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## Education

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. 1.

) f(x) = +

2

3 2x 4

x 6x 11x 6 ii) f(x) = 3 2 x 1 iii) f(x) = +32 log (x 2)

iv) f(x) = log x ( log4(3-x2)) v) =2

f 5x x(x) log 4 vi) f(x) =

24 x12

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)=+

24 - x(x - 1) x 1

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

+ 34 - x - x

) f (x) =2x - x

x - 2 - 1 + 1

3x - 8 - x

) f (x) = 5x - 3 - 1

) f (x) = log (x2 + x - 2) + log +x 33 - x

) f (x) = xe - 1 + 1 - lnx ) f (x) = x2x - 1

+ 1x - 1

, x [0, 2]

. 4. C f , : I) f(x) = x2-5x+6 ii) 3xf(x) e 1= iii) 5f(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) = x 1

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

7. ( ) ln( 1) f x k x= + + 2 1e 2 . : i) , R

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ii) fc 3 .

8. , :f g R R , : = + 2( ) ( ) 4f x g x x x R . fc gc .

9. , :f g R R : + + = 2 2( ) ( ) 1 2( ( ) ( ))f x g x f x g x , x R

10. x R f g

( ) ( )( ) ( )( ) ( ) ( ) ( )

2 2

2

3 2

x +2x+3 -x -x-2

-x-1 2x

i) f x = x +2x+1 g x = x +2x+1

ii) f x = e g x = e

v) f x = ln e +1 g x = ln e +1

11. :f R R : 2 2( 2) f(3 x) 0f x + + = , x R . f .

12. ) + += =

2

2 3x 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

13. f (x) = x + 1. ) f.

f1 (x) = 2x - 1

x - 1 f2(x)=

++

3

2x 1

x - x 1 f3 (x) = ( )2+x 1

f4 (x) = x (x 1 + 1) f5 (x) = lnex+1 f6 (x) = eln (x+1)

) R .

15. . f (x) = +x 1x - 1

, g(x)=2+ +2

22x 2x

(x - 1),R, x > 0.

) f, g ) f = g; . f=g .

.

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f g , : =( ) ( )f x g x .

i) =

2

21( ) xf x

x x = + 1( ) 1g x

x

ii) =

2( ) ln

1xf x

x = ( ) 2ln ln(1 )g x x x

16.

+ += =

2

2x 3x 1, x [0,2] , x ( 2,1)3x 6f(x) g(x)2x 3 , x (2,6) x 1 x [1,13]

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

17. f (x) =x,

+

>

2x 1, x 2 x 2

g (x) = ,

<

0, 0

g(x)1, 0

= >

: ) f g !!!!!!!!!!!!!!

.

19. = +f(x) x 3 = g(x) ln(x 2) . f g . g f . f f v. g g 20. f g, g f, f f, g g

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

.

+ > += =+ +

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

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

22. f [0, 2].

: ) f (x2) ) f (x - 4) ) f (lnx) ) 2f( 1 x ), ) f(3 x 2) ) f( 3 x)

x 3+

23. [ ]: 2,1f R

: = ( ) (2 3)g x xf , 2( ) ( )h x f x= ( ) (ln )q x f x=

24. : (0,1] Rf . ( ) ( 2) (ln )g x f x f x= +

25. 2( )2

xf xx

=

( ) 1g x x= .

: ,f g g f 1ff

26. x R + =3 3f(x) x ( ) =f f (x) x

27. f ,

+ = f(x) 12f 3, x Rx x

f.

28. f(x)

( ) ( ) ( )( ) ( )

( )

2 3 6 3

2 x 2

I) f 2x -1 = x -3x+2 II) f x = x -2x +1 x >0

III) f lnx = x - x+2 IV) f e -1 = x -3x+1

V) f(ln2x) = x+3 x >0

29. = 2 1( ( )) xf f x x R

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

30. f : R R, + +f( ) f() f() , . ) .f(x) 0 ) )f() f() f( ,R.

31. ( ) = + x ef g (x) e x = > .g(x) lnx 1, x 0 f 32. = f(x) g(x) x = 2g(x) x f(x), x R,

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= (f g)(x) (g f )(x) xf(x) 33. 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

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

) = x2, x0, f (2).

35. = +2f(f(x)) x x 1 .f(1) 36. = f(f(x)) 3x 2 = f(3x 2) 3f(x) 2, x R

37. : 3 1, 1( )2 , 1x x

f xx x

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) f gc c .

) ( )( ) , 0( )

f xh x xg x

= > hc

,x >0 .

44. f : R R (0) 0f =

=

( )( ) ,1x

f xg x x Re

. ( ) 0g x < 0x .

45. (1,5)A (5,-2) . : ) f . ) f f ( (e )) 2xf f < .

46. f : R R :

+ = + >( ln ) ( 1) ln 3, 0f x x f x x x . f , ( ) 2f x = ( 1) 2xf e < .

47. f : R R f ( ,0] f . :

) + = +( ) (5 ) (3 ) (7 )f x f x f x f x ) + = +5 3 7( ) ( ) ( ) ( )x x x xf e f e f e f e

48. f :R R . :

( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( )2

. f x > f 3 . f 2x+1 0 . f f 3x -1 < f f 2x+5

49. ( ) 1-x 1f x = e + -2 , x >0x

.

. f ( )0,+ . ( )0,+ : 1-xxe -2x +1>0 50. . ( ) ( )xf x = e +ln x+1 -1 . . . . : ( )2x 2e +ln x +1 >1 51. [ ) f : 0,+ R ( ) ( )2f x =3x+ln x +1 x>0 . . .

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