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.

1.

) f(x) =2

3 2

x 4

x 6x 11x 6 ii) f(x) = 3 2 x 1 iii) f(x) = 32 log (x 2) iv) f(x) = logx( log4(3-x2))

v)25x x

f(x) log4

vi) f(x) =

24 x

1

2 vii) f(x) 5 x 4 viii) 3 2f(x) x 1 x

2. R f(x) = ln ( x2+2x+9) = R

3. R :

)f (x)=1x1)-(x

x-4 2) f (x)=

1-2-x

2+

x-x-4

3)f (x) =

1-2-x

x-x2

+x-8-3x

1 ) f (x) =

1-3-x

5

)f (x) = log (x2+ x - 2) + log x-3

3x) f (x) = 1-e

x

+ lnx-1 ) f (x) = 1-2x

x+

1-x

1

, x [0, 2]

.

4. Cf , :

I) f(x) = x2-5x+6 ii) 3xf(x) e 1 iii)5

f (x) log (x 3) 1 iv) f(x) x 3 2

5. f (x) = x2- 3x + 2. ) f (1), f (0), f (-3), f (2)

) Cf ) f (t), f (xt), f (x + h), x, t, h R.

6. f (x) = lnx1

x . ) f. ) f (x) = e

x . )

.

7.

)2

2 3

x 2 x 2xf(x) , g(x)

x 1 x x )

x 3 x 3f(x) , g(x)

x 2x 2

) f(x) x 4 x 5 , g(x) x 5 4 x

8. f (x) = x + 1.

) f.

f1(x) =1-x

1-x2

f2(x)=1x-x

1x2

3

f3(x) =2

1x

f4(x) = x (x

1+ 1) f5(x) = lnex+1 f6(x) = eln (x+1)

) R .

9.

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f1(x) =1x

1-xf2(x) =

1x

1xf3(x) =

x

11

x

1-1

f4(x)=1-x

1)-(x2

2

f5(x)=1x

1x

2

f6(x) =1x-1x

1x

) . ) .

) R .

10. f (x) =1-x

1x, g(x)=

1)-(x2

2x2x2

2

, R, x > 0.

) f, g ) f = g;

.

11. 2

2x 3, x ( 2,1)x 1, x [0,2]

3x 6f(x) g(x)2x 3 , x (2,6)

x 1 x [1,13]

2f+3g, f.g , f 2g

12.

f (x) = 2 x,x

2x1,2x

g (x) = 3 x,32x-

3x0lnx,

: )f + g )f g

.

13. f(x) x 3 g( x) ln( x 2)

. f g . g f . f f v. g g

14. f g, g f, f f, g g

15. f i) 2(f g)(x) x x 3, g(x)=x-1

ii) 2 4(f g)(x) 3 x x , g(x) = x2 iii) (g f )(x) 5x 4, g(x) 7x 6

16. f : [0,2] R .

I) 2f( 1 x ), ii) f(3 x 2) iii) f(3 x

)x 3

17. f [0, 1]. ) f (x2)

) f (x - 4) ) f (lnx) :

18. : f (x) = 1-x

1

, g (x) =1x

1-x

. ) . )

f + g, f g. ) f, g (gof) (x) g (x) f (x).

2x 3, x 0x 1, x [1, )f(x) , g(x)

2X 3. x ( ,1) 3x 1 x, x 0

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) f, g fog gof .

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

1;

20. f (x) = xx, x > 0 .

21. x R 3 3f(x) x f f (x) x

22. f , f(x) 1

2f 3, x Rx x

f.

23. f , f( + ) = f(x)+ f() , , R. I) f(0)=0 ii) f( kx)=kf(x) , k N iii) f

24. f : R R, f( ) f () f () , .

) f (x) 0. ) f() f () f ( ) , R.

25. x e

f g (x)e x

g(x) ln x 1, x 0. f

26. f (x) g( x) x 2

g(x) x f(x), x R, (f g)(x) (g f )(x) xf (x)

27. f : R R 2f(x 1) 2f(3 x) x 1, x R.

) 2f(x) 2f(2 x) x 2x 2 ) 2f(2 x) 2f(x) x 6x 10. ) f

28. f : R R f (x + y) + f (x - y) = 2f (x) + f (y) x, y R.

) f .

) f .

) x R f ( x ) = f (x).

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

1) = x2, x 0, f (2).

30. 2f(f(x)) x x 1 f(1).

31. f (f (x)) 3x 2 f(3x 2) 3f(x) 2, x R

.

32. f 3 2f(x) x 3x 4.

) f = 0,

) f(x) 0 .) f (x) 0 .

33. R x

2f(x)

3 ) R

) R

34. f, g R,

( ).

) fog .

) fof gog.

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

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35. f R . f o

[, ] , > 0, f [- , - ].

36. f, g ,

x . g

1

f

1

.37. f: . x1, x2,.x100 1 2 100x x . x

: 1 2 100 1f(x )+ f(x ) . (x ) 1, : f(x ) 0,01f

38. f : (0, ) R f (0) 0,

f(x)

g(x) , x 0.ln(x 1)

g( x) 0 > 0.

. 1-1

39. 2f(f(x)) x x 1, x R . ) f(1)=1 ii) g(x)= x 2x f(x)+1 , g 1-1

40. f(x) = ex-1+ + 2. ) f 1 - 1.) f(x) = 4. ) ex-1+ - 2 > 0

41. 1 -1 f: R>R f2(x) < f(x)f(a - x), x R, .

42. f: R g : f(A) R. g f 1 -1, :

) f 1 - 1 ii) g 1-1

43. f , g 1-1 g f 1-1 .

.

44. f R, (fof) (x) - f (x) = x,

x R. f.

45. f (x) = x + , 0, :

) f = f-1

)f = - f-1

)f = f-1

+ c (c 0, )

46. f (x) =1x

x. ) f 1 - 1. ) f -1.

47. f(x) = 2 4 x

48. f(x) = lnx -e

xx

I)

) 1f (x) x

49. )3x 2

f(x)x 3

)2

xf(x)

4 x

) f(x) 3 2 x v) x 2, x 3f(x)2x 5, x 3

v) x, x [0,1]f(x)2 x, x [1, 2]

50. f :R+ R f(+)=f(x)+f(). f

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1 1f ().f () 1f ( )

51. f: IR IR (f f)(x) + f2003(x) = 2001, R. f .

52. f(x) = 2-- ln x. i) f , .ii) , , f(x) = f(1) iii) + l > 1.

53. f(x) = ex+ 3+ + 1. i) f , ii)

e 2x x

+ ( 2x -x )3 + 2x - 2 = e x+3+ ( + 3)3+3

54. f: R R f (f(x)) = x + f(x) x R, :

i) f ii) f(0) = 0 iii) 1f (x) f(x) x x f(R)

55. f : R R (3,2) (5,9) ) 1 2f(2 f (x x)) 9

) 1 2f(f (x 8x) 2) 2

56. 3( ) 1f x x x ) f

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

57. f, g : R R (g f)(x) = x3+ 3f(x) + 2 R.

f .

58. f . )

f 1 . )

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

. ) g1

(x) = g(x) , g(x) = lnx + x 5 .

59. f R, (fof) (x) - f (x) = x, x R.

f.

60. ) f .

f(x) = 0 . ) .

) : i) 2 3xe x , ii) xxx 1086 .

61. f : f(0)=1 f(x+y) exf(y) x ,y , f(x)=ex x

62. f: (1,+ ), f 2(x) +1 = 2f(x) + e2x .i) f 1-1.ii) f -1.

63. f : ln[f(x)] + ex1 = 0, f(x) 0 x .i. f . ) f -1.

iii) f -1 = 2.

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