(1)

: f :

2 2( ) 2 ( ) 2 0f x f x x + = , fx D , :

:

( )22 22

( ) 2 ( ) 2 0 ( ) 1 1

( ) 1 1

f x f x x f x x

f x x

+ = = =

2

, ( ) ( ) 1g x f x= ( , 1] [1, + ) 1 1x x= = .

,

(1 .

( , 1 )

)

, )+

(1( , 1 , )+ , :

( ) ( ) 1g x f x=

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

(1( , 1 ) , )+ , :

( ) ( ) 1g x f x=

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

,

, :

. , ..., 2008 1/11

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

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

. , ..., 2008 2/11

) 2

2

1 1, ( ,( )

1 1, (1,

x xf x

x x

+ = +

1)

)

) 2

2

1 1, ( ,( )

1 1, (1,

x xf x

x x

=+ +

1)

)

. , ..., 2008 3/11

(2).

f : 3( ) 3 ( ) , f x f x x x = \ .

) f 1-1

.

) f

;

R

:

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

3 3( ) 3 ( ) ( ) 3 ( )2x f x f x x f x f x= = . f 1-1. 1x x= 2 f ( )y f x= x\ , x :

3 3x y y= f , :

11 3( ) 3 , ff x x x x D =

1f , R 1f 1-1,

3( ) 3 ,g x x x x= R 1-1.

) f . ,

f ,

,

R

R1f ,

f, .

. , ..., 2008 4/11

1f

, R 1f

f. 1 3( ) 3 ,f x x x x = 1-1, o

: R

( ), 1 , 1, 1 1, + . ,

3( ) 3g x x= x . f R .

R

f

, ,

( ) (0 , 2 2,x +)[ ]0 2, 2x .

) f:

0x R . 0, x x xR , : 3 ( ) 3 ( )f x f x = x 0 3 0 0( ) 3 ( )f x f x x = ,

. , ..., 2008 5/11

, , :

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

( )2 20 0( ) ( ) ( ) 3h z z f x z f x= + + , ( )2 03 ( ) 4f x = :

( )2 0( ) 44

3min ( ) 4

f xm h z

= = = , :

( )2 20 0( ) ( ) ( ) 3 , h z z f x z f x m z= + + R :

( )2 20 0( ( )) ( ) ( ) ( ) ( ) 3 , h f x f x f x f x f x m x= + + R ,

A , 0 0( ) 2 ( ) 2f x f x< > ( )1 , , , (1), :

0 02 2x x< >0min ( )m h z >=

( )2 20 0( ) ( ) ( ) ( ) 3 0, (2)f x f x f x f x m x+ + > R :

( )0 0 02 20 01 1( ) ( ) (3)

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

mf x f x f x f x = + +

0

0lim - 0x x x x = . 0 0lim ( ) ( )x x f x f x = , f

0x .

f ( ) ( )0 , 2 2,x + . ( )1

3( ) 3 , g x x x x= R( ], 1 [ )1,+ [ ]1, 1 . :

0 0 0 0

0 0

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

2 2f x f x g f x g g f x g

x x

< > < > < >

. , ..., 2008 6/11

A , 02 ( )f x 2 2 02 x ,

, f

min ( ) 0m h z= 0x .

f

, [ 2, 2] f

[ 2, 2] .

) f:

A , , (1) (2), : 0 02 x x< > 2

( )0 2 20 0 0( ) ( ) 1

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

x x f x f x f x f x = + +

f 0x , :

( ) ( )0

0

00

0

22 200 0

( ) ( )( ) lim

1 1lim3 ( ) 1( ) ( ) ( ) ( ) 3

x x

x x

f x f xf xx x

f xf x f x f x f x

= = = + +

f ( ) ( ), 2 2,x + . , ,

f

02 x 20x .

f

, [ 2, 2] f

[ 2, 2] .

. , ..., 2008 7/11

(3).

f :

f x f x x+ = 3( ) 3 ( ) , x R . f:

) 1-1 .

)

, R R ( ], 0 [ )0, + .

:

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

3 3( ) 3 ( ) ( ) 3 ( )2x f x f x x f x f x= + = + . f 1-1. 1x x= 2 f ( )y f x= x\ , x :

3 3x y y= + f , :

11 3( ) 3 , ff x x x x D = +

f

.

0y R 0x R , 0( ) 0f x y= . . 0y R 0 03 3 0x y y= +

. , ..., 2008 8/11

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

30 0 0 0 0 0

3

0

0

0 0

3 3

0

3

3(,

( ) 3 ( ) ( ) 3 ( )( ( )

1 1( )) ,

( )) g f g

f

f x f x x f x f x y yx y g x

x y g

++

+ = + ==

=

=x x

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

1) f:

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

, , :

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

( )2 20 0( ) ( ) ( ) 3h z z f x z f x= + + + , :

( )2 0( ) 4min ( ) 3 04f xm h z += = > , :

( )2 20 0( ) ( ) ( ) 3 0, h z z f x z f x m z= + + + > R :

( )2 20 0( ( )) ( ) ( ) ( ) ( ) 3 0, h f x f x f x f x f x m x= + + + > R , :

( )0 0 02 20 01 1( ) ( ) (2)

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

mf x f x f x f x = + + +

. , ..., 2008 9/11

0

0lim - 0x x x x = . 0 0lim ( ) ( )x x f x f x = , f . 0 Rx f . R

2) f:

0x R . 0, x x xR , , , :

( )0 2 20 0 0( ) ( ) 1

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

x x f x f x f x f x = + + +

f 0x , :

( ) ( )0

0

00

0

22 200 0

( ) ( )( ) lim

1 1lim3 ( ) 1( ) ( ) ( ) ( ) 3

x x

x x

f x f xf xx x

f xf x f x f x f x

= = = ++ + +

f R

( )21( ) 0 (1)

3 ( ) 1f x

f x = >+

, f , R (0) 0f = , :

( )( ) 0, , 0f x < , (0) 0f = & ( ) (2)( ) 0, 0,f x > + f , (1), R f R :

( )322 ( ) (3)

9 ( ) 1( ) f x

f xf x =

+

. , ..., 2008 10/11

, (2), f ( , 0 , )0, + (0,0)

13

y x= .

. , ..., 2008 11/11