenby f(x) = 1 x 4 andg(x) =x +1. (a)findanexpressionforh(x)whereh(x) =g ( f(x) ). Giveyouranswerasa single fraction. 3 (b)stateasuitabledomainforh. 1 Part Marks Level Calc. Content Answer U1 OC (a) C NC A P1 Q11 (a) 1 A/B NC A4 (b) 1 A/B NC A1 [SQA] 7. Functions f andg,definedonsuitabledomains,aregivenby f(x) = xand g(x) =sinx +cosx. Findf ( g(x) ) andg ( f(x) ). 4 Part Marks Level Calc. Content Answer U1 OC 4 C NC A4 1997P1Q3

4 8. Thefunctions f andgaredefinedonasuitabledomainby f(x) =x 1and g(x) =x +. (a)findanexpressionforf ( g(x) ). (b)factorisef ( g(x) ). Part Marks Level Calc. Content Answer U1 OC (a) C CN A P1 Q6 (b) 1 C CN A6 (b) 1 A/B CN A6 [SQA] 9. Functionsfandgaredefinedbyf(x) =x+3andg(x) = x +5 x 5 wherex R, x = ±5. Thefunctionhisgivenbytheformulah(x) =g ( f(x) ). Forwhichrealvaluesofxisthefunctionhundefined? 4 Part Marks Level Calc. Content Answer U1 OC C CN A4,A1 1989P1Q19 A/B CN A4,A1

5 10. Functionsfandgaregivenbyf(x) =3x +1andg(x) =x. (a) (i)findp(x)wherep(x) =f(g(x)). (ii)findq(x)whereq(x) =g(f(x)). 3 (b)solvep (x) =q (x). 3 Part Marks Level Calc. Content Answer U1 OC3 (a) 3 C CN A4 3(x ) +1, (3x +1) 009PQ (b) 3 C CN C1 x = 1 1 ss: substituteforg(x)inf(x) ic: complete 3 ic: sub.andcompleteforq(x) 4 ss: simplify 5 pd: differentiate 6 pd: solve 1 f(x ) 3(x ) +1 3 (3x +1) 4 p(x) =3x 5,q(x) =9x +6x 1 5 p (x) =6x,q (x) =18x +6 6 x = 1

6 11.Functionsf,gandharedefinedonthesetofrealnumbersby f(x) =x 3 1 g(x) =3x +1 h(x) =4x 5. (a)findg(f(x)). (b)showthatg(f(x)) +xh(x) =3x 3 +4x 5x. 1 (c) (i)showthat (x 1)isafactorof3x 3 +4x 5x. (ii)factorise3x 3 +4x 5x fully. 5 (d)hencesolveg(f(x)) +xh(x) =0. 1 Part Marks Level Calc. Content Answer U OC1 (a) C CN A4 3(x 3 1) PQ (b) 1 C CN A6 proof (c) 5 C CN A1 (x 1)(3x +1)(x +) (d) 1 C CN A, 1 3,1 1 ic: interpretnotation ic: completeprocess 3 ic: substituteandcomplete 4 ss: knowtousex=1 5 pd: completeevaluation 6 ic: stateconclusion 7 ic: findquadraticfactor 8 pd: factorisecompletely 9 ic: interpretandsolveequationin (d) 1 g(x 3 1) 3(x 3 1) (x 3 1) +1+x(4x 5) =3x 3 +4x 5x 4 evaluatingatx= =0 6 (x 1)isafactor 7 (x 1)(x +7x +) 8 (x 1)(3x +1)(x +) 9, 1 3,1

7 1. (a)thefunctionfisdefinedbyf(x) =x 3 x 5x +6. Thefunctiongisdefinedbyg(x) =x 1. Showthatf ( g(x) ) =x 3 5x +x (b)factorisefullyf ( g(x) ). 3 (c)thefunctionkissuchthatk(x) = 1 f ( g(x) ). Forwhatvaluesofxisthefunctionknotdefined? 3 Part Marks Level Calc. Content Answer U OC1 (a) 4 C NC A P Q6 (b) 3 C NC A1 (c) C NC A1

8 13. Part Marks Level Calc. Content Answer U OC1 (a) 4 C NC A P Q4 (b) 7 C NC A17, A18, A6

9 14. Functions f andgaredefinedonthesetofrealnumbersby f(x) =x 1and g(x) =x. (a) Find formulae for (i) f ( g(x) ) (ii) g ( f(x) ). 4 (b)thefunctionhisdefinedbyh(x) =f ( g(x) ) +g ( f(x) ). Showthath(x) =x xandsketchthegraphofh. 3 (c)findtheareaenclosedbetweenthisgraphandthex-axis. 4 Part Marks Level Calc. Content Answer U OC (a) 4 C NC A P Q6 (b) 3 C NC A4 (c) 4 C NC C16

10 15. Functions f(x) =sinx,g(x) =cosxandh(x) =x + π 4 aredefinedonasuitable set of real numbers. (a) Find expressions for: (i) f(h(x)); (ii) g(h(x)). (b) (i)showthatf(h(x)) = 1 sinx + 1 cosx. (ii) Find a similar expression for g(h(x)) and hence solve the equation f(h(x)) g(h(x)) =1for0 x π. 5 Part Marks Level Calc. Content Answer U OC3 (a) C NC A4 (i) sin(x + π 4 ), (ii) 001P1Q7 cos(x + π 4 ) (b) 5 C NC T8,T7 (i)proof,(ii)x= π 4,3π 4 1 ic: interpretcompositefunctions ic: interpretcompositefunctions 3 ss: expandsin(x + π 4 ) 4 ic: interpret 5 ic: substitute 6 pd: startsolvingprocess 7 pd: process 1 sin(x + π 4 ) cos(x + π 4 ) 3 sinxcos π 4 + cosxsin π 4 and complete 4 g(h(x)) = 1 cosx 1 sinx 5 ( 1 sinx+ 1 cosx) ( 1 cosx 1 sinx) 6 sinx 7 x = π 4,3π 4 acceptonlyradians

11 16. Functions f and g aredefinedonsuitabledomainsby f(x) = sin(x ) and g(x) =x. (a) Find expressions for: (i) f(g(x)); (ii) g(f(x)). (b)solvef(g(x)) =g(f(x))for0 x Part Marks Level Calc. Content Answer U OC3 (a) C CN A4 (i)sin(x ),(ii)sin(x ) 00P1Q3 (b) 5 C CN T10 0,60,180,300,360 1 ic: interpretf(g(x)) ic: interpretg(f(x)) 3 ss: equateforintersection 4 ss: substituteforsinx 5 pd: extractacommonfactor 6 pd: solve a common factor equation 7 pd: solvea linear equation or 1 sin(x ) sin(x ) 3 sin(x ) =sin(x ) 4 appearanceofsin(x )cos(x ) 5 sin(x ) (cos(x ) 1) 6 sin(x ) =0and0,180,360 7 cos(x ) = 1 and60,300 6 sin(x ) =0andcos(x ) = 1 7 0,60,180,300,360 [END OF QUESTIONS]