Συναρτήσεις, επανάληψη καλοκαιρινής προετοιμασίας

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Μαθηµατικά Γ’ Λυκείου Οι λύσεις στην σελ 12 mathhmagic.blogspot.gr ΕΠΙΜΕΛΕΙΑ:ΜΗΤΑΛΑΣ Γ.,∆ΡΟΥΓΑΣ Α.,ΧΑ∆ΟΣ Χ.,ΓΕΡΜΑΝΟΣ Ξ.,ΠΑΤΣΗΣ Σ. 1 ΕΠΙΜΕΛΕΙΑ: Μήταλας Γ , Δρούγας Α. ,Χάδος Χ. ,Γερμανός Ξ., Πάτσης Σ. f(x)=lnx 2 g(x)=2lnx f=g???? ΜΑΘΗΜΑΤΙΚΑ Γ ΛΥΚΕΙΟΥ ΕΠΑΝΑΛΗΨΗ ΣΥΝΑΡΤΗΣΕΙΣ ΟΜΑΔΑ ΠΡΟΣΑΝΑΤΟΛΙΣΜΟΥ ΘΕΤΙΚΩΝ ΣΠΟΥΔΩΝ, ΟΙΚΟΝΟΜΙΑΣ ΚΑΙ ΠΛΗΡΟΦΟΡΙΚΗΣ Μια προσφορά του Σ.Ο.Κ.Ο.Ν Ο ΤΣΕΛΕΜΕΝΤΕΣ ΤΟΥ ΥΠΟΨΗΦΙΟΥ ΣΤΑ ΜΑΘΗΜΑΤΙΚΑ ΘΕΤΙΚΟΥ ΠΡΟΣΑΝΑΤΟΛΙΣΜΟΥ
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Transcript of Συναρτήσεις, επανάληψη καλοκαιρινής προετοιμασίας

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 1

    : , . , . , ., .

    f(x)=lnx2

    g(x)=2lnx

    f=g????

    ,

    ....

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 2

    1) , ,,....

    2) , .,

    3) , .,

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    11)-, . ,

    12)-, .. ,

    13)-, ..,

    14) 1,2,3, .,

    15) 1000+1 , .,

    16) , & .,

    17) , ....,study4exams,

    18) , ,

    19), .

    20) . ,.,

    21) ,-,

    22) .. .

    23),.,

    24), .

    25) , Spivak M. ,.

    26) . ,

    27)Problems in Calculus ,..Maron,Mir Publisher

    28) , .,

    29) , ...

    30) , ,

    31) , .,

    32) , . ., . , . , .

    33)Problem book:Algebra and Elementary functions, Kutepov A.,Rubanov, MIR Publishers

    34) ,

    35)The theory of functions of a real variable, R.L.Jeffery

    36)A Problem book in mathematical analysis,G.N Berman

    37) Bad problems in Calculus, A.G .Drolkun

    38) 1,2,3 .,.

    39) , .,

    40) , ..

    41) ,.

    42) .

    43) ,

    44)Differential Calculus ,.Ball

    45)Calculus,E.Swokowski

    46)Problems in Algebra ,T.Andreesku, Z.Feng

    47) , .. .

    48)O (Lisari team)

    49) , ,,,

    ( qr-code)

    .

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 3

    , , , , , 1-1, ..

    1. i) ,

    ii) g(x) 2 ) +0, . v) Cg Cf y=x ) +0, . ;

    2. f 0,1 .

    g : = + +g(x) f(x 2) f(ln x) 1

    3. ( )+ f : 1, =

    1f( ) ,

    3.

    ff .

    4. ( )

    i) f :

    = 2(fog)(x) 9 x x 3,3 , = 2g(x) x

    ii) f :

    = +(gof)(x) 4x 13 >x 0 , =f(x) ln x .

    iii) f , :

    =(gof)(x) x x , = 2g(x) 1 x .

    5. f : (5, 13), (7, 11).

    i) f.

    ii)

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 4

    6. . f . f

    , , > 0 f , .

    . g : = 0x x ,

    g = 0x x .

    7. f : .

    ) f .

    ) :

    =g(x) f(x) =f(x)

    h(x) lnx

    8.

    + .

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 5

    14. , :f g :

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

    2( )4

    af x x= , x 2

    Cf 4y = .

    :

    i) ( ) 3g x x= , x

    ii) 1a = iii) , Cf Cg , ().

    15. 1-1.

    i) = +

    xf(x)

    3 x ii) = + xf(x) 2x ln(2 e ) iii)

    =

    +

    3x 2f(x)

    2x 1.

    16. 1-1.

    i) =+2

    2xf(x)

    x 1 ii) = f(x) x 1 2 iii) = +f(x) (x 3)(x 1) 2017

    17. 1-1;

    i) . f

    .

    ii) . f

    .

    iii) 1000 . f

    .

    iv) .

    , . f

    .

    18. i) f : ff .

    ii) + =2f(f(x x)) f(f(2)) 0 .

    iii) + >f(f( x 1) f(2)) f(0) .

    19. f ,g : .

    i) gof 1-1 f 1-1.

    ii) fof 1-1, f 1-1 .

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 6

    20. f : 1-1 :

    (1) 2( ) (3 ) ( ) f x f x f x x

    21. f 1-1 :

    (1) ( ) (1 ) ( )f x f x f x = + x :

    i ) 0 = ii) (1 ) 1f =

    22. f ,g :

    (1) = + +5(gog)(x) 13g(x) 14f(x 3) x

    f 1-1

    i) g 1-1.

    ii) + + = + 3g(f(x) x x) g(f(x) 4x 2)

    23. f : , . :

    i) =f(0) 0

    ii) + >2(x x)f(x) 0 x 0 .

    24. 1-1

    (, ) .

    i) = f(x) 3x 1 ii) = +f(x) x(x 2) 1 iii) = +2x 3f(x) e 4

    iv)

    =

    2x 9f(x)

    x 1 v) = +f(x) ln(x 1) vi) =

    xxf(x)

    2

    25. f : : (1) = f(f(x)) x x

    :

    i) f 1-1.

    ii) = f( ) = 1f (x) f(x) x .(* = f( ) ;)

    iii) H f .

    iv) bonus =f(0) 0 f .

    26. f : = f( ) ,

    (1) = +f(f(x)) f(x) 3x x

    i) f

    = 11

    f (x) (f(x) x)3

    x

    ii) =f(1) 2 1f(2), f (5) .

    27. f : = f( ) ,

    + = +3(f(x)) 4f(x) x 8 x

    i) f 1-1.

    ii) f-1.

    iii) f-1 .

    iv) + =3x 4x 8 0 1,36. Cf xx , yy

    v) >3f(x ) x

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 7

    28. f :

    = + +2xf(x) e x 2 , x

    i) Cf yy.

    ii) f 1-1.

    iii) 1f (3) .

    iv) Cf =y 3 ( ) +x 0, . 29. = + + + + 2f(x) (x 2) lnx ( 2) 2 , ( ) = +x A 0, , = f(A) .

    f (1,) :

    i) ==1.

    ii) f .

    iii) >1f (x) x .

    30. f : : f

    f

    f

    i) f .

    ii) f-1 .

    iii) f-1 .

    iv) Cf-1 xx.

    31. f. f 1-1.

    .

    i) f-1

    ii) =1f (1) ........

    iii) = 1f (....) 3

    iv) f-1 =1f (.......) ........

    v) f-1 =1f (.......) ........

    v) f-1

    32. f : :

    + + + + x x 1f(x) e 1 x 1 f(x 1) e

    x (1)

    i) = + xf(x) e x,x

    ii) f .

    iii)

    ) =1f (x) x ) = 1f (x) x 3

    iv)

    ) 1f (x) x ) 1f (x) x 1

    v) 1Cf ,Cf .

    vi) 1f . vii)

    + = +x x x xf(2017 ) f(2019 ) f(2018 ) f(2020 )

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 8

    33. f : :

    + + =3f (x) 2f(x) x 3 0 x (1)

    i) f .

    ii) Cf xx 3.

    iii) =1f (x) 3 .

    iv) f .

    v) f , :

    ) f.

    ) :

    + > x 1f(e lnx) f(2 x)

    34.

    +

    =2x 2x

    f(x)

    , x

    , 2

    . +

    =

    3 7

    2 3 :

    i) = 1

    2

    ii) = + f(x) x x 1,x

    B. f

    0,

    2.

    . f .

    i) 1fC =y x .

    ii) 1fC fC .

    35. f :

    (1,1),(2017,2017) =1f f . i) f.

    ii) =f(x) x x .

    35. (bonus) ( )+ f : 0, >x 0 :

    (1) =f(x)f(e ) lnx . f.

    36. f ,g,h : :

    +

    = +

    2

    10x f(x) ,x

    x 25

    = g(x) x ,x

    . i) ( ) + 2226 f(1) 20( 5) 0 . ii) H Ch Cg 5 1

    h.

    . = 0 , : i) f =x 5 . ii) h . (; )

    iii) f h .

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 9

    37. ( )+ f : 0, ) +2, f(x) 2 >x 0 x 2 . =f(f(x) 1) 2 .

    38. ( )+ f : 0, ) +2, =f(2) 2 f ( 0,2 ) +2, . =f(f(x) 1) 2 .

    39.

    f : .

    i) f 1-1

    .

    ii) f-1. iii) :

    +

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 10

    44.

    = >e

    f(x) ln ,x 0x

    , = +

    x

    x

    eg(x) ,x

    e 1

    i)

    = +

    x

    1g(x) 1 ,x

    e 1 g .

    ii) g 1g .

    iii) 1g of .

    iv) >1(g of)(x) 0 .

    v) + g(x) g( x) x.

    vi)

    + + + + + =1 1 1

    g(ln 2017) g(ln 2016) g(ln 2015) g(ln ) g(ln ) g(ln ) 32017 2016 2015

    vii) * , :

    + + +

    + + +

    ++

    2 2

    2 2

    4 g( 2020) 2 8 4 g(2020)

    2 8 4 g(2020) 4 g( 2020)

    e e

    e 1e 1

    45. = + xf(x) g(x) e 1 , x

    . g : :

    = g( )

    x g(g(x)) g(x) x

    = + xf(x) x e 1 , x

    B. i) = xe x 1 ii) + >f(f(x) 1) f(1)

    . h : = + h(x)(x) h(x) e ,x

    =(0) 1 , :

    i) h .

    ii) Ch .

    iii)

    = +(hof)(x) h(x 3)

    46. f: (1,2) (3,-2)

    i) f(1) f(3)

    ii) f.

    iii) +

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 11

    48.( 2006) f: 2f(x) ( 2) 2x= + 2x

    i) f 1-1.

    ii) -1f f .

    iii) f -1f y x= .

    iv) fC 1fC . fC

    y x= 1

    6, fC -1fC .

    49*. f :

    ( )+ +33f(x x) x f(x) f(x) x (1) i) = +3g(x) x x

    i) = 1f g .

    50. f : :

    >f (x)e x 0 , x

    =f(x)f(x)

    1e 2x

    e, x

    i) = + +2f(x) ln(x x 1) .

    ii)

    =x xe e

    g(x)2

    .

    ) g .

    ) = (gof)(x) x,x .

    ) () f .

    iii) (Bonus) f (ii).

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 12

    1. i) B Cg y=2.

    = =x 0

    g(x) 2 ... x 4 B(2,4)

    Cf y=2.

    = = + + = >

    f fff

    x 1

    D {x D / f(x) D } {x 1, / f(x) 1, }

    f(x) 1

    x 1 , f (x) 1> x > 1, =ffD .

    ff .

    hx D

    ( ) fh x D

    hx D

    ( ) fh x D

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 13

    4. i) x 3,3 = = 2 2 2f(g(x)) 9 x f( x ) 9 x .

    = 2u x 0 : = +f(u) 9 u

    f = +f(x) 9 x [ ]= Df 9,0

    ii) ( )= +fD 0, : ( )= +gofD 0, ( )= + = > g ggof fD {x D f(x) D } 0, {x 0 ln x D }/ / f(A) f , gln x D g gf(A) D D = gD .

    >x 0 = + = + = +(gof)(x) 4x 13 g(f(x)) 4x 13 g(ln x) 4x 13

    : = = uu l n x x e , = +ug(u) 4e 1

    g, = +xg(x) 4e 13 = Dg

    .

    iii) = gD 1,1 =g fD

    = gfg f ff(x) DD {x D } D/

    =g fD , :

    =g f f f fD D D D

    x :

    = = =

    =

    22

    2 2

    1 f (x) 0(gof)(x) x g(f(x)) x 1 f (x) x

    1 f (x) x (1)

    (1) : = = = = 2 2 2 2 2 2 2 21 f (x) x f (x) 1 x f (x) x f (x) x 0

    + = = = (f(x) x)(f(x) x) 0 f(x) x f(x) x (2)

    (2) .

    :

    =

    x,xf(x)

    x,x .

    .

    5. i) f = =f(5) 13,f(7) 11 >f(5) f(7) f

    .

    ii) < < < f(f(x) 6) 2 f(7) f(f(x) 6) 2 11 f(f(x) 6) 13

    < > > > < f f

    f(f(x) 6) f(5) f(x) 6 5 f(x) 11 f(x) f(7) x 7

    iii) + + + f1 1 1

    f(x ) f(f(7) 9) 0 f(x ) f(f(7) 9) x f(7) 9x x x

    + + + + >

    2 21 1 1 x 2x 1 (x 1)x 11 9 x 2 x 2 0 0 0 x 0

    x x x x x

    6.

    . 1 2x ,x , > > > >

    < < >x 0f(x)

    0 xf(x) 0x

    :

    > >x 0 f(x) 0 >xf(x) 0

    < xf(x) 0

    h = hD {0}

    8. i) f = + 21f (x) x 2x 2 , ,

    ( , ) + ,

    0 f . :

    0 0

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

    = 0 . < f

    1 2 1 2 1 2 1 2 1 2x x x x 2 x 2 x f(2 x ) f(2 x ) f(2 x ) f(2 x ) (3)

    (1)+(2)+(3): + + < + + f(0) 0 .

    .

    .

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 15

    ii) x 2x1

    f(x) 0 f(f(x)) f(0) e6

    (2), (2)

    x . 1x

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    : ., ., ., ., . 16

    (1)2 2

    0

    22

    1 1 1( ) 3 4 12

    42 4 2

    1 ( 2) 8 0 8( 2) 8 2

    4 2 4 2 4 2

    = = + =

    + += + = =

    = 2 = 2 = 1 = 0(2,1) : 3 0x y = , Cg .

    : 0( ) 2 ( )AB = ,.

    15. i) f . 1 2x ,x =1 2f(x ) f(x ) (1)

    = =+ + + +

    1 2 1 2

    1 2 1 2

    x x x x

    3 x 3 x 3 x 3 x 1 2x ,x .

    :

    1 2x ,x 0 = = + = + =+ ++ +1 2 1 2

    1 2 1 2 1 2 1 2

    1 21 2

    x x x x3x x x 3x x x x x

    3 x 3 x3 x 3 x

    x2 e 0 x )

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

    + = + + = + 1 2 1 1 2 2x x 2x x 2x x1 22x ln(2 e ) 2x ln(2 e ) lne ln(2 e ) lne ln(2 e )

    + +

    + + +

    = = + = + + + + +

    + = + =

    1 2 1 2

    1 1 2 2 2 1

    1 2 1 2

    1 1 2 2 2 1 1 2 1 2 1 2

    2x 2x 2x 2x2x 2x x 2x 2x x

    x x x x

    2x 2x x 2x 2x x 2x 2x x x x x

    e e e eln ln 2e e 2e e

    2 e 2 e 2 e 2 e

    2e e 2e e 0 2(e e ) e (e e ) 0

    ++ + + + + + = + + =

    x x x x1 2 1 2

    1 2 1 2 1 2 1 2 1 2 1 2 1 2

    2(e e ) e ) 0x x x x x x x x x x x x x x2(e e )(e e ) e (e e ) 0 (e e )(2(e e ) e ) 0

    = =1 22x 2x 1 2e e x x , f 1-1.

    iii) f 1

    { }2

    . ( + 1

    2x 1 0 x2

    )

    1 2x ,x

    ( )( ) ( )( ) = = + = + + +

    1 21 2 1 2 2 1

    1 2

    3x 2 3x 2f(x ) f(x ) 3x 2 2x 1 3x 2 2x 1

    2x 1 2x 1

    + = + = =2 1 1 2 2 1 2 1 1 2 1 26x x 3x 4x 2 6x x 3x 4x 2 7x 7x x x , f 1-1. 16. i) . ;

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

    = + = + + = + = + +

    2 2 2 21 21 2 1 2 1 2 1 2 2 1 1 2 1 2 2 1 1 22 2

    1 2

    2x 2xx x x 2x x x x x x x x x 0 x x (x x ) x x 0

    x 1 x 1

    = = =

    = =

    2 1 2 12 1 1 2

    1 2 1 2

    x x 0 x x(x x )(x x 1) 0

    x x 1 0 x x 1 (2)

    (2)

    1 2x ,x ,

    = =1 2

    1x 3,x

    3 :

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 17

    = =+

    = = = = ++

    2

    2

    6 6f(3)

    3 1 102 2 2

    1 63 3 3f( )1 103 101 119 93

    ii) = =f(3) 0,f( 1) 0 .

    iii) =f(3) f(1)

    17. i) 1-1

    ii) 1-1

    iii) 1-1

    iv) 1-1

    18. i) =fofD

    1 2x ,x < > < f f

    1 2 1 2 1 2x x f(x ) f(x ) f(f(x )) f(f(x )) ff

    .

    ii) + = + = 2 2f(f(x x)) f(f(2)) 0 f(f(x x)) f(f(2))

    + = + = + = = = fof fof 1 1

    2 2 2(fof)(x x) (fof)(2) x x 2 x x 2 0 x 1 x 2

    iii) + > + < + < + > f f

    f(f( x 1) f(2)) f(0) f( x 1) f(2) 0 f( x 1) f(2) x 1 2 > x 1

    x

    19. i) 1 2x ,x

    = = = =gof 1 1

    1 2 1 2 1 2 1 2f(x ) f(x ) g(f(x )) g(f(x )) (gof)(x ) (gof)(x ) x x f :1-1.

    ii) fof 1-1

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

    = = = =fof 1 1

    1 2 1 2 1 2 1 2f(x ) f(x ) f(f(x )) f(f(x )) (fof)(x ) (fof)(x ) x x f 1-1.

    f 1-1.

    1 2x ,x ( ) ( )=1 2fof (x ) fof (x )

    ( ) ( ) ( ) ( )

    = = = =f 1 1 f 1 1

    1 2 1 2 1 2 1 2fof (x ) fof (x ) f f(x ) f f(x ) f(x ) f(x ) x x fof 1-1.

    20. x=0 : 2(0) (3) (0)f f f (1)

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

    (1)+(2) 2 3 2 2 22 (0) (3) (0) (3) 0 (0) (3) 2 (0) (3) 0 ( (0) (3)) + + f f f f f f f f f f

    (0) (3) 0 (0) (3)f f f f = = , f 1-1.

    21.

    i) H (1) 0x = : (0) (1) ( )f f f = H (1) 1x = : (1) (0) ( )f f f = +

    1 1

    ( ) ( ) 0f

    f f

    = + = + =

    ii) (1) : ( ) (1 ) ( )f x f x f = x (2) x = : ( ) (1 ) ( ) ( ) (1 ) ( ) 0 ( )( (1 ) 1) 0f f f f f f f f = = =

    ( ) 0f = (1 ) 1 0 ( ) 0 (1 ) 1 = = =f f f ( ) 0f = (2) ( ) (1 ) 0f x f x = x f x ( 0 1) 1-1 ,

    (1 ) 1f = .

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    : ., ., ., ., . 18

    22. i) x = + + = +5 5(gog)(x) 13g(x) 14f(x 3) (gog)(x) 13g(x) 14f(x 3) (2)

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

    >>> = = 1 2 1 2g(x ) g(x ) 13g(x ) 13g(x ) (+)

    = 1 1 2 2

    g(g(x )) 13g(x ) g(g(x )) 13g(x ) ( ) ( ) = (2)

    1 1 2 2gog (x ) 13g(x ) gog (x ) 13g(x )

    + = + + = + f 1 1

    5 5 5 5

    1 2 1 214f(x 3) 14f(x 3) f(x 3) f(x 3) + = + = =5 5 5 51 2 1 2 1 2x 3 x 3 x x x x

    ii)

    + + = + + + + = + + g 1 1

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

    + = + = = =3 3x x 4x 2 x 3x 2 0 .... x 1 x 2

    23. i)

    ii) f

    > > >

    < < x 0 + = + >2 2(x x)f(x) x(x 1)f(x) 0 ( (1))

    2 2(x x)f(x) x(x 1)f(x) 0 ( (2))

    x 0 : + >2(x x)f(x) 0

    24. i) 1-1 , +

    = 1x 1

    f (x) ,x3

    ii) 1-1, =f(0) f(2)

    iii) 1-1 , ( ) += +1 ln(x 4) 3f (x) ,x 4,2

    iv) 1-1, = f(3) f( 3)

    v) 1-1 , = 1 xf (x) e 1, x

    vi) 1-1, =f(0) f()

    25. i) 1 2x ,x = = = =1 2 1 2 1 2 1 2f(x ) f(x ) f(f(x )) f(f(x )) x x x x f 1-1.

    ii) x 1f (x) = = 1 1 1f(f(f (x))) f (x) f(x) f (x) x .

    (*) 0y . = 0x y : =0 0f(f( y )) y

    = 0 0x f( y ) 0y 0x =0 0y f(x ) .

    f = f( )

    iii) f . 1 2x ,x

    < < < < > f f

    1 2 1 2 1 2 1 2 1 2x x f(x ) f(x ) f(f(x )) f(f(x )) x x x x

    f .

    iv ) (1) x=0 : =f(f(0)) 0

    H (1) =x f(0) : =

    = = = =f(f(0)) 0

    f(f(f(0))) f(0) f(0) f(0) 2f(0) 0 f(0) 0

    (1) x f(x) =

    = = f(f(x)) x

    f(f(f(x))) f(x) f( x) f(x) x

    x 0 x, f(x) .

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    : ., ., ., ., . 19

    26.i) x : = + =f(f(x)) f(x) 3x f(f(x)) f(x) 3x (2)

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

    = = 1 2 1 2f(x ) f(x ) f(x ) f(x ) (2)

    (1) +(2): = = =1 1 2 2 1 2 1 2f(f(x )) f(x ) f(f(x )) f(x ) 3x 3x x x f 1-1

    = = 1fD f( ) . (1) x 1f (x) :

    ( ) = + = + = = 1 1 1 1 1 1 1f(f(f (x))) f(f (x)) 3f (x) f(x) x 3f (x) f(x) x 3f (x) f (x) f(x) x3

    ,

    x .

    ii) x=1 (1) = + = + =f(f(1)) f(1) 3 1 f(2) 2 3 f(2) 5 =1f (5) 2

    27. i) 1 2x ,x = =3 3

    1 2 1 2f(x ) f(x ) f (x )) f (x )

    >> = =1 2 1 2f(x ) f(x ) 4f(x ) 4f(x ) (+)

    + = + + = + =3 31 1 2 2 1 2 1 24f(x ) f (x ) f (x ) 4f(x ) x 8 x 8 x x f 1-1.

    ii) = = = 1 ffD f(D ) f( )

    (1) y=f(x)

    + = + = + 3 3y 4y x 8 x y 4y 8 , = y f( ) = + 1 3f (y) y 4y 8 , y = + 1 3f (x) x 4x 8,x .

    iii) = = = 1 ffD f(D ) f( )

    1 2y ,y > + > > >f

    x 0 f(x) f(0) f(x) 3 Cf =y 3 >x 0 .

    29.i) = + + + + = + + + = 2 2 2fA C f(1) (1 2) ln1 ( 2) 2 2 2

    + + = + + + = + = 2 2 2 2 2 2 2 2 2 0 2 1 2 1 0 ( 1) ( 1) 0

    = =

    + = = =

    2 2

    1 0 1

    ( 1) ( 1) 0

    1 0 1

    .

    ii) = = 1 :

    = + + + + = + + + + = +f(x) (x 2) ln x 1 (1 2) 2 x 2 ln x 1 1 2 2 x ln x = + >f(x) x ln x,x 0

    >1 2x ,x 0 <

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 20

    >> 1f (x) 0 . >x 0 ,

    > > > >> < > + >>

    f1

    1

    x 0 x 0 x 0 x 0f (x) x 0 x 1

    x f(x) x x ln x 0 ln xf(f (x)) f(x)

    >f f(0) 0f

    1 1f (x) 0 f(f (x)) f(0) x 0

    =

    < < >yxe 0,e 0 )

    1 ffC ,C .

    vi) = = = 1 ffD f(D ) f( )

    1 2y ,y >f

    x x x x2019 2020 f(2019 ) f(2020 ) (+) :

    + > +x x x xf(2017 ) f(2019 ) f(2018 ) f(2020 )

    x = 0

    33. i) (1 )

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

    1 2x ,x = =3 3

    1 2 1 2f(x ) f(x ) f (x ) f (x )

    >> = =1 2 1 2f(x ) f(x ) 2f(x ) 2f(x ) (+)

    + = + = =(2)

    3 3

    1 1 2 2 1 2 1 2f (x ) 2f(x ) f (x ) 2f(x ) 3 x 3 x x x f 1-1 .

    ii) H (2) x =x 3 +

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

    3 2f (3) 2f(3) 0 f(3)(f (3) 2) 0 f(3) 0 Cf (3,0)

    xx 3.

    iii)

    = = = =f 1 1

    1 1f (x) 3 f(f (x)) f(3) x f(3) 0

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    : ., ., ., ., . 22

    iv) 0y 0x =0 0y f(x ) . 0x ;

    ( (2) x 1f (x) + = + = = 3 1 1 1 3 1 1 3f (f (x)) 2f(f (x)) 3 f (x) x 2x 3 f (x) f (x) 3 x 2x f.)

    0y = 3

    0 0 0x 3 y 2y (1)

    + + = + + = + = 3 3 3 3 30 0 0 0 0 0 0 0 0 0 0

    f (x ) 2f(x ) x 3 0 f (x ) 2f(x ) 3 y 2y 3 0 f (x ) 2f(x ) y 2y 0

    + = + + = 3 3 2 20 0 0 0 0 0 0 0 0 0 0 0

    f (x ) y 2f(x ) 2y 0 (f(x ) y )(f (x ) f(x )y y ) 2(f(x ) y ) 0

    + +

    + + = = =2 2

    0 0 0 0f (x ) f(x )y y 2 02 2

    0 0 0 0 0 0 0 0 0 0(f(x ) y )(f (x ) f(x )y y 2) 0 f(x ) y 0 y f(x ) .

    v) ) 1 2x ,x > < (2)

    3 3

    1 1 2 2 1 2 1 2f (x ) 2f(x ) f (x ) 2f(x ) 3 x 3 x x x , f

    .

    )>

    + > + < + + x 1g(x) e ln x x 2,x 0 g

    ( )+0, (3) : >

    + + < < < >> < < 1 2 1 2x x x 1 x 1 :

    1 10 0 0 0f(f (x )) f(x ) x f (x )

    =f(x) x x .

    35. 35

    g ( ) =1g g =g(x) x .

    (1) = 1foEof E = =1 1 1Eof f oE (Eof) Eof [ = =1f E ln x

    36. .

    i) ( ) ( )+

    = +

    + + + +

    10 f(1) 2

    262 22 2 10 26 f(1) 20( 5) 0 26 20( 5) 0 10 20( 5) 026

    + + =2 2100 20 20 100 0 0 0 = +

    2

    10f(x) ,x

    x 25

    ii) = + h(x) x 5 1,x

    . i) x f(x) f(5)

    +

    2

    2

    10x 50f(x) f(5) .... (x 5) 0

    50x 25 x

    ii) x : = + =h(x) x 5 1 1 h(5) h x=5

    iii) x = = h(x) h(5) 1 f(5) f(x) =h(x) f(x)

    x = 5 f h (5,1).

    37. :

    > >

    > >

    f(x) 2, x 0f

    f

    x D x 0x 0

    f(x) 1 D f(x) 1 0

    k ) +2, f(x) 2 x 2 : =f(2) 2 : = =f(x) 2 x 2 (1)

    : = = =(1)

    f(f(x) 1) 2 f(x) 1 2 f(x) 3 3

    =1f f = f( )

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 24

    ( ) +1x 0, =1f(x ) 3 .

    38. A. :

    > >

    > >

    f(x) 2, x 0f

    f

    x D x 0x 0

    f(x) 1 D f(x) 1 0

    ( 0,2 :

    = = = =f 1 1

    f(f(x) 1) 2 f(f(x) 1) f(2) f(x) 1 2 f(x) 3 3

    ( 1x 0, 2 =1f(x ) 3 1x 2 . f ( 0,2 . ) +2, 39. i) xx Cf f 1-1.

    f Cf ,

    = f( )

    ii) Cf, Cf-1 =y x

    1f ,

    =y x .

    (-3,0),(0,3) (3,5) =y x

    (0,-3), (3,0) (5,3). ,.

    = =

    ' '0 ( 3)

    13 0

    =

    =

    ' ' : y ( 3) 1(x 0)

    y x 3

    x 3

    = =

    ' '3 0 3

    5 3 2

    =

    = =

    3' ' : y 3 (x 5)

    23 15 3 9

    y 3 x y x2 2 2 2

    x 3

    x 1 0x 1

    x 1 1 0 )= +Df 1,

    ii) ) +1 2x ,x 1, =1 2f(x ) f(x ) .: = + + = + + + = +

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

    O

    y=x

    (0,-3)

    (5,3)

    x

    Cf

    (-3,0)

    (0,3)

    (3,5)

    y

    (3,0)

    Cf-1

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 25

    + = + + = + = 1 2 1 2 1 2

    ln( x 1 1) ln( x 1 1) x 1 1 x 1 1 x 1 x 1

    ) + = =

    1 2x ,x 1,

    1 2 1 2x 1 x 1 x x f 1-1.

    iii)

    = = + + = +

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

    2

    = + = y 3 y 3

    2 2e x 1 1 e 1 x 1 (

    y 3 y 3 y 302 2 2

    y 3e 1 0 e 1 e e 0 y 3

    2)

    = = +

    2 2y 3 y 3

    2 2e 1 x 1 x e 1 1 , y 3

    = +

    2x 3

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

    iv)

    + = + = + = + + = +f1 1

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

    41. )= +Df 2, ( 9x 18 0 x 2 ) f

    ) +1 2x ,x 2,

    = = =1 2 1 2 1 2

    1 1f(x ) f(x ) 9x 18 9x 18 .... x x

    3 3 f 1-1

    =y f(x) (1), .

    = 1

    y 9x 18 (y 0)3

    = 3y 9x 18 = 29y 9x 18 = 2y x 2 = +2x y 2

    = + 1 2f (y) y 2,y 0 , = + 1 2f (x) x 2 , x 0

    f-1 y=x

    42. 1 2x ,x = + = + = 1 2 1 1 2 2 1 2 2 1g(x ) g(x ) f(x ) x f(x ) x f(x ) f(x ) x x .

    ( ) >

    = = =

    1 2 1 2 1 2 1 2

    0,1 :1 0

    1 2 1 2 1 2

    1 2 1 2 1 2 1 2

    f(x ) f(x ) x x x x x x

    x x x x 0 x x (1 ) 0

    x x 0 x x 0 x x 0 x x

    g 1-1.

    2 2y x= +

    19 18

    3y x=

    1Cf

    Cf

    y x=

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    : ., ., ., ., . 26

    43. i) f . (1) x x f(x) :

    =

    = = (1):f(f(f(x))) x

    f(f(f(f(x)))) f(x) f( x) f(x) x f

    ii) 1 2x ,x = = = = =(1)

    1 2 1 2 1 2 1 2 1 2f(x ) f(x ) f(f(x )) f(f(x )) f(f(f(x ))) f(f(f(x ))) x x x x

    f 1-1.

    iii) y =y f(x) x.

    :

    = = = = = f 1 1 f 1 1 (1)

    f(x) y f(f(x)) f(y) f(f(f(x))) f(f(y)) x f(f(y)) x f(f(y)) (2) , y

    y f(f(y))

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

    44. i) + +

    = = = = + + + + +

    x x x

    x x x x x

    e e 1 1 e 1 1 1g(x) 1

    e 1 e 1 e 1 e 1 e 1, x

    1 2x ,x < < + < + 1 2 1 2x x x x

    1 2x x e e e 1 e 1 > <

    + + + +1 2 1 2x x x x1 1 1 1

    e 1 e 1 e 1 e 1

    < +

    x

    x

    ey g(x) y (y 0)

    e 1

    + = + = = = x x x x x x xy(e 1) e ye y e y e ye y e (1 y) (1 y 0 y 1)

    = > < <

    x y ye ( 0 0 y 1)1 y 1 y

    ( )= =

    1y yx ln( ) f (y) ln( ) , y 0,11 y 1 y

    ( ) =

    1 xf (x) ln( ) , x 0,11 x

    iii) ( ) ( ) = = > = > < < =*

    1 1 e eDg of {x Df / f(x) Dg } {x 0 / ln 0,1 } {x 0 / 0 ln 1} 1,ex x

    *( < < < < < < > > > >e e e x 1

    0 ln 1 ln1 ln ln e 1 e 1 e x 1x x x e e

    )

    = = = =

    1 1

    eln

    1 ln x 1 ln xx(g of)(x) g (f(x)) ln( ) ln( ) ln( )e 1 (1 ln x) ln x1 lnx

    ( ) = 1 1 ln x(g of)(x) ln ,x 1,eln x

    iv) ( )x 1,e > > > > > > 1

    1 ln x 1 ln x 1 ln x 1 ln x 1 ln x ln x(g of)(x) 0 ln 0 ln ln1 1 1 0 0

    ln x ln x ln x ln x ln x ln x

    >

    1 2 ln x0

    ln x

    ( )x 1,e >ln x 0 > > <

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 27

    v)

    + = + = + = + =

    + + + + ++

    x x x xx

    x x x x x

    x

    1e e e e 1eg(x) g( x) 1

    1e 1 e 1 e 1 e 1 e 11

    e

    vi) + + + + + =1 1 1

    g(ln 2017) g(ln 2016) g(ln 2015) g(ln ) g(ln ) g(ln )2017 2016 2015

    + + + + + =g(ln 2017) g(ln 2016) g(ln 2015) g( ln 2017) g( ln 2016) g( ln 2015)

    ( ) ( ) ( )+ + + + + =(vi)

    g(ln 2017) g( ln 2017) g(ln 2016) g( ln 2016) g(ln 2015) g( ln 2015) + + =1 1 1 3

    vii) + + +

    + + + + + +

    ++

    2 2

    2 2

    4 g( 2020) 2 8 4 g(2020) g2 2

    2 8 4 g(2020) 4 g( 2020)

    e eg( 4 g( 2020)) g(2 8 4 g(2020))

    e 1e 1

    + =

    + + + + + + + g( 2020) g(2020) 1

    2 2 2 2 4 g( 2020) 2 8 4 g(2020) 4 g( 2020) g(2020) 2 8 4

    + + + + + + + + + 2 2 2 2 2 2 4 1 2 8 4 4 2 8 4 1 0 2 1 4 8 4 0

    ( ) + 2 2 1 (2 2) 0 = 1 0 =2 2 0 = = 1

    45. . x

    x g(x) (1)

    g(g(x)) g(x) (2)

    (2) =g(x) y g(y) y = y g( ) .

    g(x) x x (3). (1),(3): =g(x) x , x

    . i) f 1-1

    = + =0f(0) 0 e 1 0 . :

    = + = = = =1 1

    x xe x 1 e x 1 0 f(x) 0 f(x) f(0) x 0

    ii) + > + > > > > f f

    f(f(x) 1) f(1) f(x) 1 1 f(x) 0 f(x) f(0) x 0

    . i) = + xf(x) x e 1 , x x h(x) :

    = + = = +h(x)f(h(x)) h(x) e 1 f(h(x)) (x) 1 (x) f(h(x)) 1 (1)

    1 2x ,x 4

    (3 1) 2 0 (3 1) 2 (3 1) (3) 3 1 33

    f

    f x f x f x f x x

    iv) H f 1-1 (*) .

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 28

    v)

    = = = = = =1 1 1 1

    1 1 1 1 0( ) 2 ( ) (1) 1 1 0 1

    xf ex x x xf e f e f e e e x x

    vi) Cf (1,2) (3,-2) = = 1(1) 2 1 (2)f f = = 1(3) 2 3 ( 2)f f

    vii) ( )=

    + + = + + = + + = 11 (2)

    1 1 1 1 1( 2 ( 2)) 2 ( 2 ( 2)) (2) 2 ( 2) 1f

    f f x f f f x f f x

    + = + = + = = 1 1 1( 2) 3 ( 2) ( 2) 2 2 4f x f x f x x

    (*)H f 1-1

    f .

    1 2, fx x D 1 2x x ,

    < >1 2 1 2( ) ( )x x f x f x

    1 2( ) ( )f x f x f 1-1 .

    f .

    47. i) + +

    = = = = =

    1 11 1 1 1 1f(x)

    2 22 2 2 2 2 2 2

    x xx x x x

    x x x x x x x

    e e ee e ee e

    ii) 1 2,x x R

    x

    1

    ef (x)

    2 =

    < < <

    1 2 1 2

    1 2

    1 1 1 1

    2 2 2 2

    x x x x

    x x (2). (1)

    (2)

    < 1 2 1 2

    ( ) ( )f

    f x f x x x

    (1)

    :

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 29

    > > >1

    1 1

    1 2 1 2 1 2( ) ( ) ( ( )) ( ( ))

    f f

    f x f x f f x f f x x x

    (1) ..

    . :

    > > 1 2 1 2 1 2

    ( ) ( ) , ,f

    f x f x x x x x Df

    > < 1 2 1 2 1 2

    ( ) ( ) , ,f

    f x f x x x x x Df

    48.i) 1 2, 2x x 1 2( ) ( )f x f x= + = + = 2 2 2 2

    1 2 1 2( 2) 2 ( 2) 2 ( 2) ( 2)x x x x ,

    1 2, 2x x 1 22, 2 0x x [ )21f (x) x 0,= + = =1 2 1 22 2x x x x f 1-1.

    ii) f 1-1 2 0 2

    2 2y=f(x) ( 2) 2 2 ( 2) 2 2 2 2, 2y y

    y x y x y x x y y

    = + = = = +

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

    iii) T fC y x= 2f(x) ( 2) 2 ... 2 3x x x x x= + = = = (2,2), (3,3)A B .

    f -1f y x=

    -1f(x) ,f (x)x x= = .

    iv) ( )2,3 -1f(x) f (x)< . ( )3,+ -1f(x) f (x)>

    , 1

    3 ( )

    3

    y=x 2f(x) ( 2) 2, 2x x= +

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

    1

    6

  • 12 mathhmagic.blogspot.gr

    : ., ., ., ., . 30

    49*. i) g

    .

    ii) :

    ( )+ +33f(x x) x f(x) f(x) x f(g(x)) x g(f(x)) (2) x x 1g (x)

    1 1f(x) g (x) g(f(g (x))) (3)

    (2)

    1g g

    1 1 1 1 1f(g(x)) x g(f(x)) g (f(g(x))) g (x) g (g(f(x))) g (f(g(x))) g (x) f(x) (4)

    (3) (4) : = 1f(x) g (x) x .

    50. i) x

    ( ) ( ) = = = 2 2f(x) f(x) f (x) f(x) f(x)f(x)1

    e 2x e 1 2xe e 2xe 1e

    ( ) ( ) + = + = + = + = +222 2f(x) f(x) 2 2 f(x) 2 f(x) 2 f(x) 2e 2xe x 1 x e x 1 x e x 1 x e x 1 x

    + + > + >

    = + = + + = + + 2f ( x) x 1 x 0 x (*)e x 0

    f(x) 2 f(x) 2 2e x 1 x e x 1 x f(x) ln(x 1 x ),x

    .

    (*) + + >2 1 0x x x .

    x : + > =2 21x x x

    , x x + > + + >2 21 1 0x x x x , x

    ii) ) 1 2,x x < > < 1 2 1 21 2 1 2

    x x x xx x x x e e e e (+)