Τσακαλάκος Τάκης - Άλγεβρα Α' Λυκείου.pdf

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( i) = 1

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{ i} ,

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0 P( i) 1 P( 1) + P( 2) + ... + P( ) = 1 .

P( i), { i}.

= { 1, 2, ... , }

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4 + 2 .

() = = 11

() = 2

() = = 6

9 (5 + 4), .

() () ()

() () ()

() = 9

2 6 9() = = () = = () = =

11 11 11

1 2 3

1 2 2 3

1 2 3

= { , , }

= { , } = { , }.

1 4 () = () = : ( ), ( ), ( ).

2 5

: ( )

: .

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: = { 1, 2, , } . 0 P( i) 1 P( 1) + P( 2) + ... + P( ) = 1 = { 1, 2, ... , } () = ( 1) + ( 2) + ... + ( )

13


1 2 3

1 2 3

1 2

1 2

3

= { , , } :

( ) + ( ) + ( ) = ()

= { , } :

( ) + ( ) = ()

1 (1) - (2) : ( ) = 1-

2

1 2 3

1 2

3

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2

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2

2 3

( 3 )

2 3 2 2

1 1

= { , } :

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2 5 5 2

(1), (3), (4) :

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10 2 10 2

2

1

(3)

3 ( ) = (4)

10

1( ) =

5

, :

( ) = 0,9 (') = 0,4 ( ) = 0,5.

: (), (), ( - ), ( - ), [( - ) ( - )] .

1- (') = 1- 0,4

( ) = () + () - ( ) 0,9 = 0,6 + () - 0,5 0,9 - 0,6 + 0,5

() - ( ) = 0,6 - 0,5

() - ( ) = 0,8 - 0,5

( - ) + ( - ) = 0,1+ 0,3

() = = 0,6

() = = 0,8

( - ) = = 0,1

( - ) = = 0,3

[( - ) ( - )] = = 0 ,4

: ( )

: .

: .

: . ( ) = () + () (, ) . () = 1 - () ( ) = () + () - ( ) ( - ) = () - ( ) ( - ) ( - ) = () + () - 2( )

14


,

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() = = 0,8

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( ) () ...

0 ( ) 1 0 () + () - ( ) 1

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: 0 < () < 1 .

1 1 : + 4 .

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0 < () < 1

0 < 1- (') < 1 - 1 < - (') < 0 .

() + (')4 () + (') 4() (')

() (')

() + 1- () 4() [1- (

(') > 0

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3 1 1 > 0 : () = , () = , ( ) = .

3

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30 < 1

1

0 < 13

0 0

0 < 3

0 < 1 3

0 < 1 11

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20


2A (), () 6x -7x + 2 = 0

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(), () .

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6 2 3

2

2 1

1,2

2

= (- 7) - 4 6 2 = 49 - 48 = 1 > 0

7 + 1 2x = = 6x - 7x + 2 = 0 : - (- 7) 1 7 1 12 3x = =

2 6 12 7 - 1 1x = =

12 2

1P(A B)A B A P(A B) P(A) 2

2A B B P(A B) P(B)P(A B)

3

2P(B) =

3

1P(A) =

2

E

1 2 P(A B) = () + () - P(A B) P(A B) = + - P(A B)

2 37 7

P(A B) = - P(A B) 0 - P(A B) 16 6

1P(A B)A A B P(A) P(A B) 2

2B A B P(B) P(A B)

0 P(AUB) 1

1P(A B)

2

1( )

6

1 1 ( )

6 2

P(A B)3

2P(A B)

3

2 , (), () :

x - 2P(A) x + P(A B) = 0 .

5A P( ) = , : (), P(A B), P(A' B), P(' ')

9

' :

() + () = 2()

() () = P(A B)

2() = ()

[()] = P(A B)

21


2 2

2

2

5 5 5 P( ) = () + () - P(A B) = 2() -[P(A)] = 9[P(A)] - 18() + 5 = 0

9 9 9

5 () = ( () < 1)

3

1 [P(A)] =

3

1 1 () - P(A B) = -

3 9

1() =

3

1 P(A B) = =

9

2 P(A' B) = =

9

P('1

[(A B)]' = 1- P(A B) = 1-9

8

') = =9

2

- 4 3 : ( ) : x +( - 2)x + = 0.

( ) :

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- 4 - 3 3

- 4 - 3 3

2 2 2

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= {( ), ( ),..., ( )} .

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2() = 8

= = - 3 - 4 + 4

2

, :

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3

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()

= - 2

= 0

1() = =

8

2

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3() 2

() = 2 =() 8

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> 0 - 1 0

1() = =

4

2

:

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3()

() = 5 ()

< 0 = - 4,- 3,1,2,3

5() = =

8

22


|() + 2|-|() - 3|= 8, .

1 :|| .

8

() - 2 () + 2 0 |() + 2|= () + 2, 0 () 1

() 3 () - 3 0 |() - 3|= 3 - ()

:

() + 2 - (3 - ()) = 8 () + 2 - 3 + () = 8 2() = 8 + 1

, 0 ()

8 + 1() = (1)

2(1 ) 8 + 1 1 1

1 0 1 0 8 + 1 2 - 1 8 1 - 2 8 8

1

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25 .

() ;

x o .

2x ,

53x

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(), " ", :

3x9 () 9 9 3x5 () = = = = 9 3x = 4525 () 25 25 25 5

15 10.

x = 15

1

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()

8() =

25

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1 ,

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1

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x o y o .

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1 1, () = , () = , () = x, N() = y.

2 6

() 11 =() =() 22

1 ()() =

6 ()

x 1=

2 = 24 + x + y x = 24 + y x = 24 + y24 + x + y 2

y1 6 = 24 + x + y x - 5y = - 24 24 + y - 5y = - 241= =

6 24 + x + y 6

x = 24 + y

4y = 48

A : 24 + 36 + 12 .

x

y

x = 36

y = 12

= 72

CD , ,

1 , , .

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50 , '

30 .

1 .

5 .

1 ,

2 70 ' ;

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25


() = 50, () = ( ) = 50 - 30 = 20.

1 ( ) 1 20 1 ( ) = = =

5 () 5 5

1 1 1 1( - ) = () - ( ) = () - =

2 2 5 2() 7 ()

, () = =() 10 100

= 100

7 () =

10

() = 70

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1

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1 2 3 4

1 2 3 4

1 3 1 4

= { , , , } .

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2 4 81 4

= { , }, = { , }, () = , () = (4 5 1

2

1) = ,

6 ( ) .

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()

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, :

1 1 5(') , () ( ) .

3 3 6 .

1 : ( ) .

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1 1

, 1 2

(') = (') = 2 3

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(' ') .

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1 2

2 2

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0,2 :

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1 6 ,B = .

1 4 ()- () = 2() + () = , > 0, :

(), () .

1 7

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1/3 1/4, .

1 8

2

|| 7 :

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: |() + 3|-|() - 3|= + 3, .

1 :|| .

2

2 0

2

, (), () :

P(A B)x -(3P(A)- ())x + = 0 .

58

A P(' ') = , :9

(), P(A B), P(A' B), P( ) .

2 1

, , .

:

P(A B) () + ()- 1

P(A B ) () + () + () - 2

2 2

= {1, 2, 3, 4} :

= {1 , 2} = {2, 3}.

5 1 2 1

()= , ()= , ( )= (-)=12 3 3 12

1, 2, 3 4.

2 3

2

x , 3x + 12

3x .

' test.

: " ".

:

() x.

x () .

().

31


2 4

2

' -

5 1 3 : f(x) = 2x - x + , x - , . A :

8 4 4

3|1- 2()|-|3() - 1|= 2, ,

() .

2 5

2 2

- .

:

1 0 ()(')

41

[()] + [(')] 1 2

( ) - ( ) = 1, :

( ) = 0 () + () = 1

2 6

40% -

, 50%

, 15% -

-

.

12 -

;

2 7

2

,B

() 5x + 9x - 2 = 0

3(A ) = 2() = ()

2 (), () ( - ) .

2 8

2

= {1,2,...,10} -

. .

f(x) = x + 4x + , f(x) = 0

-

.

32


2 9

2

= {1,2,3,4,5}

x + y = 3 , .

( + 2)x + 3y = 1

: "

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3 0

2006 :

30% .

40% .

20% .

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: " ' " .

: " " .

: " ' " .

3 1

, (), () :

6x3 + 5x2 12x + 4 = 0 () < () :

, 1 1

( )6 2

2( ) 1

3

3 2

(x 10) (x 11) (x 20) = 0.

, -

y 2 8y + = 0 .

33


:

( ) = () + ()

.

, ,

( ) = () + () + ()

(1)

() = , = +

() =

: ( ) = + = () + () (1)

( ) () + () (= =

() ()

( )

) ()

+ =() ()

= () + ()

'

() 1

' = (, ' ),

'

( ') = () + (')

() = () + (')

1 = () + (')

(') = 1- ()

() :

() = 1 - ()

34


:

( ) = () + () - ( )

:

.

N(A B) = N(A) +

N(B) - N(A B) (1)

N(A) + N(B)

(1)

A B

N(() :

A B) N(A) N(B) N(A B)= + -

N() N() N() N()

( ) N(A) N(B) N(A B)= + - =

() N() N() N()

( ) () + () - ( )

() ()

, () > 0

() ()

() ()

() ()

= ( ) ( ),

() = ( ) ( )

( - ) = ()- ( )

() ()

:

( - ) = () - ( )

35


|

36


0 1 .

:

= { 0, 1, 2, 3, . . . }

:

= { . . . , - 2, - 1, 0, 1, 2, . . . }

:

= { | =

, * }

:

{ x | x }

:

.

:

{ 1, 3, 5, . . . } { 2 + 1 | } :

{ 0, 2, 4, . . . } { 2 | }

0 2 . -

+ = + =

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

( + ) = +

+ 0 = 1 =

()

(/) + ( - ) = 0

1

= 1, 0

= + = +

= =

0 = 0

0 0 0

( - 1 ) = -

( - ) ( - ) =

= , 0

1 1 1

= , 0

: = + ( - )

: : = 1

, 0

=

= = , 0

= 0 = 0 = 0

( - ) = -

- ( + ) = - -

= , 0

= , 0

37


0 3 .

.

, : =

,,, .

, .

, .

=

.

= =

= =

= =

( 0

)

( 0)

( 0)

=

( 0)

[

+ + =

+ = = = ( + ) 0]

( 0,

+

+ + = =

- ,

- )

0 4 .

* +

-

:

= . . . , > 1

* *

: = 1 -

1 =

* +

, * +

:

=

* +

x x x > 0 .

38


+ -

= : = ( ) =

( ) = ( ) =

-

2 2 2 + 1 2 + 1

( ) =

( - ) = ( - ) = -

0 5 .

( ) = 2 + ( : + = ( + ) - 2 )

- = ( + )( )

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

= ( + )( 2 + ) = ( ) 3 ( )

( ) = 3 + 3

( + ) = + 4 + 6 + 4 +

( ) = - 5 + 10 - 10 + 5 -

= ( )( 1 + 2 + . . . + 2 + 1 )

- = ( + )( 1 - 2 + . . . - 2 + 1 ), ( )

+ = ( + )( 1 - 2 + . . . - 2 + 1 ), ( )

+ + - = 1

2 [ ( ) + ( ) + ( ) ]

+ + - = 1

2 [ ( ) + ( ) + ( ) ]

+ + - 3 = ( + + )( + + - ) =

= 1

2 ( + + )[ ( ) + ( ) + ( ) ]

+ + = 3 , + + = 0 = = (Euler)

0 6 .

, -

- ( - > 0 ) .

: >

.

- +

39


,

- ( - < 0 ) .

: <

.

- +

> > , : > .

> , : > .

> 0, : >

>

>

< 0, : > < .

> > , : + > + .

> > , : > (, , , ).

, *

, : > >

xx -

. -

.

( , )

x [ , ]

< x <

( , )

< x ,

( , ]

x < ,

[ , )

x ,

[ , + )

< x ,

( , + )

x ,

(- , ]

x < ,

(- , )

x

(- , + )

0 7 .

:

, 0

|| = - , < 0

40


| | 0, .

- | | | |

2 = | |, | - | = | |, | | = || , | | 2 = 2

| | = | | ||

||

| | = ||

0

| | | - | | | | | | | + | |

| | < | | <

| | + | | = 0 = 0 = 0

> 0 :

1. | x | < - < x <

2. | x | > x < - x >

| x | =

x = > 0

x = 0 = 0

< 0

( , 0 ) ( 0, ) xx d( , ) = | - | .

0

0 0 0 0

0 0 0

0

0

(

0, :

|x - x | < x x - , x + ) x - < x < x +

x x - x |

- , + ) x

|

d(x, x )

x' x

x - x x x -

.

0 x = 0, : |x| < x (- , ) - < x <

0

0 0 0 0 0

0

x > 0, :

|x - x | > x (- , x - ) (x + ,+ ) x < x - x > x +

x |x - x | > (x)

x'x ' (

0

0

d(x, x )

x'

x x - x x -

x ) .

0

x

x = 0, : |x| > x < - x >

0 8 .

, -

41


x x = .

x

.

: x = x = , x 0 * +

.

=

=

=

=

( ) =

=

=

=

42


:

:

( ).

:

( ) .

:

.

.

.

, , : + + 3.

, , :

= + 1 = + 2

:

+ + = + ( + 1 ) + ( + 2 ) = 3 + 3 = 3( + 1 ) = 3 ( = + 1)

+ + 3.

2

: .

:

:

( ).

:

( ) .

:

.

.

( ) .

.

,

.. , , .

43


, = 2 + 1 , .

: 2 = (2 + 1) 2 = 2( 2

2 + 2

) + 1 = 2 + 1, .

2 , , 2 .

.

2 2x - 1 x - 2x x = = ;

x(x - 2) (x - 1)(x + 1)

, .

2 2

x (x - 2) 0 x 0 x 2

(x - 1)(x + 2) 0 x 1 x - 2

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

x(x - 2) (x - 1)(x + 1) x(x - 2) (x - 1)(x + 1)

- { - 2, 0, 1, 2 }x

1 , .

: -

:

- .

:

.

:

, :

( ) -

() .

: + = 0 .

: = 1 .

: ( )

:

.

:

.

:

:

( )

( ) -

.

.

44


2 5 4 3 2 = 6 - 2 + 12 .

2

2 5 4 3 2 2 3 2 2

... ' : 2

' :

= 6 - 2 + 12 2 (3 - + 6 )

= .

= x + x - y - y .

= x + x - y -y =

= (x + x) - (y + y) =

= x( + ) - y( + ) =

=

(x - y)( + )

: ( )

:

.

:

.

:

( ) .

-

, .

.

.

: ( )

:

.

:

.

:

, = 2 - 2 .

: 2 - 2 = ( - )( + ) .

45


8 4 2 10 = 9 x - 4 y .

8 4 2 10

4 2 2 5 2 4 2 5

= 9 x - 4 y =

= (3 x ) - (2y ) = ( = 3 x = 2y )

= 4 2 5 4 2 5

(3 x - 2y )(3 x - 2y )

9 3 6 12 6 3

:

= 8 x - y , = 27 +

9 3 6 12

3 3 2 4 3

3 2 4 3 2 3 2 4 2 4 2

3 2 4

3 2 4( = 2 x = y

= 8 x - y =

= (2 x) - ( y ) =

= (2 x - y )[(2 x) + (2 x)( y ) + ( y ) ] =

= (2 x - y (4

)

)

6 2 3 2 4 4 8

6 3

2 3 3

2 2 2 2 2

2 4 2 2

2(

x + 2 x y + y ) =

= 27 + =

= (3 ) + =

= (3 + )[(3 ) - 3 + ] =

= (3 + )(9 - 3 + )

= 3 = )

: ( - )

:

.

:

.

:

, = 3 3, = 3 + 3 .

:

3 3 = ( )( 2 - + 2 )

3 3 = ( )( 2 - + 2 )

46


2 = 3 - 5 - 2 .

2

2

= 3 - 2 =

= 3 - 2

( - 2) ( -= 3 +

= (

-

3

5

- 6 +

+ 1)

2)

( - 2)

8 4 4 4 = x + 4 y .

8 4 4 4

4 2 2 2 2 2 4 2 2 2

= x + 4 y =

= ( x ) + (2 y ) = ( 2 ( x ) (2 y ) )

: ( )

:

.

:

.

:

(

) .

( -

) .

.

: ( )

:

.

:

.

:

.

.

.

( ) .

47


2 2 2

4 2 2 2 2 2

4 2 2 4 2 2 2 2 2 2 2 2 4 2

+ 2 +

4 2 2 2 4 2

= ( +

2

)

2+ 2 ( x ) (2 y ) - 2 ( x ) (2 = ( x ) + (2 y ) =

= ( x ) + 2 ( x ) (2 y ) + (2 y ) - (2 xy) = ( = 3 x , = 2y)

y )

4 2 2 2 2

4 2 2 4 2 2

= ( x + 2y) - (2 xy) = ( )

= ( x + 2y + 2 xy)( x + 2y - 2 xy)

2 + 1 , , , : = + + 3 .

4 - 1 3, .

, , , : = + 1 = + 2 .

= + + =

= + + 1+ + 2 =

= 3 + 3 =

= 3 ( + 1)

= + + 3 .

- 1 - 2 - 2 - 1

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

2 2 - 1 2 - 1 2

2 2 - 1

: - = + + ... + + (1)

4 - 1 = 4 - 1 =

= (4 - 1)(4 + 4 1+ ... + 4 1 + 1 ) =

= 3(4 + 4 1+ ... + 4

2 - 1 2

( )

2 + 1

1 + 1 ) = 3

4 - 1 3.

:

:

.

:

.

:

= ( = -

) .

48


x,y,z 1,2,3 .

3x + 2y - 2z = 20 (1), x,y,z.

(1)

( )

2y 3x + 2y - 2zx 3x 3x - 2z 20= = = = = = = = = = = 20

1 1 3 3 4 - 6 3 + 4 - 6 1

x= 20

1y

= 20 E : = 3 20 + 2 40 - 2 60 = 60 + 82z

=

z - 2z

3 - 2 3

y 2y

2 2

3

2

20

x = 20

y = 40 3x + 2y - 2z

z = 60

0 - 120 = 20.

:

:

x, y, z .

:

x, y, z ,

, , .

:

: yx z

= =

.

,

.

:y x + y + zx z

= = = = + +

.

: yx z

= , = , =

.

: E u l e r

:

E 3 + 3 + 3 = 0 .

:

.

:

+ + = 0 .

Euler : + + = 0 : 3 + 3 + 3 = 3 .

: = 0 = 0 = 0

49


3 3 3 : (x + 2) +(4x - 12) +(10 - 5x) = 0 .

3

3 3 3

3

3

E

(x + 2) + (4x - 12) + (10 - 5x) = x + 2 + 4x - 12 + 10 - 5x = 0

(x + 2) + (4x - 12) + (10 - 5x) = 0

x + 2 = 0

3(x + 2)(4x - 12)(10 -

(

5x) = 0 4x - 12 =

: x + y + z = 0 x + y + z = 3xy

0

10 - 5x = 0

z)

x = - 2

x = 3

x = 2

2

2 2 ( - ) : = - + = .2

2

2 2

2 2 2

2

( -)= - + - =

2

2 - 2 + 2 - ( -) = =

2

2 -2 =

-

2 2+ 2 - +2 2

2 2

-=

2

+ = 0

2

- 0

> = 0

= = = 0

:

:

.

:

, .

:

- .

- > 0 >

- < 0 <

50


2 2 2 2 2 2

:

+ + + + + 3 2( + + ) + 2, > 0 > 0

2 2

2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2

2 2 2 2 2 2

2

+ + + + + 2 4 + 4 2 + 2 + 4

2 2 2 4

2 + 2 - 4 0 2( + - 2) 0 2( + ) 0 , .

+ + + 3 2( + + ) + + + 2 + 2 + 2

( -

3

2 + + (1)

2 2 2 2 2

> 0 2 2 2 2 2

> 0

- 2 + ) + ( - 2 + ) 0 ( - 1) + ( - 1) + ( - 1) 0 , .

+ 2 + 2 + 2 + - 2 0 ( -) 0,

1 1

.

:

:

.

:

.

:

.

.

.

:

:

.

:

.

:

.

, .

.

( ) .

.

51


: xy 4 x + y = 4 .

x = 4 - y 2 2 2 2

E

x + y = 4 x = 4 - y (1)

xy 4 .

xy > 4 (4 - y)y > 4 4y - y > 4 4y - y - 4 > 0 - (y - 4y + 4) > 0 - (y - 2) > 0

xy 4 .

.

2 2 2 , , + = . : + < .

2 > 0 + = (+) > 0 2 2 2 2

2 > 0 > 0

2 2

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

+ = < < <

+ 0)

1+ + 1+ (1)

< ( > 0)

1+ + 1+

+ = +

1+ + 1+ + 1+ +

+ 1 1 3 < 3 + < + 4 < + 4 < ( + ) + 0.

= |2x + 6|+ 3x - 2 .

(+) (+)

> - > 0 < - < 0

> 0 > 0 > 0 > 0 + 2 > 0 2 - > 0

> 0 2 > 0 > - > 0

, :

= 3( -

) + 5[- ( - )]- ( + 2) + (2 -) = 3 - 3 - 5 + 5 - - 2 + 2 - =

2x + 6 = 0 x = - 3

x - 3 2x + 6 0 |2x + 6|= - 2x - 6 .

= - 2x - 6 + 3x - 2 =

x > - 3 2x + 6 > 0 |

9 - 11

x - 8

2x + 6|= 2x + 6 .

= 2x + 6 + 3x - 2 = 5x + 4

:

:

.

:

.

:

, -

.

,

.

, .

54


, : | - 5|= 5 - |2 - 3|= 2 - 3.

< 5 - 5 < 0 3

| - 5|= 5 - < 53 , 2 - 3 0 2

|2 - 3|= 2 - 3 2

: 2, 3 4.

:

:

.

:

.

:

A -

.

, -

.

:

:

.

:

.

:

:

|x| < - < x <

.

,

.

|x| < - < x < .

:

-

.

.

, -

.

55


:

+ || < 1 || < 1, : | | < 1

1 +

| - 1| < 2 | - 2| < 3, : | - | < 6

2 2 2 2

2 2 2 2

2 2 2 2

2 2

||< 1 || < 1 < 1 - 1 < 0 (1)

||< 1 || < 1 < 1 1- > 0

+ | + | < 1 < 1 | + | - 5

- 1 < < 3 - 6 < - < - 6 < - < | -| < 6.

- 5 < - - 1 |x + 1| > |x - 2|

2|3x - 2|+ 4 6 -|3x - 2| |2 - 3x|- > - 2

4 2 3

8

:

|x + 1|- |x - 2| 1 |x + 1|- |2x - 4| 1

2|x - 1|+ 3 |x + 2| < 0 |2 -|x + 1|| < 3

1 2

2

2

1

2

2

2

2

3 2 2

:

x - 3x + 2 > 0

( ) : x - 5x - 50 < 0

x - 2x - 15 > 0

x - 2 > 0

( ) : 6x + 5x + 1 > 0

- x + 5x - 6 0

x - 2> 0

( ) : 2x + 1

(x - 4)(x + 2x + 4 > 0

9

2 2 2 2 :

(x - 2)(x + 2x - 8)(x + x + 1) > 0 (2x - 1)(- x + 2x - 1)(x + 1) > 0

1 0

2 2 2 2 2 2 :

(x - 3)(x + 3x - 10)(- x + x - 10)(x - 4) 0 (x - 2)(x -7x + 6)(- x - 2x + 8)(x - 1) > 0

1 1

2

2 2

:

(x - 1)(x - x + 1) 10 x + 2 x - 30 + 2

f, ' .

x 0

x -1 0

y -2 -1

x 2

x 2 3

y -1 0

x 0 f

(0,-1) (0, - 1) .

0 < x 2 . f .

(0, - 1) (2, - 1), xx .

x 2 f (2,-1)

(3, 0) .

:

: .

: .

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

- (1).

.

. , .

y

-1 0 2 3

-1

-2

148


f.

f.

x 0 f O(0, 0)

A(-1, 3). O f,

f(x) = x : 3 = - = - 3

= - 3, f(x) = - 3x

0 < x 2 . f . O(0, 0)

B(2,2). O f,

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

= 1, f(x) = x

x 2 f B(2, 2) -

xx.

O f(x) = 2

- 3x, x 0

f(x) = x, 0 < x 2

2, x > 2

y

5

4

3

2

1

-2 -1 0 1 2 3 4 5 6

:

: .

: .

: 1 : y = 1 x + 1 2 : y = 2 x + 2 :

1 2 1 2

|| =

1 2 1 2

= - 1

3 : y = 3 x + 3 3 , 3 = 0 ( y = x ) . 3 xx, 3 = 0 .

149


1 2

3 4

, :

: y = |2 - 3|x + 4 : y = |1- |x - 2, .

: y = ||x + 5 : y = ( |4|- 4)x - 1, .

2

5

5

5

: y = ( - 2)x +( - 5 + 6).

, :

' .

x'x.

1 2

2 2

3 4

5

2 - 3 = 1- 3 = 4 / / :|2 - 3|=|1- |

2 - 3 = - 1 = 2

:|| (|4|- 4) = - 1 4|| - 4||+1 = 0 (2||- 1) = 0 2||- 1 = 0

1 || =

2

'

4 =

3

= 2

1 =

2

2 2

5

, :

- 5 + 6 = 0 - 2 - 3 + 6 = 0 ( - 2) - 3( - 2) = 0

- 2 = 0 ( - 2)( - 3) = 0

- 3 = 0

(

y = x, = 0)

x'x, : - 2 = 0

= 2

= 3

= 2

:

: , ,

.

: .

: ( x 1, y 1 ) , B( x 2, y 2 ) : () = 2 2

2 1 2 1(x - x ) + (y - y ) .

, , (), (), ()

.

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

, , (), (), () : () = () () () = () () () = () ()

, , ( ) , : () + () = ()

150


2

,

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

(3,0), (1,6) (2,3).

: , .

2 2 2 2 2 2

2 2

2 2

() = 1 (0 - 0) + [ - (3 - 1)] = 1 ( - 3 + 1) = 1 | - 3 + 1|= 1

- 3 + 1 = 1 - 3 = 0 ( - 3) = 0 .

( - 1)( - 2) = 0 - 3 + 1 = -1 - 3 + 2 = 0

(

= 0 = 1 = 2 = 3

2 2 2 2

2 2 2 2

2 2 2 2

) = (1- 3) + (6 - 0) = 2 + 6 = 4 + 36 = 2 10

() + () = () = 2 10 () = (2 - 3) + (3 - 0) = (- 1) + 3 = 1+ 9 = 10

() = () = 10

() = (2 - 1) + (3 - 6) = 1 + (- 3) = 1+ 9 = 10

,

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

f ' x'x ;

2

f(x) < 0 x - 3x + 2 < 0 (1)

: C f ( ) x x

: x

() xx .

: .

: : f(x) > 0 . : f(x) < 0 .

151


2

f

1,2

= 3 - 4 2 = 9 - 8 = 1

(1) : ( A = )3 1 3 1x

2 2

(1) :

x = 1

x = 2

1 < x < 2

= =

2 5 3o : f(x) = x + x g(x) = - + .4 2

f '

g x , : 2 < < 3.

f g f g

f g

2 2

2

: = = =

C ' C , x , :

5 3 5 3f(x) > g(x) x + x > - + x + x + - > 0

4 2 4 2 x , : > 0 < 0.

= 1 > 0, :

< 0

25 3- 4 1 ( - ) < 0 - 5 + 6 < 0 ( - 2)( - 3) < 04 2

2 < < 3

:

:

: .

: x, x x < x :

x < x f(x), f(x) . A

f(x) > f(x) f . f(x) < f(x) f . f(x) = f(x) f

1 2 1 2

f(x ) - f(x )

x - x (1) .

A

(1) > 0 f . (1) < 0 f . (1) = 0 f

152


f : f(x) = 5 - x + 1 .

f

1 2 1 2

1 2 1 2 1 2 1 2

f : 5 - x 0 x 5, A = (- , 5].

A x ,x 5 x < x :

- x > - x 5 - x > 5 - x 5 - x > 5 - x 5 - x + 1 > 5 - x + 1

f . (- ,5

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

1 2 1 2 1 2 1 2

1 2 1 2 1 2

1 2 1 2 1

].

A

5 - x + 1 > 5 - x + 1 5 - x > 5 - x 5 - x > 5 - x - x > - x

f . (- , 5].

A

5 - x + 1- ( 5 - x + 1) 5 - x - 5 - xf(x ) - f(x ) = = =

x - x x - x x - x

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

1 2 1 2

2 1 2 1 2

1 2 1 2 1 2

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

1

1 2

( 5 - x - 5 - x )( 5 - x + 5 - x )= =

(x - x )( 5 - x + 5 - x )

5 - x - (5 - x ) 5 - x - 5 + x - (x - x ) = = = =

(x - x )( 5 - x + 5 - x ) (x - x )( 5 - x + 5 - x ) (x - x )( 5 - x + 5 - x )

-1 = , ( 5 - x + 5

5 - x + 5 - x< 0

2- x > 0) f . (- , 5].

2 : f x = x - 4x + 3 . ( )

To f .

: ( )

: .

: .

: f(x) f(x) f(x) f(x), x f . H f x A

, f(x), :f(x) f(x), x A . , f(x), : f(x) f(x), x A .

153


E,

f(x) = x 2 - 4x + 3 f(x) = (x 2- 4x + 4) -1 f(x) + 1 = (x - 2) 2 0 f(x) - 1

E,

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

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

f : f(x) = 2x - 3 A = [0,3].

,

y + 3y = 2x - 3 2x = y + 3 x =

2 y + 3

0 3 0 y + 3 6 - 3 y 32

- 3 + 3 y = - 3 x = = 0, :

23 + 3

y = 3 x = = 3, : 2

f(A) = [- 3,3]

x = 0 f(0) = - 3

x = 3 f(3) = 3

: ( . . )

: .

: .

: f(A) = [, ] . T , , ,

: ( )

: .

: .

: x , - .

x, .

154


, , : g(x) = |x - 2|+ 1 .

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

- x + 3, x > 2

O

(- ,2] g ., = 1 > 0.

(2,+ ) g ., = - 1

(

< 0

.

A

g

x = 2,

x = 2 , g(2) = 1

g,

.

)

- x + 2, x < - 1 f : f(x) = .

- x - 2, x > 1

fTo f : = (- ,1) (1,+ ), 0.

x < - 1 - x > 1 :

f(- x) = - (- x) - 2 = x - 2 = - (- x + 2) = - f(x), f x < - 1.

x > 1 - x < -1 :

f(-

f

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

x f .

:

: .

: .

: . x , - x :

f(- x) = f(x) ( ) f(-x) = - f(x) ( ) .

155


.

H f , .

H g , .

H h

,

.

: ( . )

: .

: .

: . :

yy, . , - .

, .

y = f(x)

y = g(x)

y = h(x)

: -

: f

: O f .

: C .

f(x) = (x) + c, C c . f(x) = (x) - c, C c . f(x) = (x - c), C c . f(x) = (x + c), C c .

156


f(x) =|x - 2|+ 1 .

f

x = x , f

f x = x - 2 + 1.

C

1 2 .

C

C 2

( )

| |

(

) ( )

1 .

2 x = 2x 1.

f

:

2

1 .

3 2 .

2 1 .

3 2 .

2 2 2

2 2 2

2 2

2

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

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

3 : 2x -1 2(x - 3) -1,

2 : 2(x - 3) -1

2 2 2

, f(x) = 2(x - 3) - 32(x - 3) -1- 2 = 2(x - 3) - 3

:

: f .

: O C .

: C c , f(x) = (x) + c . C c , f(x) = (x) - c . C c , f(x) = (x - c) . C c , f(x) = (x + c) .

-2 -1 0 1 2 x

y

1

C

C f

157


2 2 2

2 2 2

2 2

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

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

3 : 2x -1 2(x + 3) -1,

2 : 2(x +

2 2 2 2

, f(x) = 2(x + 3) - 33) -1 2(x + 3) -1- 2 = 2(x + 3) - 3

158


1

2

: f(x) = x - 1.

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

y : f(y) = 1

x +|| x < 1 : g(x) =

( - 1)x, x 1

N

, : g(- 1 ) = 3 g(2) = 4.

2

2 2

N :

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

x + 1 1- x

g(x) = |x - 1|- 2

x 2xh(x) = +

|x - 1|-|3x + 1|x - 1

5

f, :

- x, x 1

f(x) = x - 2, 1 < x 3

1, x > 3

3

2E ( - 2, - 4).

N , :

x'x.

x'x y'y.

E f,

: f(x) =|x - 2|- 1.

N . f :

x'x.

y'y.

4

2

2

2

f, : f(x) = x - 2.

( + 1, + 1) . f.

g, : g(x) = x - 2x + 2 + .

2

, . g

(,- - ).

159


6

,

(,x + 6) (,2x + 3) 1.

(3,2), (4,- 2) (- 1,1).

: , .

7

1 2

3

, :

: y = (3 + 1)x : y = ( - 3)x + 5 :

2 + 3 : y = x + 1, 2.

- 2

3

3 4

3

, :

' .

: y = - 5x + 2.

5

1 : y = - x + 3.

9

8

. f.

:

f.

f.

f(2), f(4), f(6).

x : f(x) = 1.

9 N f,

.

y

5

4

3

2

1

-2 -1 0 1 2 3 4 5 6

y A 3 B

-2 -1 0 1 2 x

1 0

2 2

2

*R

:

2x - 1 f(x) = x - 9 g(x) = x -|x|+ 1 h(x) =

2x + 1

p , p(x) = ( - 4)x +7.

, p .

160


1 1

2

:

|x|- 3f(x) = 1- 4x g(x) =

x

- x + 1 , x 01h(x) = p(x) =

x + 1 , x 0(x - 1)(x - 3)

1 2

3 2 :

f(x) = 2x + 1 g(x) = - x - 2x + 1 h(x) = x - 4x + 3, x 2

1 4 3 : f(x) = 2x - 3 .

1 5

2

f f(x) = x + , . > 0 . -

< 0.

:

g(x) = (4 - )x - 2 h(x) = 2|x|+|x - 1|

1 6

4 2 :

f(x) = x + x + 2 g(x) = |x - 1|+ 2

1 3

2

:

x + 1, x 0 f(x) = ( - 9)x + 1 g(x) =|x - 2|+|x + 1| h(x) =

- 2x + 3, x > 0

N , f

f(x) = |x + 1|- 3|x - 2| . [2,+ ).

1 7

2

. :

f(x) = - 2x

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

- x + 2, x 1

|x + 1|-|x - 1|h(x) =

x

161


1 8

2

2

x,

f ' g, :

f(x) = x - 2x g(x) = x + 4

f(x) = x - 4x + 1 g(x) = x

- 8

f(x) = 4 - x g(x) = x + 2

162


(x1, y1) (x2, y2) :

() = 2 2 2 1 2 1

(x - x ) +(y - y )

.

: () 2 = () 2 + () 2 (1)

: () 2 = |x 2 x 1| 2 (2)

() 2 = |y 2 y 1| 2 (3)

O:

(1)(2)

(3) () 2 =|x 2 - x 1|2 +|y 2 - y 1|2

2 2 2

2 1 2 1(AB) = (x - x ) + (y - y )

(. 1)

1 2 1 2

|| =

1 2

|| 1 = 2 1 = 2 1 = 2

(. 2)

1 2 1 2

= -1

1 2

y = 1 x y = 2 x .

. :

() 2 + () 2 = () 2

1 2 + 1 2 + 2 2 + 1 2 = ( 1 2) 2 + (1 - 1) 2

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

y = 1 x + 1 y = 2 x + 2 y = 1 x

y = 2 x, y = 1 x + 1 y = 2 x + 2

1 2 = - 1 .

:

1 2 1 2 1 2 1 2

|| = = - 1

y

y2

y1

0 x1 x2 x

.1 1 2

1 2

.2

1

A

B

|y2 - y1|

|x2 - x1|

1 (1, 1)

2 (1, 2)

163


164


1 . f ( x ) =

x

:

= * = (- , 0) (0 , + ) ( x 0)

:

f() = * ( x 0 f(x) =

x 0)

:

x * - x * .

A

f(- x) = = - = - f(x)- x x

: f * .

:

y =x

x y = 0

x 0 y 0, .

:

f(x) =

x .

A > 0 f (- , 0) f (0, + )

A < 0 f (- , 0) f (0, + )

: f * .

. .

:

f *

:

y =x

-

.

.

> 0 < 0

165


:

x , ,

x + x + .

x , ,

x - x + .

x 0, -

, x 0 x 0 + .

x 0, -

, x 0 x 0 - .

2. : f ( x ) = 1

x g ( x ) = -

1

x

:

x + 1

x 0 x + -

1

x 0

x - 1

x 0 x - -

1

x 0

xx ,

x ( )

xx .

x 0 + 1

x + x 0 + -

1

x -

x 0 - 1

x - x 0 - -

1

x +

yy ,

x 0 yy -

.

f(x) = 1

x g(x) = -

1

x

166


3 . f ( x ) = x 2

:

= ( f(x) x )

:

f(A) = [0,+ ), > 0

f(A) = (- ,0], < 0

:

x * - x * .

A 2 2f(- x) = (- x) = x = f(x)

: f * .

:

x = 0 y = 0.

(0,0)

.

:

f(x) = x 2 .

A > 0 f (- , 0] f [0, + )

A < 0 f (- , 0] f [0, + )

:

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

< 0 x = 0, f(x) = 0 .

:

2y = x

.

y y

.

> 0 xx, < 0 -

xx.

> 0 < 0

167


:

, :

2 2

2 2

x + x + ( > 0) x ( < 0)

x x + ( > 0) x ( < 0)

-

- -

xx.

|| .

|| .

4. f ( x ) = x 2 + x +

:

= ( f(x) x )

:

f(A) = [ - ,+ ), > 0

4

f(A) = ( - ,- ] , < 04

:

x =

- 2

:

yy : H o (0, )

xx : H - - - +

( ,0), ( ,0)2 2

.

:

f(x) = x 2 + x + .

A > 0 f (- ,

-2

] f [

-2

, + )

A < 0 f (- ,

-2

] f [

-2

, + )

:

> 0 x = - , f(- ) = - 2 2 4

< 0 x = - , f(- ) = - 2 2 4

:

y = x 2 + x +

. (

-2

,

- 4

).

168


> 0 < 0

> 0

= 0

< 0

169


2 2 2N :

f(x) = 2x - 1 f(x) = 2(x - 3) f(x) = 2x - 12x + 17

g(x) = 2x2

g(x) = 2x2 o f(x) = g(x) - 1,

f

g f(x) = 2x2 - 1

1 -

. -1

g(x) = 2x2 f(x) = 2(x-3)2

g(x) = 2x2 o f(x)= g(x-3),

f

g 0 3

3 .

f(x) = 2(x2 - 6x +17

2) = 2(x2 - 2 3x + 32 -32 +

17

2) =

= 2[(x - 3)2 -1

2] = 2(x - 3)2 1

g(x) = 2x2 o h(x) = g(x - 3) f(x) = h(x) - 1 0 3

f

g - f(x) = 2(x - 3)2 - 1

3 1 .

: -

: f .

: O f .

: f(x) = (x) + c, C c . f(x) = (x) - c, C c . f(x) = (x - c), C c . f(x) = (x + c), C c .

g(x) = 2x2 h(x) = 2(x-3)2

-1

170


2 x = 2x - 1.

f

:

2

1 .

3 2 .

2 1 .

3 2 .

2 2 2

2 2 2

2 2

2

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

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

3 : 2x - 1 2(x - 3) - 1,

2 : 2(x - 3) - 1

2

2 2

2 2 2

2 2 2

, f(x) = 2(x - 3) - 32(x - 3) - 1- 2 = 2(x - 3) - 3

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

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

3

2 2 2

2 2 2

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

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

:

: f .

: O f .

: C c , f(x) = (x) + c . C c , f(x) = (x) - c . C c , f(x) = (x - c) . C c , f(x) = (x + c) .

171


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

' :

x'x.

x'x.

x'x

(2,0).

y'y (0,2).

x = 2.

1.

x = 3.

2 2 2

= [- ( + 1)] - 4 1 = + 2 + 1- 4 = - 2 + 1 =

- ( + 1) = - = =

2

2

2

( - 1)

+ 1 ( - 1)- - - =

2 2 4 4

: (f ( x ) = x 2 + x + )

: .

: O .

: xx, : = 0 . xx, : < 0 > 0 . xx, : < 0 < 0 . (, ), : , f . ( x f(x) )

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

yy (0, ), : = .

() x = ,

-2

= .

() , -4

= .

x = ,

-2

= .

(, ),

-2

=

-4

= .

172


2

2

x'x, :

= 0 ( - 1) = 0 - 1 = 0

x'x, :

< 0 ( - 1) < 0, .

, ,

= 1

2

x'x.

x'x (2,0), 2 .

: 2 - ( + 1) 2 + = 0 4 - 2 - 2 + = 0

y'y (0,2), f(x) = 2 x = 0.

= 2

2 2

x = 2, - = 2.

2 + 1

: = 2 + 1 = 42

1, - = 1.

4

( - 1) : - = 1 ( - 1) - 4 = 0 ( - 1+ 4)( - 1- 4) = 0

4

( + 3)( -

= 2

= 3

5) = 0

x = 3, - = 3.

2 + 1

: = 3 + 1 = 62

= - 3

= 5

= 5

2 f(x) = x - 2( + 2)x + 5 + 2 .

f 2

x = 4 .

2

2 ,

2 - 2( + 2)2 + 5 + 2 = 0 4 - 4 - 8 + 5 + 2 = 0 - 6 = - 4 (1)

f x = 4,

- 2( + 2) x = 4 - = 4 - = 4 + 2 = 4 (2)

2 2

(

1) (2), :

- 6 = - 4 = = 6 - 4 = 6 1- 4

+ 2 = 4 +

6 -

2 = 4 8 = 8

4

6 - 14 =

= 2

= 1

173


2 : f(x) = - x + 4x - 3

2

1,2

= - 1 < 0

4 - = - = 2

2 2 (- 1)

= 4 - 4 (- 1)(- 3) = 16 - 12 = 4

- - 4 = = 1

4 4 (- 1)

3- 4 2 x = = 2 1 =

1- 2

: f ( x ) = x 2 + x +

: .

: O .

:

: , , , f(x) = 0,

-2

-4

.

.

([- ,+ ), > 04

(- ,- ], < 0

4

) .

: x =

-2

: yy : o (0, )

xx : - - - +

( ,0), ( ,0)2 2

.

(- ,

-2

] [

-2

, + ) .

( )

> 0 x = - , f(- ) = -2 2 4

< 0 x = - , f(- ) = -2 2 4

..

174


f : A =

M o :

= -1 < 0, :

(- ,2] f ..

[2,+ ) f ..

:

f

.

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

2

:

x'x : To x'x (1,0) (3,0).

y'y : To y'y o o (0,- 3).

:

(2,1)

:

x = 2.

:

y

1

0 1 2 3 x

-3

175


1

2

2

2

N :

f(x) = 3x - 5

f(x) = 3(x - 2)

f(x) = 3x - 12x +7

2

2 (x) = 3x - 2.

f

:

1

2 .

2 3 .

2 1 .

3 2 .

5 2 f(x) = x - 2( + 1)x + .

f 1 -

x = - 1 .

3

2

2

2

1

2

:

f(x) = x - 5x + 6

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

h(x) = x - 3x + + 1, x = 1

x = 2.

.

.

4

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

' (1,3), :

.

,

f.

176


6

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

' :

x'x.

x'x .

f(3).

0.

4.

x = 2.

177


178

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