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11-Jan-2017
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• 14

• 14

1

1. :

) ( ) 1 1f x x x ) ( ) 5 5g x x x

) 1

( )3

xh x

x

)

2( )

2 15

xc x

x x

) ( ) 3 1d x x ) ( ) ln(1 ln )k x x

) ( ) 2 1l x x x ) 3( ) ln(2 )m x x

2. 2 2

2

1 , 5( )

ln( 1), 3

x xf x

x x x

. ,

f .

3.

x

2

e 1f(x)

3x 4 x 4. ,

f .

4. :

, 0 3

( ) 1, 1 0

2, 3 1

x x

f x x

x x

.

5. :

) ( )i f x x ) ( ) | |ii g x x ) ( ) | 1| 2iii h x x

,

6.

) 2 5 6

( )3

x xf x

x

) 2(x)= x +9- 6xf

) 13 3 3 2( ) x x xf x ) ( ) 1 2f x x

7. f , g g(x)=f 2(x)-2f(x)+x2+3 x , g .

8. f : : 3 2 2( ) 2 ( ) 5 ( ) x xf x f x f x e e , x . fC

xx.

9. f : R R 2

2

2 ( ) 3 ( ) 1( )

3 4

f x f xg x

x x

xR . Cg ' .

• 14

2

,

10.

2 2 4f x x x ,

2 2g x x x .

11. ( ) ( 2) ( 2)f x n x n x

x y=f(x)

ln5y .

12. f,g: , : xf(x) 1 g(x) e , x

f,g.

13. f(x) = x3+4x2+4x+1, g(x) = 2x2+5x +3. i) Cf Cg ; ii) Cf Cg

= 6..

14. f = g. f g R

f (x) = g(x).

) 3 2

( )x x

x

e xef x

xe

2

( ) 2xe

g xx

) 2

2

4( )

2 | |

xf x

x x

2( ) 1

| |g x

x

) 9

( )3

xf x

x

( ) 3g x x

) 3

1( )

1

xf x

x

3 2 3( ) 1g x x x

15. .

) 2f(x) ln(x 2) ln(2 x) g(x) ln(4 x )

)

x 1f(x) ln(x 1) ln(x 1) g(x) ln

x 1 ) 2f(x) 2ln(x 3) g x ln(x 3)

16. , ,

2x xf(x)

x 2

2x ( 1)x 2 3g(x)

x 1 .

17. f, g A :

f(x) f(x) g(x) 4 g(x) f(x) g(x) 4 8 x A . : f g .

18. f g x :

2 2 2 24 3 (x) 2 (x)g(x) g (x) 4x (x) ( )x f f f f x f g

• 14

3

19. 2 9

( )2

xf x

x

2

1( )

6

xg x

x x

.

: f g f g ,f

g.

20. f (x) 1x g(x) 6 x .

f+ g ,f- g ,f g , ,f g

g f.

21. 1, [0,2]

( ), (2,4]

x xf x

x x

2, [ 1,1]( )

2 4, (1,3]

x xg x

x x

.

: i) f g ii) f g

22. 1, [ 2,0]

( )2, (0,2]

xf x

x

0, [ 3, 1]( )

3, ( 1,1]

xg x

x

.

: i) f g ii) f g

23. fog , gof :

) (x)=x+3,x [0,2] f , g(x)=-x+2 [-2,2]

) 2

( )2

xf x

x

, ( )g x x

) ( )f x x , 2g(x)= 9-x

)2, 1

( ), 1

x xf x

x x

, g(x)=4x 2

) ( ) 2 f x x , 2g(x)= x 1

24. x

xxf

3)(

fof.

25. 2 3, 1 1

3 1, 1 10f x

x x

x x

2, 0 4

3, 4 7

x xg x

x x

.

fog.

• 14

4

26. f :

) (f og)(x) = x2-4x , x R g(x) = x+4

) (f og)(x) = 51 x , x -1 g(x) = -x5

) (fog)(x) =1

1

x

x

, x > -1 g(x) = ( 1)n x

27. g :

) 2f g x x x 1f x x

) xf g x e x x 1f x x

) 2 1f g x x lnx, 0f x x

) 8f g x x 1xf x e

28. f 0,1 .

:

) f x ) 4f x ) lnf x

29. f 5,8 .

g x f x f x

30. f(x) [1,4] ,

: ( ) ( 1) ( 3).h x f x f x

31. f : , : f x y f x f y ,

,x y :

) 0 0f ) f ) , ,f x y f x f y x y

) f

32. f : , : ( ) ( ) ( ) 1f xy f x f y xy

,x y . :

) ( )f ) 1 1

,( )

f xx f x

) ( )

1,( )

x f x xf x y

y f y y

33. f : :

) ( ) (2 23 )f x f x x ) 1 4

2 ( ) ,f x f xx x

.

• 14

5

34. f : :

) 2 ( ) 4 ( ) ,f x f x x x x ) 1 6

( ) 3 2 ,f x f x xx x

.

35. f : , : ( 1) ( )xf x e ef x ,

x .

) f ) f.

36. :f , 2 4( ) ,x f x x x x x

f.

37. :f , ln 1x

f x f xe

0x . ) f.

) fC .

) f.

38.

) ( ) 2 lnf x x ) ( ) 1 2f x x x ) 2( ) 1x xf x e e ) 1

xf x e

) 2( ) 2f x x x ) 2( ) 6 8f x x x )3

( )2

xf x

x

)

2

2( )

1

xf x

x

39. .

) 2

2

3 , 0( )

2 , 0xx x x

f xx x

)

2 3 0( )

1xx x

f xe x

40. f, g R, (

).

i) gof .

ii) fof gog.

iii) f (x) = ln [ln(x)], x > 1.

41. f . : i) -f

ii) kf k>0 k 0 f2

• 14

6

42. f : : 1 2 1 21

( ) ( )2

f x f x x x

1 2,x x 1 2x x . :

) 2

xg x f x .

) 2

xg x f x .

43. f 0, :

f xf x e x 0,x . f .

44. f

: 22

f xe x x ,0x .

45. : 0,f

: 2ln 4 1f x x x 2,x .

46. ) ( ) xf x x e

(0,)

) , 0, :

e e .

47. ) ( ) ln xf x x e

0,

) x>3 : 3 ln 3 ln3

xe .

48. ( ) xf x x , x , 0,1 .

) f .

) 3 3x x x x 0,1x

49. :f 3f x f x x x .

) f .

) 3 3 2f x f x

50. :f

3 3 22 2 5 1f x f x x x x x .

) f

• 14

7

) 3 1 1f x f x .

51. :f 3 3 3f x f x x

x .

) f

) 1 1f 3 0f . ) :

i. 2 2f f x f f x ii. 0f x iii. 1f x iv. 2 4 0f f x

52. 3( ) 5 1f x x x , x .

) f

) ( ) 7f f x .

53. ( ) lnxf x x , x >0 0,1 .

) f 0,

) : 2 2 2 24 9 ln( 4) ln( 9)x x x x x x

54. :f .

2 1fof x x fof x

55. 2 2 lnf x x x .

) ) f

.

) 0f x .

) 0f x .

56. f R

( ) ( ) ( )f x f y f xy x,yR.

(3 4) ( 2) (6 4) (2 1)f x f x f x f x .

57. f ( (3 7 0) )f x f x

x . A f

) f(x)=0 ) 2 0( )3f x

58. :

) 2( ) 2 3f x x x ) 2( ) 4 5f x x x

) 6 3( ) 4 3f x x x ) 2( ) 2 2x xf x e e

• 14

8

59. : 2y x

.

60. 2( ) 2f x x x , , x . , f

2 .

1-1

1 - 1

61. 1-1 :

) 2

(x)1

xf

x

) (x) 2f x )

3(x) 3x 2x 1f

) 2

2(x)

4

xf

x

)

2(x)

1

xf

x

) 2(x) xf e x

) 3

(x) xf e

) 2

(x) ln2

xf

x

62. 1-1 :

) ( ) 1f x x )3( ) 2 3 2f x x x ) ( ) 2xf x e x

) 4( ) 5f x x ) 23

2

xf x

x

1-1

63. ) A f : g : 1-1 , g f 1-1.

) f 1-1 (x)=[f(x)]3+2f(x)-3 1-1

64. :f : ( ) ( ) 2f f f x f f x x

x .

i) f .

ii) fC .

65. :f : 3( ) 4 ( ) 2 3f x f x x

x . :

) f 1-1 , ii) fC 1,1A .

1 - 1

66. A f : f(f(x))+f3(x)=2x+3 , x . ) f 1-1 ) f(2x3+x)=f(4-x)

• 14

9

67. f : R R