μεθοδολογια + ασκησεις συναρτησεις

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3

1.

, ,

Bolzano-2. . .3.

4.

. . (: ) .. . , , . ( ). , , . ( ).5.

. . .

6.

1-1 . , , . y x : , .

x ,

x ,

, , 1 3 =.

( , ) ()= , ()= ( , ) . .

= = .

= .

7. , :

8. = , 1-1

, .

9. 1-1 ,

10. . f ( , )(( , ).. .

.

. . . .

/ ( , , . .

/

11. ,

12. ( )

f .

f . .. (1)

(1) f 13. !!! . , ( ) , . , , .

14. . , . : .15. . f

. .

.

, ,

16. !!! . .. : , , . 0.

17. x , Bolzano ,

18. , , 0 ( 0 ).19. , .. . -.

f 1 , 2 1 2 ( f (1) = f ( 2) f ;

((((( "1 - 1". ( =2006 ;

(((((.

. ( ;

. (-1 ;

. ( (-1

. ((-1 (-1( ;

. (

(-1

1. f, g , x ( R,

f (x) = g (x).

6. 2. f 1 - 1, .

3. f g 1 - 1 R,

gof 1 - 1 R.

1. f (x) =

. R - {- 2, 2} . R . R - {- 2}. [2, + () . R - {2} 2. f (x) = ln (4 - x2)

. R - {- 2, 2} . R - {2} . [2, + () . (-2, 2) . (- (, -2) ( (2, + ()

3. f (x) = 2ex

. g (x) = ln

B. h (x) = ln

. (x) =

lnx

. (x) = E. t (x) = ln ( x-2) 4. y = f (x) xx

. y = f (-x)

B. y = - f (x)

. y =

. y = 2f (x)

E. y = - f (-x)

5. f (x) = x2004 + x2002 + x2000 + 5 xx

. 2004

B. 7

. 2002. 3

E. 0

() = + -1-1 + .+ 1 + 0 , IR. :

1. f IR = . 2. f , g : 0. 3. f : x f . 4. f [ , ] x0 ( , ) ,

, f [,] f() f(), f() f() xo(,) f(xo)=

f (x) =

- 1. ; f (x) = .

f (102006);

f(x) =

R : f(x)= , R.

f . R

:

f : R(R f(x) = 3x-7 + ex-2

)

) f(x)=0

) f

f, g R .

fog 1-1. . g 1-1.

. :

g(f(x) + x3 + x) = g(f(x) + 2

f:R(R

) R(R g(x) = f(x)-x

)

f , R 0< < 1

)

00.

( 1981)

f (1)

, R. :

() f 0=0 R () f 0=2005 R

,

1 .

, , , , F() = 5 + 15(1-2-0,2 ). ) ; ) 5 ; ;

)

(: ;)

) ;

10 .

1.

f (x) , f x0.

2.

= ( 0

3.

f (x) = - (,

EMBED Equation.3 = 0.

4.

|f (x)| = |

|( 0,

EMBED Equation.3 =

.

5.

10 . , f [,] f() f(), f() f() xo(,) f(xo)= 5 . f, [-2006 ,2006] . f [-2006 , 2006] ( ) :

1. f f(-2006) f(2006)

2. f(-2002) < f(2002) f [-2002 ,2002]

3. f f(-2002) f(2002)

4. f(-2002) >2002 f(2002) < -2002 -2002 2002 f.

5. 0,5.[f(-2002) + f(2002)] f.

10 20. .

8. :

)

)

8

.

.

9 30.

(1) .

12

. (1)

() () 13 40. f: f(1) + f(2) + f(3) = 0. f() = 0 .

8

. , . 1-1.

8

. f [0,1] f(0) = 2 f(1) = 4. f , (0,1) :

f() = . 9

2.1

2.2 2006

2.3 2.4 , ,

2.5 , , ,,

2.6 2.7 , ,,

2.8 2.9 Ox

2.10

2.11 (((

.. f =(2 , e+3)(e+3 , +)2.12 0 R ( = 2-12 0 (

2.13

2.14 ) f =R 1,2 R 1 < 2 (1) . 31 < 32 (31 -7< 32 -7

(1)( 1-2 < 2 -2 (( =ex . )

31 -7+< 32 -7 + ( . f . R.

) f(x)=0 =2

f(2)=3.2-7 + e2-2 =-1+1=0.(2)

f . R =2 .

) f(x) < 0 (( (2)) f (x) < f (2)(x - 2

f(x1) -1 > f(x2)- 2 ( g(x1)>g(x2).

g R.

) = 0< < 1 R ) f()=- R.

) f()=- R

1-1 (2) (1) (

( f(2-4)= f(5-20) ( 2-4 = 5-20 ( 2-9 +20=0 ( = 4 =5

2.17 ()

1-1 .

() .

2.18() ( (1))

EMBED Equation.3 1-1 .

() (1)

EMBED Equation.3

() (1)

() (1)

2.19() ) (2) . (2)

EMBED Equation.3 .

(3).

() 1-1

.

(3)

2.20 ()=( )=0 / 0 / 0

=

2.21

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

(1)

EMBED Equation.3 2.22

==

=

==

2.23 (1)

( (2)

(3)

(3)

2.23() (1)

(2)

(2)

(3)

EMBED Equation.3 10+0 = 10

2.24

EMBED Equation.3 (1)

, (2)

. =0

2.25 (1)

(2)

(2) . .

(2)

2.26 (1).

(2)

. . .

(1)=2004+0-2005=-1

2.27=

(1)

=

=

=

EMBED Equation.3 (1) =24+12=36

2.28 (1)

EMBED Equation.2

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

2.29 . / -6 / 0

.

.

. ,

, =6.

2.30

(1)

= .

(2) ( )

EMBED Equation.3

EMBED Equation.3 .

(2)

2.31 (1)

.

= .

2.32

=

f

, 0=0.

, f 0=0.

f .

2.33 . (1)

EMBED Equation.