14ο λύκειο...

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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.

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    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

    .

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    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

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    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

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    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)

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    9

    67. f : R R