100015997-ΦΥΛΛΑΔΙΟ-ΑΛΓΕΒΡΑΣ-2012-2013
TRANSCRIPT
-
1
:
() ().
Q ,
Q, Q Q
, .
, Q .
:
Q , .
1. :
Q
:
Q
Q , o Q .
Q .
2. :
Q
:
Q = 2=2
Q , o Q
2 2
Q (-4)2=42 -4 4.
-
2
Q ,
, Q Q
Q
O > .
1. :
:
Q:
Q ,,,
.
2. :
:=
Q:2=2
Q = 2=2.
Q ,
.
.
Q
.
-
3
1. :
:
Q:
:
: -
:
:
:
, .
Q ,
.
.
1. :
:
Q:
:
: -
:
:
:
.
-
4
2. :
:
Q:
:
:
:
:
: .
.
-
5
1.
Q Q P .
) :
Q:
)
Q:
2. () ,
.
)2=25 =5
)=4 2=16
)2=36 =6
)2=100 (=10 =-10)
)2 9 3
3. () ,
.
)2=6 2=9
)2=16 3=12
) 2=6 +3=6
)3=6 3=8
4.
:
) 2 25, 5 ...... -5.
) 2=9, =3 ...... =-3.
) , 2+2=0, =0 ..... =0
) , 2+2 0, 0 ..... 0
-
6
Cantor: ,
,
.
.
: ,
:
,
,
, ,
.
1. ,
.
2. ,:
3. ,:
4. :
:
:
: _
:
A. :
,
, .
,
. ,
.
-
7
. : ={1,3,5,7,9,11,.......}
1 100: ={1,2,3,4,5,6,.....,100}
B. :
,
.,
, :
{/ }
. : ={ / }
1 10: ={ /110}
, : =
:
. { }.
,
. .
:
.
, .
, =.
,
.
-
8
:
={/ }
.
.
:
= =.
= ( ).
( ) = ( )= ( ).
.
, =.
={/ }
.
:
= =.
= ( ).
( )=( )= ( ).
.
, =.
= , , .
-
9
={/ } ( )
:
= =.
= =.
()=.
-={/ } -={/ }
.
-
10
1. :
) / 2 5A ) / 3 2 ) 2/ 9
) 2/ 4 ) 2/ 2 6 0
2.:
) / 1 3 ) / 2 5 ) / 3 5 1
) 3/16 9 0
3. : 2 2 2( ) ( 25)(4 9)( 2)
:
) / ( ) 0 ) / ( ) 0 ) / ( ) 0
) / ( ) 0
4. 1,2,3,4,5 1,2,3 3,4
. :
, , , ( ) ' , ( ) ' , ' , '
5. 1,2,3,4,5 , 4,5,6,7 , 1,4,7 .
: , , , ( ) , ( ) , ( )
6. :
)=[1,4] =(3,5) )=[2,7] =[7,10] )=[-1,2] =(2,3)
)=[-3,1] =(-1,1)
7. ={ / 2 -4++1=0 }
={ / 2 +(+1)+2-1=0 }.)
) ={0,1,2,3,4,5}
, , () .
-
11
1
:
.
. , , ...
:
.
={1,2,.....,}
1,2,....., .
. ={,} : :
={1,2,3,4,5,6}
:
.
:
.
,
.
.
. ={1} ,={2,4} ,={1,3,5}
.
: ,
.
. .
-
12
:
.
. ={,} (
)
:
.
. ={7} .
Venn
A = ,
.
-
13
1. .
( ) :
) ,
) .
.
2. .
.
3. .
Venn :
) ,.
) .
) .
) .
) ,
4.
.
:
) , ,
.
) , ,
.
5. ,
(), . :
) .
) :
: .
: .
: .
: .
: .
6.
.
. :
: .
: .
: .
: .
7. . :
: .
: .
:
) ) )- - )( ) ( )
)(-) (-) ) ) ) .
-
14
8.
. :
: .
: .
: .
) :
) ) ) ) ) ) ) ) ( )
)
.
9.
.
) 30 , 20 15
. :
:
:
) 30 , 40% 50%
. :
:
: .
10. , ,
:
) = ) = )-= ) .
11. .
.
12. (
), :
:, . .
.
13. (
)
) .,
) :
i. :
ii. : 3 .
) -
-
15
fA: k
,
.
fA=k
f1,f2,.,f , {1},{2},.....{}
f1=k1
, f2= 2k
,......., f=k
, :
0 fi1 i=1,2,,
f=f +f +.....+fi 1 2=1
. ()
'' ()()= =
()
0 ()1
( )=0
()=1
={1,2,....,}
. {i}, i=1,2,,,
, (i), :
0 ()1 i=1,2,,
v
P( )=1ii=i
() {}.
={1,2,....,} , ,
(1)+(2)+....+()
( )=0.
-
16
( )=()+()-( )
= ( )=()+()
()=1-()
()()
(-)=( )=()-( )
(-)=( )=()-( )
)
N(AB)=N(A)+N(B)-N(AB), (1) N(A)+N(B)
AB
)
N(A)= N()=, + ,
. ,
N(A)=+= N(A)+N().
: )()()(
)(
)(
)(
)(
)()(
)(
)A()(
(simply additive law)
. , , P(AB)=P(A)+P(B)+P().
-
17
) AA'=, A'
, ,
:P(AA')=P(A)+P(A') P()=P(A)+P(A') 1=P(A)+P(A'). P(A')=1-
P(A).
)
AB :
)()()(N
)(N
)(N
)A(N)B(N)A(N .
)
A-B AB
(A-B)(AB)=A, :P(A)=P(A-B)+P(AB) P(A-B)=P(A)-P(AB).
De Morgan , , :
( )=
( )=
-
18
1. . :
):
):
) 4.
2. 52 .
:
):
):
):
) .
3. 4 ,10 6 .
, :
):
):
):
4.A 30 ,25 ,5 DVD 4
DVD. .
:
): .
): DVD.
): .
): .
5. 12 18 .
40 .
. :
): 40 .
): 40 .
): .
6. 50 .
:
0 1 2 3 4 5
6 14 13 9 5 3
50 .
:
): .
): 3.
): : 3 .
): 3 4 .
): 2 4 .
-
19
7. 50 1 50.
. :
): 6 4.
): 6 4.
8.
,
. :
:
:
()= 1
4 ()=
1
3 .
3
5 ,
:
) .
) .
) .
) .
) .
9. . ()=()
()=(), *
{1} , ()=()= 1
+1.
10. .
:
)( )() )( )( )
11. ()0,28 ()0,71. :
) ( )1,01-( ) ) , .
12. 2004 ,
. , 0
-
20
15.
.
) . ,
) :
i. :
ii. : 2,
iii. : 3 4.
16. .:
={1,2,3,....,20}
={/= 2}
={/= 5}
.
) ,, -.
) (),(),( ),(-).
17. :
()= 2
3 ,()=
7
12 ( )=
1
3
:
)() ) ( ) )(-) )(-)
18. :
()= 1
2 ,()=
1
4 ( )=
5
8
:
)( ) ) ( ) )(-) )[(-)]
19. :
()= 2
3 ,( )=
1
4 ( )=
3
4
:
)() )( ) )((-) (-)) )( )
20. , ,
()= 1
6 ()=
2
3. :
)( ) )( ) )(-) )( ).
21. .
:
60%
70%
10%
-
21
:
) ,
) ,
) .
22. .
:
, 11
15
, 1
5
2
5
:
) ,
) ,
) ,
) ,
) .
23. 55% , 40%
30% .
. :
)
)
)
)
24. 120 ,24
,20
12 2 .
. :
) ;
) ;
) ;
-
www.georgefrang-math-trip.blogspot.com
22
2
+=+ =
+(+)=(+)+ ()=()
(+)=+ (+)=+
+0= 1=
-
+(-)=0 1
aa =1 0
a 1. = =, (, 0)
a 2. = = , (,, 0)
a + +3. = = , (, 0)
a +4. = = = , (,, (+) 0)
+
a + +5. = = , (,, 0, )
- -
-
www.georgefrang-math-trip.blogspot.com
23
:
........ , 2
= ,=1
1 ,=0
: =0 00 ( ).
+ = =( )
-
=
( ) =
=
-
1 = 0
= 0
:
.
2 2 21.(+) = +2+
2 2 22.(-) = -2+
3 3 2 2 33.(+) = +3 +3 +
3 3 2 2 34.(-) = -3 +3 -
2 25.(-)(+)= -
-
www.georgefrang-math-trip.blogspot.com
24
2 2 2 26.(++) = + + +2+2+2
3 3 2 27. + =(+)( -+ )
3 3 2 28. - =(-)( ++ )
-1 -2 -3 2 -19. - =(-)( + + +.......+ )
-1 -2 -3 2 -2 -110. + =(-)( - + -.......- + )
:
.
1.
, ..
.
. (-).
2.
(
)
.
.
3.
2 2(-)(+)= -
4.
,
:
2 2 2(+) = +2+ 2 2 2(-) = -2+
-
www.georgefrang-math-trip.blogspot.com
25
5.-
:
3 3 2 2 + =(+)( -+ ) 3 3 2 2 - =(-)( ++ )
6.
) 2x +(a+)+ 2x
, 2 (+) (
, ) (
) :
2x +(a+)+ =(+)(+)
) 2x ++ :
2 1 2 , : 2x ++ =(- 1 )(- 2 )
1 : 2x ++ = 2
1(- )
.
-
www.georgefrang-math-trip.blogspot.com
26
1. :
) 3 2 2a ) 2 5 2 3 46 3y y ) 3 25 8 5 8 v) 3 25 8 5 8
v) 3 2 2 3y y y ) 8 4x y ) 2 4 664 36x y ) 38 27x ) 324 3a
) 4 4a
2.: ) 236 12 1 ) 29 12 4 ) 24 8 4 ) 4 2 29 6 1y x y
) 2 7 12x x ) 2 5 4x x ) 2 5 6x x ) 24 24 28x x
) 23 12 15x x ) 2 30x x
3.: ) 3 24 ( )a a ) 5 3 28 8 ) 2 2 2 2
) 2 2 3 3 2 ) 2 2( 9) 6 9 ) 3 3 3 3 1
4. 2 .
5. , :
) (+3)2 , .
) (-6)2 , .
6. (-3)+(+3)=2, = -=3.
7. :
)
23 3
24 4
)
22 6
2 2 312 4
)
3 320 20
2 215 15 15
y
x xy y
)
25 6
26 9
x
x x
) 2 2
2 8
24 8
y
x y x
8. :
)
2 2 + +5+6
= 2 2
-4 -1 )
2 2 +y +2xy =
2 2x -y )
2 2 +4+4 a +2a = :
2 2-y a x-a y
)
2 3 -8+16 x -16x= :
2 2 2x +2xy+y +y
-
www.georgefrang-math-trip.blogspot.com
27
: , > ,
- .:
> ->0
1. > > >
2. > >
:
.
3. > >
a>
>
+>+
: ( )
.
5. > > ,,,> >
:
.
6. > > ,>
> . , (5) :
>
>
. - =
.
>
-
www.georgefrang-math-trip.blogspot.com
28
. > . : = ,
= (),
< , < (). , > .
, : =
=
= . , ,
, =
. = . :
> , (4), > (),
< , (4), < ().
, = .
1
,
.
.
! .
: (2+1)(y2+4)(xy+2)2
:
x2y2+4x2+y2+4x2y2+4xy+4 x2y2+4x2+y2+4-x2y2-4xy-40 4x2+y2-4xy0
(2x-y)20 I ,y .
2
.
.
: >>0, 1 17 7
a - > -a
, > 7>7 (1)
-
www.georgefrang-math-trip.blogspot.com
29
, ( ) 1 1 1 1
> < - >-
(2)
(1)+(2) : 1 17 7
a - > -a
,
.
.
.: 2 2 3 3 + -
2 -
2 2 + 2 2 2 2 2
2 + + 2 0 (-) 02
2 2 3 3 2 2 2 2 2 2 + - + (-)( ++ ) + 2 2
++2 - 2 - 2
2- 0 (-) 0
2 2 2 2 2 2 + 2 +2+2 2+ .
, :
-
. :
1. ->0 >.
2. -
-
www.georgefrang-math-trip.blogspot.com
30
1.
>1 >.
2.
-
www.georgefrang-math-trip.blogspot.com
31
1. ) 2 1 2 2 ) 3
2 6 2 3
2. ) 3 9 3( )
) 2 2 2 ( 2)
) 4 24 (4 )
3. :
) ( 4 ) (2 ) ) ( ) ( 2 )
) ( )( ) (4 5 ) ) 24 ( 2 ) ( )
4.: ) 22( 1) ( 1) 8 ) 2 225 2( 3) 8 (2 1)
)2( 2) 1
6 3 2
)
2( 5)2
2 8
5.: ) 2 4 5 0 ) 22 10 25 0 ) 2 1 0
) 4 27 16 0
6. , :
) 2 2 20 4(2 ) ) 2 2( 1) 4 3(2 3) .
7. 1 2 3 2
: ) 4 3 ) 3 5
) 2 2 ) 1 1
.
-
www.georgefrang-math-trip.blogspot.com
32
: , 0 = - ,
:
.
1. ||=|-| 0
2. || || -
3. || 2
= 2
4. >0 = = =-
5. = = =-
6. ||=|| ||
| | = || ||
, :| | = || || | |2 = (|| ||)2 |
|2 = ||2 ||2( )2 = 22, .
7. =
8. - + +
| + |0
10. - >0
-
www.georgefrang-math-trip.blogspot.com
33
A
.
,
d(,)
|- |. : d (, ) = | - |
d (, ) = d (, ) .
-
www.georgefrang-math-trip.blogspot.com
34
=-(-2)+(2-4)-[-(-)]-[-(3-2-4)]=
-+2+2-4+-+3-2-4=3--6.
,
:
1.
.
2.
.
3. ,
.
. =2+|+3|-|2-|
: +3=0 =-3 2--0 =2.
:
- -3 2 +
+3 - 0 + +
2- + + 0 -
:
2: =2+(+3)-[-(2-)]=2++3+2-=2+5
-
www.georgefrang-math-trip.blogspot.com
35
1. :
) 2 21 4x x ) 2 24 4 6 9x x x x
) 2( 1)( 1) 3 10 25x x x x
2. 2 4x :
)|-4|+ |-2| )|3-6|-|2-8| )5+|-4|-|8-4| )|3-6|-|4-2|-|2-|
3. 4 1x
: )|+4|-2|+1|+3|| )|2-3|-|1-3|+|-|
4. :
)|+1|+|+3| )|-2|-||2+4| )||+|-1|+|3-|
)|+4|-2 +|-3| )|3-6|-|4-2|-|2-|
5. =-d(,-3)+d(5,)
:
) .
) ( ).
6. :
=|-|||+|+|||
7. :
a)|+y||x-2|+|y+2|
)d(,)d(,3)+d(3,)
8. :
=|-3|-|+1|+|y-1|-|y+2|
-13 -2y1
a)Na .
-
www.georgefrang-math-trip.blogspot.com
36
) .
9. :
)|+1|2-4=|-1|2
)(||-||)2+2||=2+2
10. :
) 3| +2|
=||2|| +2
) 23||+
=||||+3
11. :
) 2
12
+1 )
24 +9
||12
12. |+|=|-| =0 =0.
13. |2+|=|2|+||, , 0,
.
14. |3-2|=|3|+|2|, , 0,
.
15. d(9a,9)=d(a,0) =9 =3.
16.
-
www.georgefrang-math-trip.blogspot.com
37
: ) a , .
) -
, .
, a 2 =
, 2 =
, 0 :
:
.
=
-
, ( ) =
, =|| :
.
= , 0. :
=
, 0.
= 0.
-
www.georgefrang-math-trip.blogspot.com
38
:
.
= 0.
:
: ) 3,4
, .
) 5, ,
. ,
,
.
) 6 , = .
.
-
www.georgefrang-math-trip.blogspot.com
39
1. :
) 75 48 27 4 3 )2 45 80 245 3 5
)3 18 72 50 10 2 )3 32 128 18 7 2
2. :
) =( 12 48 27 ) 3 ) = ( 32 50 18) 8
) = ( 20 45)( 80 5)
3. :
) = 2( 3 2) ) = 2( 6 7) )= 2( 3 2)
4. :
) 7 4 3 ) 6 2 5 ) 11 6 2 ) 5 2 6
5. R :
) 2 2 225 9x x x ) 2 216 4 3x x x
6. 0 1x :
2 2 2 1A x x x
7. 2 3y : = 2 24 4 6 9y y y y
8. 1 3x : = 2 22 1 6 9
1 3
x x x x
x x
9. :
) 3 5 62 2 ) 3 5 74 123 3 )4 11
8 3
6
6 ) 6 923 137 7
-
www.georgefrang-math-trip.blogspot.com
40
10. : )
1
2 )
10
5 )
3
1
2 )
8 5
12
9 )
1
5 2 )
3
6 2
)3
5 2 )
3
5 3 )
33
1
3 2 )
3 3
1
4 2
11. 0a :
) 4 23a a a ) 3 5 74 4a a )5 3 6 3a a ) 5 6 34 8a a
12. :
) 3 6a a ) 6 4a a ) 3 9a a ) 3 44 a a
13. : ) 5 3 2 ) 5 2 3 10
14. : ) 46 6 4 ) 6 2 2 3 .
-
www.georgefrang-math-trip.blogspot.com
41
3
1
( ),
,
+=0 :
) a o
=-
) =0 0==- :
( ) =0 : 0=0 () 0 :0=
1.
3(+1) -2=2(-2) +1 3+3-2=2-4+1
3-2-2=-4+1-3 -=-6 =6
2. 3-(+1) = -2(-1) 3- -1 = -2+2
-- +2 =2-3+1 0 = 0
3. 2+3(-1)=3-2 2+3-3=3-2 3-3=-2+3-2
0=-1
1. |()|= :
,
() (
, ,
(-) , (-)
).
,
|()|= ()=
.
: ) ||=0 =0
) ||=
.
-
www.georgefrang-math-trip.blogspot.com
42
: :
4-5|-3| |3 9| 3|6 2 | 6
12 2
4-5|-3| |3 9| 3 |6 2 | 612 2
4-5|-3| |3(-3)|-3- =|-2(-3)|-6
12 2
4-5|-3| 3|(-3)|-3 - =2|(-3)|-6
12 2
4-5|-3| 3|-9|-312 -12 =12 2|-3|-12 6
12 24-5|-3|-6(3|-3|-3)=24|-3|-72
4-5|-3|-18|-3|+18=24|-3|-72
-5|-3|-18|-3|-24|-3|=-72-18-4 -47|-3|=-94 |-3|=2-3=2 -3=-2 =5 =1
2. |()|=|()|
,
|()|=|()|.
()=() ()=-().
: 2|3-|-|2+5|=0
2|3-|-|2+5|=0 2|3-|=|2+5| 2(3-)=2+5
2(3-)=-(2+5) (6-2=2+5 4=1 1=4
) (6-2=-2-5
0=1 )
3. |()|=()
() :
) () 0 ()=()
() 0.
) ()
-
www.georgefrang-math-trip.blogspot.com
43
: |x-4|=5-2x
-4 0 4,:
|-4|=5-2 -4=5-2 3=9 =3
4
-4
-
www.georgefrang-math-trip.blogspot.com
44
2 (-2)(+2) 0 -2 0 2 +2 0 -2
(-2) -2 = =
(-2)(+2) +2
) ,
.
( (1) )
=2 : 0=0
=-2 : 0=16
= :
>0, = .
0, = .
-
www.georgefrang-math-trip.blogspot.com
45
(=0)
2 2 2 +=0 =- =- 0
:
- 0, = - .
- 0, .
(=0) , 2 +=0 (+)=0
=0 +=0
=-
=- 0
2 ++=0
:
2
22
2
2
2
22
2220
2
4
4
2
4
4
2
2
2
2
0
0
= 2 - 4, : 2
2
4
2
(2)
:
> 0, :
(2), (1), :
.
-
www.georgefrang-math-trip.blogspot.com
46
= 0, (2) :
2
2
0
2
0
2
0
2
2
0
2
2
2
0 2
1,2-
2
=0 1 1,2
-
2
0 .
0
) 1 , =0
.
) ,
-
www.georgefrang-math-trip.blogspot.com
47
1. 24 +8=0 4(+2)=0 4=0 +2=0
=0 =-2
2.
32-9=0 32=9 2=3 = 3
3.
2 2
2 2
1,2
1 2
2-1=(-6)(+6) 2-1= -36 -2-35=0
= -4 =(-2) -4 1 (-35)=4+140=144
- 2 144 212 =
2 2 1 214 -10
= =7 = =-52 2
4
2
2 2 2
, :
+(+1)++4=0
.
=0
(+1) 4 1 (+4)=0 +2+1-4-16=0 -2-12
1,2
5=0
=(-2) -4 1 (-15)=64
2 64 2 8 5 -3
2 1 2
A
2++-0 0 -+
=1 2
-- =2 2
,:
S=
+ =-1 2
=
=1 2
Vieta.
-
www.georgefrang-math-trip.blogspot.com
48
2++-0 0 1 2 S= 1+ 2
= 1 2 Q
x2-Sx+P=0
:
S= 1+ 2 = 1 2
,
1 2,:
) (, ..)
) :
12+2
2=(1+2)2-212
13+2
3=(1+2)(12-12+2
2)
. 2+3-5=0.
) 1,2 .
) :
) 1+ 2 ) 1 2 ) 12+2
2 ) 13+2
3 ) 1 1
+ 1 2
) 2 1+ 1 2
) =32-4 1 (-5)=9+20=29>0 .
) ) 3
+ =- =- =-31 2 1 )
-5= = = =-51 2 1
) 12+2
2=(1+2)2-212=(-3)
2-2 (-5)=19
) 13+2
3=(1+2)(12-12+2
2)=(-3)[19-(-5)]=-72
) +1 1 -3 31 2+ = = =
-5 5 1 2 1 2
)
2 2 + 19 192 1 1 2+ = = =-
-5 5 1 2 1 2
-
www.georgefrang-math-trip.blogspot.com
49
2++-0 0 :
1. 0 S=0
2. 0 =1
. 32+(+2)+-1=0.
:
) )
)=(+2)2-4 3(-1)=2+4+4-12+12=2-8+16=(-4)20
S= +2
- =0 - =0 +2=0 =-2 3
.
) =(+2)2-4 3(-1)=2+4+4-12+12=2-8+16=(-4)20
= -1
=1 =1 -1=3 =4 3
[()]2 +()+=0 0
()=y y2+y+=0 ,
, ()=y .
(2+1)2-7(2+1)+10=0
2+1=y y2-7y+10=0 2
1,2
=(-7) 4 1 10 49 40 9
7 9 73y = = = 5 2
2 1 2
T 2 2
2 2
+1=5 1 2
4 1
= 2 = 1
-
www.georgefrang-math-trip.blogspot.com
50
2+||+=0
||2=2
,
||2+||+=0 y=||. y
y=|| .
2-6||+8=0
y=|x| ||2 -6||+8=0
=(-6)2-4 1 8=4
1,2
6 4 6 2 4 2
2 2y
||=4 ||=2
= 4 = 2
2++=0 0
=
2++=0. =
.
4-22-8=0 (
=2)
=2 2-2-8=0
=(-2)2-4 1 (-8)=36
1,2
2 36 2 6 4 -2
2 2
2 =4 2 =-2
= 2
:
.
.. (
.. ).
-
www.georgefrang-math-trip.blogspot.com
51
.. .
.
.
2
+10 2- =
+2 -4 -2
-10 2 -10 2
- = - = : +2 0 -2 02+2 -2 +2 (-2)(+2) -2 -4
-10 2(-2)(+2) -(-2)(+2) =(-2)(+2)
+2 (-2)(+2) -2
2(-2)-(-10)=2(+2) -2-+10=2+4
2 -5+6= 20 =(-5) 4 1 6 1
5 1 51 = = = 3 21,2 2 2
=2 .
-
www.georgefrang-math-trip.blogspot.com
52
1. : )
) )3 2 5 4
4 6 12
)5 14 7
3( 3)2 4 2
2. : ) )2
1 3 5
2 3 3 2
)2
1 1
5 6 3 2
)
2
3 4 2( 1)
1 2 3 2
)2 2 2
1 1 1
3 2 2 4
3.N :
) ) 1 4
23
x )
3 35
2 3
x x
)4 2 4 10
52 9
x x )
3 6 2 38
2 3 6
x x x
)10 5 5 20 4 4 6 6x x x ) 2 1 4x x
) 3 7x x )2 3 4 5 0x x )1 3 2
04 6
x x
4.N :
) 3 1x ) |||-2|=5 ) |5-|2-1||=4 ) ||-1|+4|=7
) |1-|3-2||=6 ) ||+3| -2|=|-5| ) 7 2 3 6 4x
) ) 2 24 4 4 0x x x ) 3 2 0x x x x
) 23 0x x x
5.N :
)|2-8|=2-8 )|2|-4|-+3=0 ) -2|+4|=1
5 4 5 2 1
12 3 5
2 _3 2 3
10 2 5
2
2
4 3 3 8
2 2 4
2 3 6 18 0x
2 23 9 0x x x
-
www.georgefrang-math-trip.blogspot.com
53
6.N :
) ) 2( 6) 9( 1 )
) 2(2 1) 2(1 2 ) ( 1) ) 22( 2 ) (4 ) 0
) 2( 2) ( 1) 0 ) 2 2 22 ( 1) 2 ( 1)
) 23( ) 2 ( ) ( 1) ) 2 2( 8)( 1) 2 (3 ) 2
7. : ) ) 4 81x ) 6 21x
) 5 32x ) 7 48 0x x ) 5 325 16x x ) 5 22 16 0x x
) 4 28 0x x .
8. : ) )2 4 4 0
) 2 2 8 0 ) 2 3 4 0 ) 2 5 6 0
) 2 6 5 0 ) 2 5 7 0 ) 22 8 10 0
) 29 6 1 0 ) 23 5 2 0 ) 24 12 9 0
) 22 7 6 0 ) 22 6 11 0
9.: ) 2 16 0 ) 22 18 0 ) 23 12 0
) 236 16 0 ) 2 3 0 ) 22 8 0 ) 25 30 0
) 27 35 0 ) 22 8 0 ) 23 27 0
10. : ) 2( 2)( 5 4) 0 ) 2 2( 4 )( 7 6) 0
) 2 2(3 48)( 4 32) 0 ) 2 2( 9 8)( 5 7) 0
11.: ) (3 10) 3( 2) ) 1 (3 5 ) ( 2)
) 4( 2) 4 ( 3)( 3) ) 2( 1) 4 5(2 1)
) 2 2( 2) ( 1)( 1) ( 3) 6 )2 3 ( 1)( 1) 3
6 3 2
) 2
24(1 7 ) 3 (7 1) 2 1025 5
) 2(3 2) (2 1) 2 6
13 6 2
2( 1) ( 1 )
3 64x
2 2 3 0
-
www.georgefrang-math-trip.blogspot.com
54
12. :
) 2 (2 ) 6 0x ) 22 (3 2) ( 2) 0
) 2 (2 ) ( ) 0 ) 22 ( 3) 1 0
13.
:
) 2 2 22 2 1 0 ) 2 2 2( ) 0 , 0
) 21
( ) 1 0 0.
14. 4. )
) .
15. 2 12 0 (1) 2 2(2 9) 6 0
(2). (1) (2). : ) )
(2).
16. 2 18 21 0 6. :
) ) .
17. 2 16 20 0 10. :
) ) .
18. 1 2, 2 3 1 0 ,
: ) 1 2 ) 1 2 )2 2
1 2 )3 3
1 2 )
1 2
1 1 ) 2 1
1 2
.
19. 1 2, 2 5 4 0 ,
: ) 1 2 ) 1 2 )2 2
1 2 )3 3
1 2 2 1 )
2
1 2( ) ) 2 21 2
1 1.
2 ( 1) 1 0
-
www.georgefrang-math-trip.blogspot.com
55
20. 1 2,
22 3 4 0 ,
: )1 2
)1 2
) 2 21 2
) 2 21 2
)
1 2( 1)( 1) ) 1 2(2 3)(2 3) .
21. 1 2,
22 3 4 0 ,
: )1 2
)1 2
) 2 21 2
) 1 2
2 1
)
1 2 ) 1 2 .
22. : )2
4 ) -3 5 ) -6 1 ) -4 -1 ) 1
2 -2 )
2
3 6.
23. : )-2
+2 ) 1+ 1- ) 2 .
24. 1 2,
2 3 1 0 ,
:
) 12 22 )
2
1 2
2 )1
1
2
1 ) 1
2 2
2.
25. 1 2,
2 5 2 0 ,
:
) 13 23 ) 1 2 ) 1
1
2
1 ) 21
2
2 .
26. 1 2, 2 4 2 0 ,
:
) 2
1
1
2
2
1 )
2
1
2
2
2
1
) 21 1 2 2
2 1 2
) 1
1
3 1
3 2
2
3 1
3.
27. : 2 ( 1) 2 6 0x
) .
) : )
).
-
www.georgefrang-math-trip.blogspot.com
56
28. : 22 ( 5) 3 0x .
.
29. : 2 ( 1) 0x .
.
30. : 2 4( 1) 8 1 0x . ) 1 2,
) =(1-2)(
2-2)
.) 2 21 2 1 1 2 2 64 .
31. :
) ) 2 5 | | 4 0x ) 2 4 3| |x
) 23 | | 2 3(| | 1) ) 2 | 6 | 40 0
32. :
) 4 25 4 0 ) 4 28 9 0 ) 4 24 17 4 0
) 4 27 10 0 ) 4 25 6 0
33. :
) 6 37 8 0 ) 6 38 7 1 0 ) 8 417 16 0
) 8 416 15 1 0
34. :
) 6 8 0 ) 4 3 0 ) 5 4 0
35. :
) 2( 5) 2( 5) 3 0 ) 2(2 1) 2(2 1) 15 0
) 2 2 2( 1) 7( 1) 10 0 ) 2( 1) 3 | 1| 4 0
) 2( 5) 3 | 5 | 10 0 ) 25 (2 1) 4 |1 2 |
2 2 | | 3 0x
-
www.georgefrang-math-trip.blogspot.com
57
36.:
) 2
3 61
2 2 )
2
4 1 1
1 1 1
) 2
10 2
4 2 2 )
2 2
4 4
1 1
) 2 2
4 4
1 1 )
2 2
5 20 14
2 4 2
37.:
) 1
21
1
) 1 4
1 1 41
)2
2
1 1 11 1
)
2 41 1
7
1 2 31
.
-
www.georgefrang-math-trip.blogspot.com
58
4
1
,
1 :
(
, )
(
,
).
: x-3 5 1014 2 4
x x
2x-4 x+5 10+x 2x-4 x+5 10+x-
-
www.georgefrang-math-trip.blogspot.com
59
,
.
:
7- 4-5(x-2) 13-3(x+1) 1-
4 2
:
4-5(x-2) 13-3(x+1) 4-5+10 13-3-3 -5+3 13-3-4-10
-2 -4 2
:
7- 7- 1- > 4 1-4 >4 4-(7-)>2 4-7+>2
4 2 4 2-2>7-4 ->3 10 >2.
2(-1)-3(4-)
-
www.georgefrang-math-trip.blogspot.com
60
2-2-12+3
-
www.georgefrang-math-trip.blogspot.com
61
|()|< |()|
|()|> |()|
.
|()|0 x R- x/A(x)=0
|()| 0 .
-
www.georgefrang-math-trip.blogspot.com
62
x2 + x + , 0 :
2
2222222
4
4
2
4
4
2
2)
(
:
2
22
4
2
(1)
:
>0 . , :
2
2
2
2
2
2
2
2
4
2
22
2
22
: x2 + x + = (x - x1)(x - x2) , x1, x2 . , >0,
.
= 0 . (1) : 2
2
2
, = 0,
.
-
www.georgefrang-math-trip.blogspot.com
63
2
2 ++ =2-
4 :
>0 , 1,2
- 1 2 + 2++
>0, , , : x2 + x + = (x - x1)(x - x2) , x1, x2
x1 0. , (1), .
x1 < x < x2 (), x - x1 > 0 x - x2 < 0, (x - x1)(x - x2) < 0.
, (1),
x1 < x2 < x (), x - x1 > 0 x - x2 > 0, (x - x1)(x - x2) > 0.
, (1),
=0 , 1,2
- 1,2 + 2++
-
www.georgefrang-math-trip.blogspot.com
64
= 0, : 2
2
2
,
2
,
2
0 , =32-4 1 (-10)=49
1,2
3 49 3 7
2 2x 1=-5 2=2
-
www.georgefrang-math-trip.blogspot.com
65
- -5 2 + 2+3-10 + - +
. (- ,-5) (2,+ )x
) -2+2-1
-
www.georgefrang-math-trip.blogspot.com
66
2++>0 ( )
:
0 ( )
2++
-
www.georgefrang-math-trip.blogspot.com
67
2. 42+4(2-1)+4-3>0 R.
-
www.georgefrang-math-trip.blogspot.com
68
: (-2)(2+6+9)(2-3-4)(2-+2)0 B(x)
, () 0
A(x)
-
www.georgefrang-math-trip.blogspot.com
69
2x-3 0 (-3)(
2x 3 10x 3 10) 0
xx
-3=0 =3 2-3-10=0 =5 =-2
- -2 3 5 + 2-3-10 + 0 - - 0 +
-3 - - 0 + +
- + - +
(- ,-2) (3,5).
)
2+2-3 0 1 -3 .
2 2
2 9)(x+2)(x0
2x 2 3(x+2)(x 9)(x 2 3) 0
xx
- -3 -2 1 3 + +2 - - 0 + + +
2-9 + 0 - - - 0 +
2+2-3 + 0 - - 0 + +
- - + - +
(- ,-3) (-3,-2] (1,3].
-3 1
.
A(x)
B(x)
...
:
( ).
.
A(x)
B(x) ,
-
www.georgefrang-math-trip.blogspot.com
70
: ) 2x-1
12x 4
) 4 10
2x+1 1x 1
xx
) 2-4 0 2
-1 0 022x-1 2x-1 2x-1 x 4
12 2 2 2x 4 x -4 x 4 x 4
2 22x-1-x +4 -x 2 3 2 20 0 (-x 2 3)(x 4) 02 2x -4 x 4
xx
- -2 -1 2 3 + -2+2+3 - - 0 + + 0 -
2-4 + 0 - - 0 + +
- + - + -
(-2,-1] (2,3]
-2 2
.
) 2-1 0 1
4 10 4 102x+1 1 x+1 1(x+1)(-1)x 1
x xx x
(+1)
(-1)
4(-1) 10(x+1)(-1) (x-1)(+1)
(+1)- + 0
(-1)
4(-1) 10(x+1)(-1) (x-1)(+1)
04(-1)-10+(+1)
(x+1)(-1)
20
4-4-10+ +(x+1)(-1)
20
+5-14(x+1)(-1)
20( +5-14)(-1)(+1) -1=0 =1 +1=0 =-1
2+5-14=0 =-7 =2
-
www.georgefrang-math-trip.blogspot.com
71
-7 -1 1 2
2+5-14 + o - - - o +
-1 - - - o + +
+1 - - o + + +
+ - + - +
(- ,-7] (-1,1) [2,+ ]
-1 1
.
- +
-
www.georgefrang-math-trip.blogspot.com
72
1. : )3( 2) 5( 1) 3 2( 3)
)4 3 2
23 6 4
)
3 53 2
2 2
)3 5 10
14 2 4
2.
:
)3( 1) 2 1 2( 3) 2
)1
12
4 41
4 8
)1 2 1
32 2
20 3 306
7 7
)4 3 6
5 15
5
4 2 4
3. :
)3 5 1 7 5 )2( 1) 3 3( 1)
) 5 4( 2) 2( 3) 5(2 ) 4
)4 2 3 2
2( 1)3 2
)5 5( 3)
13 6
4. :
) 1
3 13
x )
2 4 44 5
3
xx
)
15 2 98 2 7
2
xx
) 1 2 2 5 5 6x x x )6 3 4 8 3
2 1 28 5
x xx
-
www.georgefrang-math-trip.blogspot.com
73
5.:
) 1 5 3x ) 3 2 1 5x )5 3 1 8x )4 6 2 10x
6.:
) 4 1 0x x ) 4 2 3x x )3
3 22
xx
7. :
)2
( )3
xf x
x
)
5 2( )
13
xf x
x
)
5 2( )
2 2 1
xf x
x x
) ( ) 4 20f x x
) ( ) 6f x x )3
( )5
xf x
x
)
3( )
3
xf x
x
) ( ) 2 6f x x x )2 2
1 6( )
2 9
xf x
x x x
)
1( ) 5
3f x x
x
8. :
) 2 2 3 0x x ) 2 3 4 0x x ) 2 4 12 0x x
) 23 4 4 0x x ) 2 4 4 0x x ) 22 4 2 0x x
) 2 8 16 0x x ) 2 2 3 0x x ) 23 3 1 0x x
) 23 6 3 0x x .
9.:
) 2 5 0 ) 2 3 0 ) 22 5 0 ) 2 6 0 ) 2 9 0
) 2 7 ) 22 4 ) 2 .
10.:
) 2 ) 4( 5) ( 4)( 4) 0 ) 22( 3)( 3) ( 1) 4
11. :
) 2 2 3 0 2 2 0 ) 2 2 0 2 2 8 0
) 2 4 5 0 2 4 0 ) 2 6 0 2 2 1 0
) 2 5 4 0 2 2 8 0 .
-
www.georgefrang-math-trip.blogspot.com
74
12. :
) 23 2 8 )4( 1) ( 7) 2(2 5)
13. 2 14 50
5 26.
14.a :
) 2( ) 4 5f x ) 2( ) 9f x )2
2( )
7
xf x
x x
) 2
2
3 10( )
4 3
x xf x
x x .
15. , f
:
) 2( ) 3 10f x x x ) 2( ) 3 4 4f x x x .
16. , f
:
) 2( ) 3 4f x x x ) 2( ) 2 3 5f x x x .
17. , f
g:
) 2( ) 4 6f x x x ( ) 2 3g x x
) 2( ) 4 8f x x x 2( ) 3 4g x x x .
18. 2 ( 3) 6 0x 2
.
19 . 2 ( 5) 3 7 0x
.
20. 2( 1) 4 2 0, 1x
.
21. 2 ( 3) 0, 0x
.
-
www.georgefrang-math-trip.blogspot.com
75
22 . 24 4(2 1) 4 3 0
.
23. : 2( 2) 2 2 3 0 .
24. :
) 2( 2)( 2 3) 0 ) 2( 3)( 4 3) 0
) 2 2( 4)( 3 10) 0 ) 2 2( 3 4)( 3 2) 0
25.:
) 2 2( 6)( 2) 0 ) 2 2(9 )(2 ) 0
) 2 2(16 )(4 2 )( 2 1) 0
26. :
)2
04
)5
01
) 1
04
)2
30
4 5 )
2250
2
)2 2 3
02
.
27.:
) 2
2
( 2)(9 )
2 30 )
2 2
2
( 1)( 6)
2 80 )
2
2
( 3)( 1)0
4
28.:
)2 5
42
) 8
2 1 )
2
2
2 4 51
2 )
2
3 13
3 3
) 1 2
11 1
.
-
www.georgefrang-math-trip.blogspot.com
76
5
A
: *
R.
a: * R
H
:
+1,.-1,
.
(.)
: ,
.
.
+1=+ +1-=
- . : =1+(-1)
:
=1+(-1)
-
www.georgefrang-math-trip.blogspot.com
77
. : ,, .
:
, ,
, :
, , ,
, , .
: ,,
:
1. ,
+
2.
2. O 1,2,3,........., + +......+1 2
2.
. :
- ,
:
= ( + ) 12S
,
, :
= [2 +(-1)] 12S .
.
i) - , +1
+1- .
+2=+ =
2
-
www.georgefrang-math-trip.blogspot.com
78
S :
ii)
-1S
1 2 -1 -1 -1S = + +....+ + S =S + S -S
.
. : ) =3-5 .
) ()
2S =2 - , ..
) +1 +1=3(-1)-5=3-2.
+1- =3-2-(3-5)=3-2-3+5=3
. =3 =1: 1 =-2
) 22=2 -1 -(-1)=2 -5+3-1S
2 -5-3) =4-3= -1
2 S -S =2 --(2
+1=4(+1)-3=4+1 +1-=4+1-(4-3)=4
. =4 1 =1
.
.
=1+(-1) 1,.
.
. 7 9, 4
9 16. .
1
3
+3 +8 +11 1 1
= +6 +6=9 +6=9 7 1 1 1 1 + =16 + =16 2 =16 =294
,
,, .:
i. , 1=, +2=
ii. =1+(-1)
.
iii.
.
-
www.georgefrang-math-trip.blogspot.com
79
.
7 2 50
..
2,1, 2, 3,........, 7,50 1=2 9=1+8 50=2+8 8=48
=6.
2,8,14,20,26,32,38,44,50.
.
i. 1, 3,5,...... .
=2
ii. 2, 4,6,...... .
=2 .
.,
i.
. :
.....,-3,-2,-,,+,+2,+3,......
ii. .
-, +
:
.....,-5,-3,-,+,+3,+5,......
.
) .
20 -560.
) .
4 105.
) : -2,-,,+,+2
-2+-+++++2=20 5=20 =4.
(-2)(-)(+)(+2)=-560 (4-2)(4-)4(4+)(4+2)=-560
4(16-2)(16-42)=-560 16(16-2)(4-2)=-560 (16-2)(4-2)=-35
4-202+99=0.
2= 2-20+99=0
-
www.georgefrang-math-trip.blogspot.com
80
=11 =9 2
11 = 11
. 2
9 = 3 . =3 =4 -2,1,4,7,10
=-3 =4 10,7,4,1,-2
-2,1,4,7,10
) -3,-,+,+3
-3+-++++3=4 4=4 =1
(-3)(-)(+)(+3)=105 (2-92)(2-2)=105 94-102-104=0
2=26
9 2=4 = 2
=2 =1 -5,-1,3,7
=-2 =1 7,3,-1,-5
-5,-1,3,7
: ,
. .
+1
=
= +1
- . : =1-1
:
, =1-1
-
www.georgefrang-math-trip.blogspot.com
81
. 1 0( 0)
0 0( 0)
. : ,, .
:
2=
, , ,
, , , 0 2 = ,
, ,
. .
: , ,
, 2 =
,.
.
.
4 . ,,, 2= 2=
=.
. :
-1S = 1 1 -1
. 1 , :
- -1 - -+1 1 11 1S = = = = 1 -1 -1-1 -1
(1)
(1)
(2)
-
www.georgefrang-math-trip.blogspot.com
82
(2) (1) :
, 1, :
: -1,
Sv = 1 1.
.
- , +1
+1
.
S :
-1S
1 2 -1 -1 -1S = + +....+ + S =S + S -S
.
. ) - +1
3 = -1
2.
. 1,.
) S=2(3-1).
. 1,.
) +1 +2
3 = +1 2
.
3-(-1) -13 2
2
+23
+2 -1 3 2 +2-(+1)+1 2= 3 2+1 +1 3 3 2-1
2
. = 3
2 1=9
) -1 S-1=2(3-1-1)
-
www.georgefrang-math-trip.blogspot.com
83
=S-S-1=2(3-1)- 2(3-1-1)=
3 4
2 3 2 33 3
.
4 +13+1 3 3
4 33
. . =3 1=4.
,
, ,
.,:
1,2,...., .
. , 1,2,....,,.
1= +2= +2=1+1 =+1
=
+1 ,
+1 ,
=1-1
.
. 2
9 162 5 ,
.
1,2,3,4,5 2
9, 1,2,3,4,5,162
. 1=2
9 7=162 , :
7=162 16=162
2
9 6=162 6=729 = 3 .
=3 : 2
9,2
3,2,6,18,54,162
=-3 : 2
9,-
2
3,2,-6,18,-54,162
-
www.georgefrang-math-trip.blogspot.com
84
.
() . ,:
1,3,5,....., .
=2.
2,4,6,....., .
=2.
1,2
,3,..... . =
.,
., .
: 2
....., , , ,,, ,....3 2
.,
2 ., 3
....., , , ,, ,....5 3
.
) 3 .
8, 21/8.
) 4 .
64, .
)
,,
3 =8 8 =2
:
12 2+4+4=21 -17+4=0 =4 =
4
1 1 21 1 1 21+ + = + + = 4 4
8 2 2 2 8
=2 =4: 1
2,2,8
=2 = 1
4 : 8,2,
1
2.
1
2,2,8
-
www.georgefrang-math-trip.blogspot.com
85
1. 15 : 7, 15, 23, 31, ..
2. () : 8 = 22 14 = 40.
.
3. 2 92.
20 .
4. 1 = 23, = - 2 = 5.
S.
5. : 1 = 3, = 4 S = 136.
.
6. : 1 = 1, = 25 S = 91. .
7. () :
1 42 + 43 5 = 0
8. . : 4+12=0 7+19=-30.
:
) ,
) =.
9. . 3+4+12=10 13 8.
:
) ,
) 23
) =2.
-
www.georgefrang-math-trip.blogspot.com
86
10. R + 1 2 + 3
1 .
11.
16. .
12. , :3+, 2(-), -5,
.
13. ,, .
(+)2-2 , (+)2-2, (+)2-2.
14. ,, :
+ +, + ,
+ + + +
.. :
2 , 2 , 2
.
15. :
)1+4+7+....+94 )-5-1+3+7+....+135 )-6-9-12-.....-114
16. :
) 50 3.
) 40 5.
) 21 153.
) 5 31 206.
17. . 16 50, 6 11
40. 31 .
-
www.georgefrang-math-trip.blogspot.com
87
18. . 1=45 6+11=0. .
19. . =4 21=316.
.
20. . 6=8 11=23. :
) ,
) 14,
) S=15+16+...+25
21. - : =5-8,
., .
22. N*, () :
S = 22 5, () 1 .
23. N* S = 32 ( ),
() .
24. 6 50 10 ,
.. .
25. . 55,
695.
26. 4 ., 20,
25
24.
27. :
) (x + 2) + (x + 4) + (x + 6) + .. + (x + 60) = 1050
) 1 + 7 + 13 + .. + x = 280, x >0
-
www.georgefrang-math-trip.blogspot.com
88
28. 20 .
. 1
16 , 7 28 .
) 10 ;
) 4 10 ;
) 6 , 9
, 12 ,
;
) ;
29. 12
1200/m2 100/m2
.
) m2 ;
) 82 m2 ;
) 200 m2,
;
30. . 1=48 5=3. 5 .
31. . 6=-108 3=4. :
)
) 5
32. : 1 = 6, = 2 = 3072. .
33. x R : 1x , 4 1x5 , 3x
.
-
www.georgefrang-math-trip.blogspot.com
89
34. , , 23-2 , ., ,, 0,
, , , ..
35. 7 :
192,96,48,......
36. :
)5+10+20+....+640
)4-8+16-.....-8192
37. x, , z 147, x, , z
, x, z,
.
38. :
) 3 + 32 + 33 + .. + 3x = 120, x N*
) 1 + x + x2 + x3 + .. + x39 = 0
39. 21
27.
40. - =3 2.
) .
.
) 6 .
41. - = 24-, R .
) .
) 7 127, .
-
www.georgefrang-math-trip.blogspot.com
90
42. 2 486 4 ,
.
43.
42 512.
44. 4
4096 .
45. .
) 10 ,
6 .
)
33 . 10 .
i. 20 .
ii.
;
46.
.
20 .
) 7 ;
) 5 ;
) 7
1000 ,
.
-
www.georgefrang-math-trip.blogspot.com
91
47.
.:
)4,2,1,... )9,-3,1.... )18,12,8,...
48. . 1=18 , ||
-
www.georgefrang-math-trip.blogspot.com
92
6
:
()
.
,:
.
y f(x) f .
, y .
=f(A) .
:
f:
f(x)
f:
, :
A
A .
( ),
f(x) ( ).
-
www.georgefrang-math-trip.blogspot.com
93
:f()=+-1
-1+......+1+0
, = .
: P(x)
f(x)=Q(x)
() Q(x) ,
,
: ={ /Q(x) 0 }
: f(x)= A(x) ,
A= x /A(x) 0
: 1
f(x)=A(x)
,
: A= x /A(x)>0
:
,
,
.
: :
) x-5
f(x)=2
x +2x-3 ) f(x)= 2x-6+x-1 )
3-x-2f(x)=
2x+8 )
1f(x)= x+2-
x-3
) 2+2-3 0 1 -3.
= -{-3,1}
) 2-60 26 3
[3,+ )
) 3-0 3 2+8>0 2>-8 >-4
-4
-
www.georgefrang-math-trip.blogspot.com
94
().:
3x-2, a 1f(x)=
2-x +2x+8,
-
www.georgefrang-math-trip.blogspot.com
95
1. :
) 3 2
( )12 4
xf x
x
)
2
1( )
4f x
x
)
2
2
9( )
3 4
xf x
x x
) 3 6
( )1
xf x
x
)
3( )
1
xf x
x
)
2
2 3 4( )
3 4 3
xf x
x x x
) 2 2
2 3 4( )
4 9 4 3
xf x
x x x
)
3 2
5 3( )
4 4
xf x
x x x
) 4
5 3( )
27
xf x
x x
)
6 2
2( )
16
xf x
x x
2..:
)2
( )3
xf x
x
)
5 2( )
13
xf x
x
)
5 2( )
2 2 1
xf x
x x
) ( ) 4 20f x x
) ( ) 6f x x )3
( )5
xf x
x
)
3( )
3
xf x
x
) ( ) 2 6f x x x )2 2
1 6( )
2 9
xf x
x x x
)
1( ) 5
3f x x
x
3.:
)
2f(x)= x -4x-5
)
2f(x)= 9-x
)
x+2f(x)=
2x -7x )
2x-1f(x)=
2-2x +x+6
)
2f(x)= 2x -4x+3
)
5-xf(x)=
29x -6x+1
4.:
)
2-x +3x+10
f(x)=x-2 )
32 2f(x)= x -x-2+ -x +2x+15 )
2-x -3x+10
f(x)=2x +4x+3
)
3 2x +2x-3f(x)=
2x -16
-
www.georgefrang-math-trip.blogspot.com
96
5. :
2x-5, 0f(x)=
2-x +3, 0
x
x
) f;
) :
)f(-3) )f 5
2 )f(-1) )f(0) )f
1
2 )f( - 3 ).
6. :
3x-1, x
-
www.georgefrang-math-trip.blogspot.com
97
a)Na f.
) f(-2) f(13).
9. f: :
f(x+2)=3f(2)-2(x-3)
. :
) f(2)
) f() .
10. f: :
f(2-1)=42-6 -f(3)
. :
) f(3)
) f() .
-
www.georgefrang-math-trip.blogspot.com
98
-
: yy
y,
, .
yy .
(,) ,
.
yy xy
.
y=0,
(,0).
yy =0,
(0,y).
(x,y) ,:
y=0
yy =0.
(0,0).
-
www.georgefrang-math-trip.blogspot.com
99
4 ,
:
(,y) :
1 , ,y>0.
2 , 0.
3 , ,y0 y
-
www.georgefrang-math-trip.blogspot.com
100
1. (2+4,3-) .
.
2.
: )( 2 9a , 2 2 3a a ) )(||-2, 2 5 6a a )
)( |2-5|-3 ,|-3|-2).
3. (3,5) (-1,2) (-1,-2)
:) ) yy ) )
1 3 .
4. (2-7,3-y) (-5,1). ,y ,
: ) ) yy )
) 1 3 .
5. :
)(2,4) (5,8) )(-3,-4) (3,4) )(-5,1) (7,6) )(-7,-3) (-3,0)
6. (-1,-2) (,1) =5.
7. (1,4) ,(5,2) (-2,-2)
.
8. (-2,3) (1,5).
) .
) .
) .
9. (2,6) 10.
10. yy (-2,1)
13 .
-
www.georgefrang-math-trip.blogspot.com
101
: f A .
y, (,f(x)),
f.
H f Cf.
: (,) f
, : f()=
;
f
f(x). f
. (
) f .
. (
).
:
:
y=f(x) y=0 f(x)=0
( ) .
yy:
y=f(x) =0. y=f(0)
.
: 0 ,
yy.
:
Cf , yy
( ).
Cf f(x)>0 k f(x)
-
www.georgefrang-math-trip.blogspot.com
102
. ) f(x)=2x-4.
)
:
) )
) .
: f(x)=0 2x-4=0 2x=4
x=2.A (2,0).
yy: =0 f(0)=-4.A
yy B(0,-4).
) Cf f(x)>0 2x-4>0 2x>4 x>2.
Cf f(x)
-
www.georgefrang-math-trip.blogspot.com
103
1.
:
) 2( ) 3 4f x ) ( ) | 2 5 |f x x )6
( )2
xf x
x
)
3
5( )
xf x
x
) ( ) 4f x x ) ( ) | 1| | 2 |f x x x 2. :
2x-6, x 2f(x)= 2
x -3x-10, x
-
www.georgefrang-math-trip.blogspot.com
104
7. :
f(x)=x2-x+2 g(x)=x2+x+8
Cf 2, Cg
(-2,2). :
) .
) Cf Cg.
-
www.georgefrang-math-trip.blogspot.com
105
f(x)=ax+
: ,
.
=0, .
: 000) (
-
www.georgefrang-math-trip.blogspot.com
106
y=0 .
f(x)=ax
(0,0).
=.
y=ax+.
=1 (=45 ) y=x
.
=-1 (=135 ) y=-x
.
1:y=a1x+1 2:y=a2x+2 , :
1=2 .
1=2 1=2 .
12 =-1 .
. ) , 1:y=(22+4)-7 2:y=(
2+4)+13
.
) , 1:y=(+1)-3 2:y= -2
2+4 .
) 22+4=2+4 2-4+4=0 =2
) +1-2
2=-1 (+1)(-2)=-2 2--2=-2 2-=0 (-1)=0
=0 =1
-
www.georgefrang-math-trip.blogspot.com
107
1. y=(+2) +2+8 .
) ) .
2. y=(2-6)+-1 .
) ) .
3. y=(3-)++1 045 .
) ) .
4. :
)f(x)= 5
2 1
2
2
x
x
)
2 2 3( )
2 8 3
xf x
x x
)f()=||-2 )f()=|+2| -|-1|
5. :
) :y=(3-5)+2 :y=4-1 ): y=(-3)-1 :y=(5-)+4
): y=( 2 4 ) -7 : y=(+10)+21 ): y=(-3)+2 : y=(43)
) : y=|+2|-1 : y=-5 ) : y=|-3|-2 : y=|3+7|
6. :
) :y=(+1) : y=(+3)-2 ) :y=(+8) : y=(+8) -4
): y=(-1) -2 : 2
62
y
): y=(-2)-1 : y=
3
3
) : y= 22 4 : y= 43 5 .
7. : y=(+3)+5 : y=(+8)+2+1.
: ) ).
8. :
)(-4,0) (0,2) )(0,-2) (3,1) )(0,3) (2,1) )(0,-1) (4,-1)
-
www.georgefrang-math-trip.blogspot.com
108
9. :
) :y=-3+2 (2,-1)
) :y=2+5 (2,-13).
10. :y=2+7
(-4,-1).
-
www.georgefrang-math-trip.blogspot.com
109
-
f,:
f(x)=(x)+c, c>0,
c .( 1)
f(x)=(x)-c, c>0,
c .( 2)
( 1)
( 2)
f,:
f(x)=(x-c), c>0,
c .( 3)
f(x)=(x+c), c>0,
c .( 4)
-
www.georgefrang-math-trip.blogspot.com
110
( 3)
( 4)
:
f, :
f(x)=(x+c)+d, c,d>0
c
d .
f(x)=(x+c)-d, c,d>0
c
d .
f(x)=(x-c)+d, c,d>0
c
d .
f(x)=(x-c)-d, c,d>0
c
d .
:
-
www.georgefrang-math-trip.blogspot.com
111
1. f
, 1,2 :
1
-
www.georgefrang-math-trip.blogspot.com
112
f(1)-f(2)0, f .
5- - 5-
-
www.georgefrang-math-trip.blogspot.com
113
) 1,2( - ,2) 1 2
.
f(x )-f(x )1 2=
x -x1 2=
3x +1 3x +11 2 (3x +1)(x -2)-(3x +1)(x -2)- 1 2 2 1x -2 x -2 (x -2)(x -2)1 2 1 2= =
x -x1 2x -x1 2
3x x -6x +x -2-3x x +6x -x +21 2 1 2 1 2 2 1 =
(x -2)(x -2)(x -x )1 2 1 2
7x -7x -7(x -x )2 1 1 2= =(x -2)(x -2)(x -x ) (x -2)(x -2)(x -x )1 2 1 2 1 2 1 2
-7
(x -2)(x -2)1 2. 1
-
www.georgefrang-math-trip.blogspot.com
114
A
:
1. f 0,
f(x)f(x0) 0
f(x0) f.
2. f 0,
f(x)f(x0) 0
f(x0) f.
, :
f(x)M f(x) .
f(x)=M f(x)= ,
.
. 2x
f(x)=2
x +1 -1
1.
H f =
:
-
www.georgefrang-math-trip.blogspot.com
115
2 22x 2x 2x+x +1 (x+1)
f(x) -1 -1 +1 0 0 02 2 2 2
x +1 x +1 x +1 x +1
.
E -
1 0 0
2 22x 2x 2x-x 1 -(x-1)
( ) 1 1 02 2 2 2
x +1 x +1 x +1 x +1f x
.
:
= = =0 =0 x=-1
2 22x 2x 2x+x +1 (x+1) 2
f(x)=-1 -1 +1=0 0 (x+1)2 2 2 2
x +1 x +1 x +1 x +1
-=1 =0 =0 =0 x=1
2 22x 2x 2x-x 1 -(x-1) 2
( )=1 1=0 -(x-1)2 2 2 2
x +1 x +1 x +1 x +1f x
f =-1 -1 =1
1.
:
1) f ,:
f(-x)=f(x) .
2) f ,:
f(-x)=f(x) .
,
0. :
(-,)
[-,]
(- ,-) (,+ )
(- ,-] [,+ )
-
www.georgefrang-math-trip.blogspot.com
116
(-,-) (,)
[-,-] [,]
,a a
*
,
yy.
,
.
;
,
:
0,
.
f(-x),
:
f(-x)=f(x), f ,
f(-x)=-f(x), f .
f(-x) f(x) f(-x) -f(x) .
: ,
) - ) f(-x) f(x)
f(-x) -f(x).
, .
-
www.georgefrang-math-trip.blogspot.com
117
1.
:
f(x)=|x| , g(x)=|x|+2 , h(x)=|x|-2
2.
:
f(x)=|x| , g(x)=|x-2| , h(x)=|x+2|
3.
:
a)f(x)=|x| g(x)=|x+2|+1
)f(x)=|x| g(x)=|x-1|-2.
4. :
) 3( ) 5f x x x ) ( ) 2f x x x ) ( ) 12 3 4f x x x ) ( ) 4 3f x x
5. f
i)f(x-4) f(3x) ii)f(|x|) f(2).
6. f
)f(|-2|) f(4) )f(|2-3|) f(5).
7. :
) 2( ) 3 2f x x ) 2( ) 1f x x ) 4( ) 2 6f x x ) 2( ) 3( 5) 11f x x )2( ) 5 2( 1)f x x
8. 2
4x -4x+5f(x)=
24x +3
. f 2
3
2.
9. :
) 4 2( ) 2f x x x ) 3( ) 3f x x x ) 2( ) 5 4f x x x ) ( )f x x x
-
www.georgefrang-math-trip.blogspot.com
118
)2
( )4
xf x
x
)
2
3( )
9
x xf x
x x
) ( ) 3 3f x x x
)2 4 1
( )3 3
x xf x
x x
) 2 2( ) 4 2 4 4f x x x x x
10. :f ( ) ( ) ( )f y x f y f x ,x y .
) f(0) ) .
11. :f ( ) ( ) 2 ( ) ( )f y x f y x f x f y
,x y . ) f(0) ) .
12. f: . :
g(x)=xf(x)-|f(x)|
.
13. f: . :
g(x)=f(f(x))-x4f(x)
.
-
www.georgefrang-math-trip.blogspot.com
119
7
f(x)=ax2
: =
: ,
yy.
: (- ,0] [0,+ ).
: =0 f(0)=0. :
, + f(x) + .
A , - f(x) + .
: f(x)=ax2,
(0.0) .
f(x)=-ax2
: =
: ,
yy.
: (- ,0] [0,+ ).
: =0 f(0)=0. :
, + f(x) - .
A , - f(x) - .
: f(x)=ax2,
(0.0) .
-
www.georgefrang-math-trip.blogspot.com
120
:
f(x)=ax2 g(x)=-ax2 0
.
|| yy (
)
a
f(x)= >0x
: =(- ,0) (0,+ )
: ,
(0,0). : y=x y=-x
: (- ,0) (0,+ ).
: . :
, + f(x) 0
A , -
f(x) 0.
-
www.georgefrang-math-trip.blogspot.com
121
( ,
).
0 ,
0 , f(x) .
0 ,
0 , f(x) .
( ,
).
: af(x)= >0x
, , 1
3 .
a
f(x)=- >0x
: =(- ,0) (0,+ )
: ,
(0,0). : y=x y=-x
: (- ,0) (0,+ ).
: . :
, + f(x) 0
A , -
f(x) 0.
-
www.georgefrang-math-trip.blogspot.com
122
( ,
).
0 ,
0 , f(x) .
0 ,
0 , f(x) - .
( ,
).
: a
f(x)=- >0x
,
, 2 4 .
-
www.georgefrang-math-trip.blogspot.com
123
1.Na , :
)f(x)=(8-2)2 .
)f(x)=(||-3)2 .
2. , :
)f(x)=(3-)x2 (- ,0].
)f(x)=(||-1)2 [0,+ ).
)f(x)=(2+3+2)2 ( -,0]
)f(x)=(62+7+1)2 [0,+ )
3. , :
a) 3+6
f(x)=
(0,+ ).
) 2 5-14
f(x)=
(- ,0).
) 24 4+1
f(x)=
(0,+ )
) 2 4+10
f(x)=
(- ,0).
4. , :
) 1-|+3|
f(x)=
1 3 .
) 2 +4+3
f(x)=
2 4 .
5. :
a)f(x)=x2 g(x)=x2+1 h(x)=(x-1)2
)f(x)=-x2 g(x)=-x2-2 h(x)=-(x+1)2
-
www.georgefrang-math-trip.blogspot.com
124
6. :
a) 4 4 4
f(x)= g(x)= -3 h(x)=x x x+1
7. :
|a-1|-1f(x)=
x
(-2,-1) g(x)=ax2 .
) .
) f g.
8. :
2x -2 x
-
www.georgefrang-math-trip.blogspot.com
125
f(x)=ax2+x+ >0
: =
: =-
2.
: (- ,-
2]
[-
2,+ ).
: =-
2 f(-
2)=
-
4.
To K(-
2,
-
4) .
- :
: ,
f(x)=ax2+x+ >0
:
yy: (0,)
. f(x)=x2-4x+3.
: =
: =-
2=
-4- =2
21
: (- ,2]
[2,+ ).
: =2 f(2)=-1
To K(2,-1)
:
: y=0 f(x)=0
x2-4x+3=0 x=1 =3
(1,0) (3,0).
yy: =0 f(0)=3.
(0,3).
-
www.georgefrang-math-trip.blogspot.com
126
:
f(x)=ax2+x+ 0
:
-
www.georgefrang-math-trip.blogspot.com
127
. f(x)=-x2-2x+3
: =
: =-
2=-1
: (- ,-1]
[-1,+ ).
: =-
2 f(-1)=4.
To K(-1,4) .
:
: y=0
f(x)=0 -x2-2x+3=0 x=1 =-3
(1,0) (-3,0).
yy: =0 f(0)=3.
(0,3).
:
-
www.georgefrang-math-trip.blogspot.com
128
.
1. :
)f(x)=x2+4x-2
) f(x)=-x2+6x-4
) f(x)=-4x2+4x-1
2.Na
:
) f(x)=x2-2x-3
) f(x)=2x2+4x+3
) f(x)=-3x2-4
3. f(x)=x2-6x+5.
a)Na .
) .
) .
) .
4. f(x)=-2x2+8x-6.
a)Na .
) .
) .
) .
5. f(x)=x2++ (3,-5).
.
6. f(x)=2+(+2)-2(2-1) =-1.
) ,
) .
7. f(x)=-2+(+2)-2 . :
) ,
) .
8. , :
f(x)=x2+(-2)x-+1
:
)
)yy.
-
www.georgefrang-math-trip.blogspot.com
129