algebra 10-klas-merzlyak-pogliblene
TRANSCRIPT
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. . . . .
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I
10
2010
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[373.5 : 372.85512.1 + 517.1]
22.141721.6
52
( 09.08.2010 1/11-7525)
. .
52 : . 10 . -
/ . . , . . -
, . . , . . . X . : , 2010.
415 .: .
ISBN 978-966-474-103-0. [373.5 : 372.851512.1 + 517.1]
22.141721.6
. , , . . ,
. . . , . . 2010
. . , , 2010
ISBN 978-966-474-103-0 TOB TO , -, 2010
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8-9
8-9
1.1.
, ( a + 5 + 7 Va-9\ 7 + 1.2. - + ? - + . ^ -9)( + 9) (a~9)2JW + 3/ 9 +
1.3. * f 2 f -+- l\ * {( + )2 U 2 J (x + yf U J ) 1.4.
( -1
b 2 + b
1
: - 1 + b-b s-b 4
1-
1.5.
~ + + ( x - y f + x y f 6+ + + 2 3 1 ^ ^ (X + )2 - {(X3 - *)(* + 3 + + 2))
1.6.
----* + 5
-
1. 8~9
1- 1- 1-
1.7. + ^ = 1. xy + yz + zx 1 1
1.8. , + r +; r-(x 2+lf-x 2 *( + 1?-1 X -( + 1) 2
2 -+1 , 2(-1) 2 2 2 ( 3 -1) 2 1.9. +; 5 + 5 - .
+ + 1 + +1 8+4+1 1.10.
, , ab 2abc
(a + )(a + c) (b + c)(b + ) ( + ) ( + ft) (b + c)(c + a)(a + b)
1.11.
b-c . - . a-b 2 + _ 2 _ + 2 ( - ft) ( - ) (ft - ) (b - ) ( - ) ( - ) -b - -
1 1 2 4 2" 1.12. ! + = + + -.
1-6 1 + 6 1 + 1 + ft4 i + b2n
1.13. , 2 - - 1 = 0. ,
1.14. , + b + = 1, + + - = 0, a b
2 + + 2 = 1.
1.15. , +- + - = 1 - + + - = 0, a b
2 2 2
1.16.
je 2 (~) + 2 ( - X) + z 2 ( - ).
1.17.
( - Z 2) + ( 2 - X 2) + { 2 - 2).
1.18. , , , + = & + - = . b
, I abc j = 1.
,
1.19. (>/28 - Vl2) /lO+-s/84.
1.20. * U / S 11). W 6 + 1 n/6-2 -/
-
1. 8 ' 9
, - - 1.21. , - -j . V2V2 + 3 272-3
1.22.
t .23.
1.24.
1.25. +^/l3-4>/3.
1.26. (2-/)\/ + 4 + /7-3 /.
1.27. , (4 + /) =
1.28. , / ^ ( + ^)(/0-72) = 8.
1.29. , + . / 9 , 4 ^ 4 + V5
1.30. , f p j ^ ^ H ^ S ^ . V27 -3v l8 + 3v l2-v8
1.31.
Js + 2 10 + 2 ^ + ^ 8 - 2 ^ 1 0 + 2 .
_ r) . aJ + byfb , 2>/& 1.32. 7-jj=r +j= -. We+VfeJ(a-ft) /a + V
1.33.
VWa + Vi/ a-Ja-b-Ja
1.34. [Va8-2a2 + a + j .
, , , . ( 1 + Vl-* , 1-1 + Y -1 . -, 1.35. ; = + : - + 1.
U - * + Vl-JC 1 +-vl + / 2
1.36.
"** --^- ) , 0 < X < 1, ,/ + JC Jl-x
1.37.
-X 2 +X-IJ \
a + V2a+2 J
-
1. 8~9
1.38. ^ ^ .. ^
~ 2 + 0 < < 1.
1.39.
+ + ^ yja + b-2 yfab
yf + yfb\ a~ b b-yfb yfb+a) 2
l + [a + Ja 2-lf ib + ylb 2-lf 1.40. (a + Vaz-l)(t> + Vf>2-l)
1.41. b 2 - B b - ( b - l ) ^ l + 2 ? > 2 f2 + 3&-( +1)Vb2-4+2 Vfc-2
1.42.
I + 2 \fa~l \
+ 2 yja-l
a-2yja~l Va2 -4a+ 4
n . Jx + 4\lx-4 + Jx-4.yJx-4 1.4d. v f M X
1.44. ' :
1) _J> 4 1 2-4 + 4 2-4 + 2
2 )1+-^- + 2 7 + 4 2xz + 7 - 4 2-1'
+ 1 , -2 , -3 , + 4 . 1 1 1 = 4;
-1 + 2 + 3 -4
X 2 + 4 + 4 2 + 6 2 + * + 1 2 + 9 4)
+4 +2 +1 + 3 '
5)| + 2| + | - 3 | = 5;
6) j 2* + 5 = je + 2;
7 ) | * - 1 | - 2 | * - 2 | + 3 | * - 3 | = 4.
1.45. ' :
5 1 1)
*3-1 4+4 + 4 2 (1 - )'
X | + 1 X + 34 _Q.
22 +12 + 10 42 +16-20 3+52--5~ '
8
-
1. 8~9
, 2-1 , -1 - 1 ) + = - + 4; +1 +2 - 1 . . xz+2x + 2 , +8 + 20 2+4 + & , 2+6+12 4 1 }- *
+ 1 + 4 + 2 + 3 * ) ~2 | - 1 ' 6 ) | | 8 - | - * + 1 | + * = 6.
1.46. ' :
11 4 2 - 2 4 - 22 _ ' -3 2 ' ' 2 + 2- 42-2
1.47. ' - = .
2-16 -4 4 + 1.48. ' :
1) (2 - 6 + 5) (3* - )2 > 0;
2) (X2 - 6 + 5) ( - )2 < 0; 3) (2 X ~~ 2) (2 - 4 + 3) > 0;
( + -2)4 (+3) } (-7)(1-3) >{'
3 + 2+3 + 3 ^ -6 + 7
(-2)3 6 ) \ + 3 \ ( - 4 ) > 0 ' 7) (; + 7) /jc + :2-20 > 0;
9) + 1 - 4
) + 1 1 JC-
11) (X 2 + + 1) ( 2 + - 3) > 5;
2 + 3
2+3 12) ( + 3)(2 + 3)-16. 2 f +3 > 0 ;
13) a g + ' v 1 | > n
14) (14-3; X J 2
15) 1 2 - | + - 2 < 0;
16) ] 2 + ] > 2 - X 2 .
9
-
1. 8~9
1.49. ' :
1) (X - 10* + 9) ( + )2 > 0;
2) ( 2 - IOjc + 9) (4* + )2 < 0;
3) ( 2 - 6* + 8) (X 2 - 4) > 0;
4) * ! + 2x2 + 5* +10 ,.. 0 . 2-~6
5) -2 X +2
6) ( 2 - X - 1) (X 2 - X - 7) < -5;
7) (x 2-2x){2x-9)-9-^f^ 1 ;
10) ^ ^ - *
11)\ 2 - 2 - 3\< ~ 3;
12) I X2 + + > ; + 3.
1.50.
( + 4) X 2 + ( + 4) + 3 = 0
?
1.51.
( + 3) 2 + (2 + ) + 1 - 0
?
1.52. , - X 2 ~ (2 - 4 + 3) X + - 2 = 0 .
1.53.
( - 2) X 2 - ( - 4) * - 2 = 0 ?
1.54.
2*2 - ( a + l ) jc + o 1 = 0 ?
1.55. ( + 4) 2 -- 2 + 2 6 < 0 ?
1.56. (2 - 1) 2 + + 2 ( - 1 ) ; + 2 > 0 - ?
1.57. ( - 3) 2 -
- 2 X + - > 0 ?
10
-
1. 8~9
1.58.
2 - 2 - - 2 = 0 1, 1?
1.59. 2 -
2 2 - 2 (2 + 1) X + ( + 1) = 0
X v < < 21
1.60. 2 - 2 +
+ 2 - = 0 [-2; 6}?
1.61. 2 - 4 + 4 > 0 ?
1.62. 2 + - 7 < 0
X (1; 2)?
1.63. '
2 - 2 - ( 2 + 2) < 0 2 < 9?
- + 1
2 + + 1
-
1. 8~9
1.68. :
1) = V l 7 - l 5 x - 2 x ^
+ 3
2) = 12 2-4 3-9-]2-\\.
1.69. :
7 - , . \ \4 2 -19 + 12 '
2) = ij\ X -11 (6) + - 2+4-2'-
1.70. :
1) = ~~ 3) = 2 + 2 + 2; X
2) = + -\ 4) = 5-/-6 + 10.
1.71. : 1 } 3 )
2) = + 4) y = 3-Vx a-2x + 2.
1.72.
1
X -4 + 10 1.73.
2 J X 6 +11
1.74. :
1 X2 1) max 2) min , = (-; -1) U (1; +).
2 + 2 , / 2 - 1
1.75. :
1 1) min- -'-
2) max {yf2x + /+ ) , = [ - 1 ; 2].
1.76.
f : 1) f () = X2 + 4 + 5 , = [-1; 1];
2) f () = X2 - 4, = [-1; ], > -1.
12
-
1. 8~9
1.77.
f :
1) f () = -2 + 6* - 2 , = [0; 4];
2) f () = 2X - X 2, = [; 2], < 2.
1.78. ' -fx +2 + 3 +Jx + S =8.
1.79. ' 2 + Vx-1 + 2 -Jx + 2 =17.
1.80. ' | | + j -2 \ + !-1=2.
1.81. ' 2^- 2 = + 4.
1.82. :
2 - 1 | 1 ] ! , | ,
* -4 /1-jc-Vl + jf
1.83. :
14 X5 1) y~-f==
V2-je - /2 +
2) = ^ + j-; (4:-2)5 (4x + 2)s
2:+ 1 2-1 3) = ; j .
- + 1 +3 + 1
1.84. , D (f) = 3L , = f () + f (-) = f () f (-) , - f () - f (-) .
1.85. :
1)/ = (|*|-1) 2 ; 4) =\+2\-1;
2) y = Jl-\x j; 5) =\ VI* 1-2-11;
3) 6) ,=|>/-2|.
1.86. :
1} V = W T 2 ; 4) = ( | - 2 | + )2;
2) 1
4 5) 1 - 2
1
) =+\|_2; 6) /=|72^-2|
13
-
1. 8~9
1.87. 1.1 = 2 + + . a, .
%
) )
1.1
1.88. 1.2
= 2 + + . , .
1.89. , -
j 2 - 6 | \ + 8 | = .
1.90. , -
|jc2 + 2 | j c - 2 j - 4 | = a.
1.91. , 1.3
= 2 + + = + ?
' 1
1 [ ) )
1.2 1.3
1.92.
4 24
(* + 1) 2 4+\ + 1.
1.93. ' :
1) 2 - 12 + 4 2-4+1^0; 2) | \ + 2 = 4- 2.
14
-
1. 8 ' 9
1.94. ' :
1) 2 + 2 2 - 6 - 24/ + 9 = 0; 2) 9- 2 = V 3 + I -
1.95. :
1) (X - 3) = ( + 5)2; 5) - + I X | = 1;
2 ) 2 = \\-, 6)\\-3 = 9- 2;
3) x + 2 = J\y\-l; 7) =
1-
4) {/- [ = /;
1.96. :
1) (X - )2 = (X + 2)2; 5)| + 11 + | - 2 | = 2;
2)\\ = 2-, 6) (I X - )2 + ( ^ - )2 = 4;
3) x+a-JfFH: =
4) |jr|-l = > ;
/ 2 - 1
1.97. :
1) X > + 2 - 2; 3) (X + ) | \ > 0;
2) I X [ < 2 - 2 ; 4) ( 2 + 2 - 1) 2 < 0.
1.98. :
1) < I X - + 1; 3) (X - ) [ X < 0;
2 ) | - 2 | - | / + | > 2 ; 4) * + ' ~ 1 > 0 .
1.99. '
: \ + 2>1, |
2 + 2
-
1. 8~9
1.103. ,
1.2 .3 + 2-3-4+3-4.5+ . . . + ( + 1 ) ( + 2 )= ( + 1 ) ( " + 2 ) ( + 3>-
4
1.104. , - + + - + ...+ + : = 3 . ^ 2 4 8 2 2" 2"
1.105. ,
1.106. , 5" + 2 + 62 + 1 31, N.
1.107. , 7 52" + 12 6" 19, N.
1.108. , 14-3" + 9 - 7 : 23, N.
1.109. 2" > 2, N, > 3.
1.110. 2" + 4 > ( + 4)2, N.
1.111. 3" > 3, N, > 4.
16
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2.
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1. ,
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2. . *Intermezzo
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3. .
4. .
5. .
1 4 , 3 . -
2 5 , .
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5 .
-
: , , , D .
17
-
2.
, :
= { };
s {5 > 7};
= { 2 }.
- , . -
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1, 0.
, ,
f, -
, {0, 1}.
f . ,
= {10 5}, f () = 1.
', , , ...,
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{5 >3}, {5 = 3},
= {5 > 3} -
* .
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{ },
N = { }.
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: { ,
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= {10 5 10 2).
: = {10 : 5} = {10 : 2} -
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18
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2.
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1 X X
X 0 0
0 X 0
0 0 0
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0 1 -
{, X},
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-
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= { },
{ }.
V = {
}.
':
(1815-1864)
,
V X X 1
1 0 X
0 X X
0 0 0
19
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2.
:
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' ..., ,
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( )
=> (-
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:
=
1 1 1
1 0 0
0 1 1
0 0 1
, -
(-
): , (
); ,
( ).
, -
,
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,
.
,
{ 2 X 2 - 5, }
{ 2 x 2 = 5, " }
.
.
X : 10, : 5
.
20
-
2.
= 1, :
1 : 10, 1 5, -
.1(11 .
~ 5, :
5 10, 5 : 5, .
. ( -
) , -, , , , .
: (-
: , ).
:
1 1 1
1 0 0
0 1 0
0 0 1
:
= {2 = 5} = {2 > 5}.
= {2 - 5 , 2 > 5}
,
.
,
.
. -, , , ,
.
: (:
>> , ).
:
1 0
0 1
,
.
21
-
2.
, ( V ) , => , -
.
. ( ) V . '. :
( ) V
1 1 1 1 1
1 1 0 1 1
1 0 1 0 1
1 0 0 0 0
0 1 1 0 1
0 1 0 0 0
0 0 1 0
0 0 0 0 0
. -* , .
= .
: = , => -
.
, , A=>B = A v - -
.
V :
=> V 1 1 1 0 1
1 0 0 0 0
0 1 1 1
0 0 1 1 1
, Av i i ,
. , .
,
. -
,
U = U + ^ +
22
-
2.
\ = \ ^
n ( U ) = ( ) U ( ) (b + ) = ab +
-
. , V
V ,
V = V .
,
V V .
, , ,
= ,
( AB) = ( ).
,
' 2.13.
, -
.
, , ,
.
, , ,
' (-
, ,
, ). -
.
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.
A v -
:
V 1 0 1
0 1 1
, f -
, - / ( A v A ) = 1.
, A v A , -
.
. , -
, , .
2
-
2.
, ' -
2.14.
,
, .
2.1. :
1) 5 > 5;
2) X < 5;
3) , sin 30 cos 45?
4) ABCD , AB = CD;
5) 1 ;
6) , 5 ;
7) ;
8) g ?
2.2." f . f (), :
1) = { 2 };
2) { X 2 + X - 1 = 0 };
3) = {- };
4) A = {V5Q};
5) A s {Q };
6) { = [;] };
7) { = }.
2.3. :
= {5 < 6}, {6 }.
, :
1) ; ) = > ; 5) ;
2) V ; 4) => ; 6) .
2.4. :
A s {2 = 3}, = {2 }.
, :
1) ; 3) => ; 5) ;
2) V ; 4) ; 6) .
24
-
2.
2.5.* f , ,
/ () = 1. , , f:
1 ) / < ) ; 3 ) f ( A = > B ) ; 5 ) / .
Z ) / ( A V B ) ; A) f { );
2.6.* / , ,
f()=1 . , , f:
1) / ( ); 2) f ( V ); 3) f ( => ); 4) / ( ).
2.Y.* f , .
f (), :
1) f ( ) = 1; 3) f ( => ) = 1 f () = 1;
2) / ( V ) = 1 f () = 0; 4) / ( ) = 0 / () = 0.
2.8." / , ,
/( ) = 1. :
1) f ( v b ) ; 2) ^ ( => ) ; 3 ) / ( ) ; 4 ) = > ) .
2.9.* :
1) =>; 3) ( ) => ; 5 ) ()=>.
2) ( V ) ; 4) ( ) ( V );
2.10.* :
1) V ; 3) ()=> ;
2) =>; 4) (VB)a (BVC) .
2.11.* N
, 2.1. :
= { };
= { }.
, ,
f:
1) / ( ) = 1; 3) / ( V ) = 0; 5 ) / ? (VB) = 0.
2) f ( ) = 0; 4) / ( ) = 1;
0
2.1
25
-
2.
2.12." N -
, 2.2. :
{ };
= { }.
, ,
/:
1) /{ab)=1; 2) /(vb) = 0; 3) f ( ) = 1.
0 0 ( -0N 2.2
2.13.* , :
1) = ;
2) = ;
3) A V A = A ;
4) V = V ;
5 ) V ( V ) = ( V ) V ;
6 ) ( V ) = ( ) V ( ) ;
2.14.' , :
1) => ; 5) ( => ) V (
7 ) V ( ) = ( V ) ( V ) ;
8 ) V = ;
9) = V ;
10) => = =>;
11) A B = (AAB)V (AB) .
);
2) ; 6) (=) =>;
3) => ; 7) { => ) => ;
4) => ( => ); 8) ({ => ) ( => )) ( => ).
2.15.** ' :
1) '; 2) .
2.16." ' '
.
2.17.* * :
*
1 1 1
1 0 1
0 1 0
0 0 1
* V, -
.
26
-
2,18.* * :
*
1 1 1
1 0 0
0 1 0
0 0 1
* V, .
2.19.* :
1 1 0
1 0 0
0 1 0
0 0 1
, ' J- ',
' .
2.20.* .
1 1 0
1 0 1
0 1 1
0 0 1
, ',
' .
',
,
'.
, ' -
, ,
,
1. -
1 :
, .
27
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2.
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, 2,11 2.12 (
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).
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, , F,
, , :
F 1 1 1 0
1 1 0 1
1 0 1 0
1 0 0 0
0 1 1 0
0 1 0 1
0 0 1 0
0 0 0 0
, F
( ), . -
, F, ',
' .
28
-
3. .
1 , 1 , 0 ( )
: AB . -
, -
, ,
1,1, 0. -
(-
) : AB .
, F
' :
F = ( ) V ( ).
, -
-
.
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(1897-1954)
0 - - 0
,
,
,
.
,
.
.
:
;
5;
N 2;
X + = 1.
, -
, .
29
-
2.
,
.
, . -
.
, , :
() = [ };
() = { 5};
() = {| < 2 };
D (; ) {* + = 1}.
, .
- -
, . :
(2) ;
(4) ;
(5) ;
D (0; ) .
(), X .
, ()
().
, ,
() -
() Z, () , -
D (; ) .
, (),
, ,
() . -
()
.
, () ,
. () :
= [-2; 2].
, ,
, -
. , () { - = 0}
, a Q () = {2 + 1 ='0} ,
R.
. , -
X () 20092010 pp.
17 ,
(), . -
30
-
3. .
{
}.
A () () .
. () () -
, , - .
{) ().
, () {%/ >\ () = {| | = }, (je) s
s (). , = = [0;
-
.
.
. ' () () , .
' () () : () ().
, () = { > 5}, () = { < 7}, '
* >5,
' , (5; 7], -
() (je). () () = {5 < JC < 7}.
. ' (*) () , A U .
' () () :
() V ().
, () { < -5}, () = { > 5}, '
5,
' , (-; -5) U (5; + 5}.
. () ()
,
, ()
, () .
31
-
2.
() () -
:
() => ().
, () = { > 5}, () -
= { > 3},
() () .
, ,
A (jcJ => () . ,
, , -
(. 3.1).
. () () - ,
,
() () .
() () :
() ().
,
{) = {* - X - 2 = 0),
() = {( + 1) ( - 2) (2 + 1) = 0},
() () .
. () -
, \.
() :
). ____ , () = { > 5}, () = {
-
3. .
A (je) ( () V ()) s ( () ()) V ( () ()).
I (,
> 5 ,
X < 7,
>11
>5,
|5,
1 > 1 1 .
() .
, .
A ()
X .
()
.
:
(Vx ) ().
:
( ) ().
V ( All
) . -
: ,
, .
3 (
Exist ) .
: , ,
.
, () = {| \ > 0},
R, , :
( ) () = { | > 0} -
;
( ) () = { , X > 0} -
.
, -
.
-
.
,
.
() N. -
(Vn N) ()
(1) A ((VA N) A (k) => A (k + 1)),
:
(Vn () s (1) A ((VA N) A (k) A (k + 1)).
33
-
2.
, , -
:
- ()
().
,
(Vx ) () => ().
, .
, -
3.
, -
N:
() = { 6};
() = { ': 3}.
:
(Vn N) () => ().
,
(Vx ) () => (X ) ,
() , ()
. '-
.
'
. ,
: ABC Z = 90, AB 2 = AC 2 + 2.
, -
, .
. -
, :
( ABC) s { Z C = 90};
Q ( ABC) {AB 2 = AC 2 + 2}.
(VA ABC )(A ABC) Q ( ABC),
: - ABC, -
Z = 90, AB2 - 2 + 2.
() (), ,
. ,
, () => ()
. ,
(Vx ) () => ().
34
-
3. .
, , 6,
, 3.
(Vn N) () => ().
1.
(V* ) () => ()
(Vx ) () => ()
.
, -
.
7-9 .
(Vx ) () () ()
(), ()
().
()
() .
(Vx ) () ()
. :
() ()
()
, (). -
.
:
,
,
;
- ,
.
2. (Vx ) () => () -
(Vx ) () => ().
, :
-
.
.
: -
.
: -
.
35
-
2.
, : - -
.
, =>
=> (. 2.13 (10)),
{ => ) = ( => ) . ,
,
(Vx ) () =t> () = (Vx ) () ().
. -
,
.
3.1. :
1) ( + 1)" - 1 , N;
2) - X R X 2 + X + 1 = 0 ;
3) ;
4) , 5, N;
5) -
0 1;
6) , ;
7) .
3.2. [-2; 3)
A (JC) = { }.
.
3.3. [0;
() s{x 3 - = 0}.
.
3.4.
R (; ) ^ (2 + 2 = 1} .
, -
.
36
-
3. .
3.5."
() {[X] > },
.
"3.6.0
B(x;y) = {jx 2y 2 =\,
.
3.7.* () = { - 5 = 0},
Q () = { + 2 = 0}. , :
1) (X) A Q (X); 2) (X) V Q ().
3.8."
() s{x* 5}, Q () = {* -2}.
:
1) P(x)AQ (X); 2) () V Q ().
3.9.* () = { 10}, () = { : 5} N,
() => ().
3.10.* () = {| | = -1}, S () = { + 3 = 0} -
R.
() = S ().
3.11." () { > 2}, Q () =
s { > 5}. :
1) () => Q (); 2) Q () => ().
3.12.* () = { > 2}, Q () =
{ < 5}. :
1) (X ) => Q (X) ; 2) Q (X) => ().
3.13." ()
(), (. 3.2).
:
1) () (); 2) () V (); 3) () => ().
0
)
. 3.2
37
-
2.
3.14/ , R, -
:
() = (- = 2); (X) = {[] > };
() = {( - )2 > 0}; D () {sgn (2 - 6 + 9) = 1}.
3.15." ,
, :
(; =\ab\}; (; b) = {| ab \ = ab);
(; b) = [ab > 0}; D (; V) = {ab > 0}.
3.16.* (), () () . -
, :
1) ( () ()) () s () ( () ());
2) ( () V ()) V () () V ( () V ());
3) () V ( () ()) = ( () V ()) ( () V ());
4) () () = () V ();
5) () V () = () ().
3.17.* :
1) ( R) > ; 4) (Vn N) (3 - ) 6;
2) ( R) j < 0; 5) ( R) sin 2 = 2 sin .
3) (Vn N) < 2;
3.18." () { }
. :
1) (Vp ) (); 2) ()(); 3) ( )
3.19.* :
N > 1 ( N) (( : ) => ( = 1 V = ))?
3.20.* -
5, 25
, , -
.
3.21."
,
, ,
.
38
-
= = 2 . = ", N, .
N, - , - - EL
, = 0.
- ", N, - : - .
: = 2k, Pi,
, k = 1 = 2, -
8 .
- X 2 '-
,
' .
, - > 0 2k X, = .
, = ,
, [0; + 0.
' , (->; 0) (0; +)
' = , .
39
-
3.
- , , .
, -
(-)2* - X 2".
Xj 2 , ( 0 ] ,
2 (- ~ > 0. -
, (Xj)2* > (2)2\
x f > x f .
, - ", , -
(-; 0]. ,
[0; +=).
= ",
1
. 4.1
1 1
1
J if X
. 4.2
= ", (. 4.1). ,
= X 4.2.
: .
, = 1 = ,
7 .
= 2k + 1, k N.
, - Ob 4
X, = .
> , = ",
, R.
X < 0, 2 + 1 < 0; > 0, 2 + 1 > 0.
, 0) (0; +>) -
= ", .
4j> = ", , .
, -
(-)2* + 1 = ~x 2 k + \
40
-
4.
2 , < 2. -
,
( ^ 2
, = , ,
.
= xf, , > 1 (. 4.3). , = *
4.4.
= X -
= >
, > 1
_\
7
/
1
3 J L X
/ /
VI
- ) L 1 X
. 4.3 . 4.4
~
= ", N, N, > , [0; + 0.
, (0; 1) = - , - (1; -f) (. 4.5).
m i n , , , - 4.5,
> > 1, >0
. 4.5
41
-
3.
, 4,6. m i n
(. 4.7).
, m i n > , > > 1
. 4.6 . 4.7
= ", N, -
.
R
[0; +)
= 0 X = 0
> 0
(-; 0) (0; +)
< 0 {-; 0),
> (0; -)
/
(->; 0],
[0; +)
42
-
4.
1.1. = *
: 1) (2; -12); 2) (-3; -3)?
4.2. = 3
: 1) (3; -18); 2) D (-2; 64)?
1.3. f () = 19. :
1) f (1.4) f (1,8); 3) / (-6,9) f (6,9);
2) f (-7,6) f (-8,5); 4) f (0,2) f (-12).
4.4. f () = 2 1. :
1) / (20) f (17); 2) / (-44) / (1,5); 3) f (-52) / (-45).
4.5. f () = 20. :
1) f (3,6) f (4,2); 3) / (-2,4) f (2,4);
2) f (-6,7) / (-5,8); 4) f (-15) f (2).
4.6." f () = 50. :
1) / (9 ,2) / ( 8 ,5 ) ; 3) f (19) f (-19);
2) / (-1) f (-1,2); 4) f (-7) f (9).
4.7. " = 1600, :
1) ;
2) ?
4.8. ' :
1) X 6 = 2; 2) = -3; 3) 7 - 9; 4) 6 = -10? 4.9. ' :
1) X5 = 32; 2) = ~~-\ 3) X = 81; 4) " =-16.
4.10. ' :
1) X = -27; 2) = 0,00032; 3) = 64; 4) 8 = -1.
4.11." :
1) = = 2 4; 2) = *1 = -27.
4.12. = 5 = 3.
4.13. :
1) X 8 = X + 1; 3) X 4 = 0,5 - 2;
2) X5 = 3 - 2; 4) 3 = 2 - 3.
43
-
3.
4.14.* ' -
:
1) 2) = % = 2-0,5 X 2.
= , [2--3 = 0;
4.15. ," =, x t = 2, : 1) -
; 2) ?
4.16.* ? >, x t > , :
1) ; 2) ?
4.17. 1 > 2, ">, :
1) ; 2) ?
4.18.' :
1) 12 = - 6; 2) 24 = 2 + 7 - 8?
4.19. - X6 = 9 3?
4.20. /, R,
: f (0) = 0, f (-1) = 1, f (2) = 1024? 4.21. /, R, -
: f (0) = 0, f (-1) = -1, f (3) = 243? 4.22.* :
1) = 3 - 1; 4) = (X - ) 4;
2) = (X + 2)3; 5) = ( + )4 - 1;
3) = X4 ~ 4; 6) = -X3;
4.23/ :
1) = X 3 + 3; 4) = (X + )4;
2) = (X - )3; 5) = (X - )3 + 2;
_L . * ., _ ^..
7) = -| 4 ;
8) = j X31;
9)y = (|xj + )4.
7) = - X 4 ; 8) = (I X j - 2)3;
6) = ^" 9) = I X + 1 3 ) = X 4 + 2;
4.24/ :
[4, 0 ;
|xJ, -1.
, -
.
44
-
4.
3, 0 .
[ , -
. 1.26.* :
1) = 1 * *4; 2) = I X I X 4 + 5.
1.27.* :
1) = X X ; 2) = I X I X 4 - X 5 .
4.28." f () X s
:
1) [0; 2]; 2) [-2; -1]; 3) [-1; 1]; 4) (-; -2]; 5) (-2; 1).
4.29.* f(x) - 6
:
1) [-13; -1]; 2) [-2; 1]; 3) [1; 4) (1; +-).
4.30.* -
f () = , :
1) f ( - 4 ) > f ( - 2 ) ; 4 ) f ( 4 ) > f ( 2 ) ;
2) f (-4) < f (2); 5) f (-4) > f (2);
3) / (-4) < f (-2); 6) f (4) > f (-2)?
4.31." f , f (3) = 21 -
X R.
4.32." f , f (6) = 24
X R.
4.33." f , f (4) = 2 0
X .
4.34." ' :
1) X 1 1 + X 3 = 2; 2) 24 + 10 = 3.
4.35." ' :
1) 43 + X 7 = -5; 2) X 6 + 8 = 4.
4.36."
f () = X8 (-1; ], > 1,
4.37."
f () = X6 [; 2], < 2.
4.38.* ' 517 - 8 = 2.
45
-
3.
4.39.* ' 15 + 24 = -9.
4.40.*
/ , f2, ..., /, , -
f (fn ()) = fkn ().
4.41.* R
fv f2, .... N , n e N , -
f (.) fa () fk + n ().
' ,
, -
.
, ' . -
, ' , .
, .
. -
,
f(x) = f (-X), D{f),
, -
. ' - .
:
/ ( + ) - / () + X, X R, ; (1)
f (je + ) = 2 f {) + X - , X R, R. (2)
' (1).
f ( + ) = f () + X -
= 0, :
/ ( * ) = / (0) +
f (0) , ,
' (1)
f () = X +- , .
,
, f () = +
(1). .
f () = + -
(1), :
( + ) + - { + ) + .
: f () = + , - .
46
-
, ' -
',
'.
. '
(2).
, = 0.
f () = 2/ (0) + .
, ' (2)
f () + , .
. -
f (JC) = X + (2),
+ + = 2 ( + ) + - ;
= 0.
, f () = + -
(2) : f (,) = .
: f (JC) = JE.
.
, -
.
. .
' :
f ( + ) = f(x) + f (),
/ () = / ( * ) + / (), f ( +) = f(x)f (),
f () -/
-
3.
, , , -
f () = 5.
: f, IR,
:
1) f ;
2) f (2) = 32;
3) f () = f () f () > 0, > 0.
-
,
f () = X 5.
4.42. /, R, -
f () + f () = X + .
4.43. f, , R -
f ( + 1) = f () + 1.
4.44. f, , -
/ ( + f ()) = ( - 1) / ().
4.45. f, R f () = 5,
:
1) (2) = 32; 2) f () = f () f () X > 0, > 0?
4.46. /", R, R -
f (2 - 3) + 12 = f (2) + f ().
, = ", Z, - .
. ,
' .
= (-=; 0) U (0;
{1}. -
5.1.
48
-
4.
1
1 , = *
0 X
= X
II < N .
,
= 1, -, X
. 8 .
= ~ -
- . , - - 5 - 1 "
. = ~ , N, (-; 0) U (0; +>).
, .
= ", N,
:
.
: = 2k, k N.
: ' 2" -~ \-
,
' , 0.
, - > 0 -2 k X, = .
, = ~",
, (0;
^ , 0) (0; +>)
= ",
.
= X , .
, -
(-)- 2* = = 4 = ( - ) *
2 , ^ (->; 0),
2 (-; 0) Xj < . - > - > 0. -
, 0
-
3.
^ , = ", ,
(-; ),
^ , = ~ ,
, (;
, X
k N, . -
= - k N,
, .
,
,
.
-
= X ", (. 5.2).
, y = ~ j 5.3.
\ = ' ,
-
- 1 0 1 *
. 5.2
/ \ / \ X L 1
/ mm -1
X -(1 0 1 X
. 5.3
: n. = 2k - 1, k N.
, - * 0
X, < 2 1 1 = .
, = ",
, ( 0 ) U
U (0; +).
X < 0, * , 0, }. , >0. X X2" 1
^ , (->; 0) (0; +)
= X .
= X , .
50
-
4.
, -
(-)-'2*-" = = (-) - * 2 * 1
, 1 0),
2 0) < 2. -1 > -2 > 0. -
. 1 ^ 1 , < ;
Xj 2 -' ^ 0 (->; 0). -
, (0;
^ , ", ,
0) (0; +>).
= ", (. 5.4). -
, -~ 5.5.
= = ", N, N,
> . , ,
51
-
3.
= = ", m N, ( N,
> , , 5.6, 5.7.
, >
. 5.6
m i n , >
. 5.7
= N,
.
0) (0; +) 0) U (0;
(0; (-; 0) (0; +=)
_ -
-
> 0
(-;' 0) (0; +-)
< 0 0),
> (0;
/
(-; 0),
(0;
(-; 0) (0; +)
52
-
5.
5.1." = 4 :
1 > ( * ) ; 2) |-2; 3) ( | ; 8); 4) - ) 7
5.2." = 5 :
1) (0; 0); 2) (-1; -1); 3) ); 4) D (-3;
5.3. = 3
: 1) (-5; 20); 2) |2;
5.4. = 4
: 1) (3; -3); 2) (-2; |)?
5.5. f (:) = ~19, : 1) f (1,6) f (2); 3) f (-9,6) f (9,6); 2) f (-5,6) f (-6,5); 4) f () f (-10).
5.6." f () = ~25. : 1) / (18) f (16); 2) f (-42) f (2,5); 3) f (-32) f (-28).
5.7. f () = "16. : 1) f (1,6) f (2,2); 3) f (-3,4) f (3,4); 2) f (-4,5) f (-3,6); 4) f (-18) f (3).
5.8. f () - 40. : 1) f (6,2) f (5,5); 3) f (24) f (-24); 2) f (1,6) f (-1,7); 4) f (-8) f (6).
5.9. " = 2500, :
1) ;
2) ?
5.10. ' :
1) 6 = 2; 2) X 5 = 0,3; 3) ~7 = -3; 4) ~8 = -2?
5.11. :
1) = "1)1; 2) = ({X - 2 ) 2 )
5.12. :
1)
-
3.
5.14. :
1) = X ~ X ; \ -2 1 2) = X = -.
5.15," = 7 / = 4.
5.16." :
1) = 2 + 2 ; 4> = jc 3 - 1;
2) = (* - )"2; 5) / = ( ~~ )"3;
6) = * 3; )
7) = "31; 8 ) = - 1 3 ;
9 ) = - | .
5.17." :
1) =
2 ) = 4 * 5 ;
3; 3) = 1 .
,
.
54
-
6. -
.24.'
f (JC) = JC :
1) f (-2) > f (-1); 3) / (-2) < f (-1); 2) / (-2) < f (1); 4) f (2) < f (1)?
5.25." R\{0} f ,
= X \{0}.
5.26." \{0} f , / (4) = l JC R\{0}.
5.27." \{0} f ,
= JC30 JC R\{0}.
5.28.* 1R f ,
f (X ) = jc28 X \{0}.
-
, ( -
) , .
- ,
N, > 1.
. .- , N, > 1,
, - .
, ' 32 2,
25 = 32; -64
-4, (-4)3 = -64; 81
3 -3, 4 = 81 (-3)" = 81.
, - " = ,
N, > 1, - ,
- .
, = " -
, ,
" = - .
6.1 : -
= " =
.
55
-
3.
:
, 1,
- - , .
, > 1, :
(: - ). \ - . ,
, .
, ^32=2, ^ 6 4 =-4, V = 0.
. , /2 : -
2.
, 2"*, k N, - ,
- , -
0 - , , - [0; +) .
, > 0 , .
(. 6.2). < 0, = = ; - 0, ; > 0, , .
:
, < 0
- ; = 0 -
0; > 0 ,
- .
56
-
6. -
~.> 0 =
J . t
X
= , < 0 1
1 ' =,> 0\
1 11 2L
V -fo 0
= , < 0 X
_
, > 1
. 6.1 . .2
, " = > 0
' ' . -
- .
. - (
' , N, > 1, '
, - .
- ' -
:
, / = 3, 3 > 0 4 = 81;
/4 =2, 2 > 0 2 = 64;
'3/0 = 0, 0 > 0 10 = 0.
, > 0 = , N, > 1, trf = .
, -
- '
: \[.
2i[, k N, -
. , -
.
-
' " - , N, > 1.
^ , - -
x = \fa.
^ > 0,
: = rf, 2 = -yfa.
57
-
3.
= 0, X = 0.
, = 7
X = 5 :
- , - ' :
yf>0 (%/) =.
, (/7) =7.
, - N
2k + XI 2* + 1/ v - a = - va
2"*[ = , .
=
: va j =-1, va ; =-a.
' .
, = ^ 2 = ->/2.
6.1. ( ):
1 ) \27 = 3 ; 3 ) V - 2 7 = - 3 ; 5 )
2 ) = 4 ) V L 6 = 2 ; 6 ) ^ 3 2 = 2 ?
6.2. , :
1) 2 8;
2) 3 -
81;
3) -3
81;
4) 10 ' -
10 000.
6.3. :
1 ) V 2 5 ; 3 ) ^ 0 , 0 0 1 6 ; 5 ) ^ ; 7 ) 4 < / 0 , 1 2 5 ; 9 )
2 ) / 2 1 6 ; 4 ) V - 0 , 0 0 0 0 1 ; 6 ) 8 ) | ^ - 2 4 3 ; 1 0 ) .
58
-
6. -
.
1
. 2
11.5.
1
0.0.
1
0.7."
1
2
0.8."
1
2
6.9.
1
2
0.10
1
0.11 1
2
0.12
1
2
7- 4) -8 5 . V 1024'
5) ^27*;
6)
5) l ^ t f ;
6)
:
\/343; 3) 0,5 /64;
58 8 1 '
:
(/); 3) / ; 5) -*; 7)(-3*/l)4; 9 ) ^ ^ 48 ^ .
3 ; 6 ) (5^ ) 3 ; S ) ^ ) 6 ;
:
) 8 ; 3) ;
4) (|^45)%
:
0,3 ?/000-5^/256 + 6 - ^ ;
+ (-2 /)2 - V128;
V 256 32 16 \2 / :
200 ,001 - /-0,00032 - (-4 V f ;
v/8000 - 4 /7^ - (-^8 f +
-
3.
6,13." ' :
1) [ - 9; 4) \[ = -6; 7) %2 + 7 = 0;
2, 5) [ = 2; 8) /2 + 7 = 0;
3) 3; 6) \[ =0; 9) ^/2* + 7 = 7.
6.14.' ' :
1) /* =-2; 3) 2; 5) -2 = 0;
2) tfx =-2; 4) /* - 2 = 0; 6) -2 = 2.
6.15/ :
) 2) - . 6.16/ ' :
1) - 82 4 + 81 = 0; 2) *6 + *3 - 56 = 0; 3) 1 2 + 6 - 12 = 0.
6.17.* ' :
1} - 25 3 - 54 = ; 2) *8 + *4 - 48 = .
6.18." :
)
6.19." :
) ^ s 2) ^ - . \x2-S6 v + 4 6.20." ' :
1) ( 2-4)[ + = 0; 2) (-1) 1$ 2-2-3=0.
6.21." ' :
1) = 2) ( + 2) \/*2 + 2* - 3 = 0.
6.22." :
1) = 2) =
6.23.* :
1) ~ (/*)4; 2) y = {^2 + ^f
6.24." , : 1) \2; 2) ^/.
6.2 V , : 1) :[\ 2) /2.
6.26." - :
1) (-)[ + 1 = 0; 2) (~){[ +1) = 0; 3) ( * - ) ( ^ - ) = 0.
60
-
7. -
.27." -
:
1) ( + 1)/- -0; 2) (-1)({/-) = 0.
-
, -
.
7.1 ( ). - k N :
0 2" = X. : \\>0 ( j o |)2* = 2".
7.2 ( ). >0>0,, > 1,
. \=, X > 0,
, > 0 " = .
: 0 >>0. yf-yfb> 0. ,
(/ /)" = (/)" {tfb)" = ab.
, < 0 < 0, N, > 1,
'ifab = yj. yjb.
7.3 ( ). > 0 > 0, N, > 1,
V b /
.
, < 0 b < 0
61
, N, > 1 , V& ~ sf^b'
-
3.
7.4 ( ) . > 0, N, k N,
> 1,
. k = 1, , , -
.
k > 1. : = ^ ^ . . . . V= qfo ... = tfcf. ^ - ' \ j
* h
7,5 ( ) . > 0, N, k N, > 1, k > 1,
. : / >0.
, =(V)*=o.
7.6. > 0, N, k N, > 1,
. k - 1, , , -
.
k > 1. : = ]* =
1 : 1) ^/16 /2; 2) 3J 375
'
1) , :
^ . ^ 2 = ^16-2=^32=2.
2) (), -
: /4 . 2 = 3 _ 2
. V375 \25 5' *
2 : 1) l\f*; 2) 3) 4) / V , ; > 0 < 0.
'. 7.5 7.1.
1) 3 , > 0.
62
-
7. -
2) = |.
3)
* ) , > 0 < 0, :
=() = \\ = \\\\ = (-) = -.
3
= +.
IJC |,
- X + X.
X > 0, = X + X = 2.
X < 0, = -X + X - 0.
\2, >0,
[,
-
3.
7.4. :
1) >/25 /;
2) W
3) ^ 2 1 5 - 5 3 . # - 5 4 ;
4) 102
5)
6) ^2/7+10.^2/7-10;
8) Vl25-Vl8-V2. 0 b > 0:
1) N/252;
2) V m ;
3) ^625a 'V ;
4) \729 5 1 .
7.7. , m > 0 > 0:
1) \49 ;
2) \125 1 5;
7.8. :
3) V0,000064m3V
4) ^ V 7 .
1) 3) 2\/ft; 5) ' ^ V 7 ; 7) ^81; 9)
VTTIV' 2) 4) 6) 8)
7.9. :
1) 3) ^
2) 4)
5) 7) \/4;
6) \/27; 8)
7.10." %/a :
1) ; 3) ;
2) ; 4) .
7.11. sfb , b > 0, : 1) ; 3) ' ;
2) ' ; 4) .
64
-
7. -
7.12.' :
1) $f*=a;
2) tfa*=-a;
3) \Ja* = ;
4) =
5) fa-5)3 )';
6) yj(a - 5)4 = (/^)4;
7) 2)4
8) ^/ ( -1) = / - ^(1 - );
9) = 4;
) =
7.13.' :
1 2 ) ^ = -5; = 4) =
7.14.* :
1) 3) = 5) =
2) ifb = $P-iPb; 4) =
7.15." :
1) $ -4 = ]~2- + 2;
2) %](-3)(7-) = -3-$7-;
3) / ( *-- ^^^-/-;
4) ^( + 1)( + 2)( + 3 )=^^-^ + 2-^/ + 3?
7.16." ,
:
1) 3) V18; 5) yjm16;
2 ) 4 ) , 4 ^ ; 4 ) ^ " ; 6) ^ (-5) ' 2 .
7.17." ,
:
2) 3) 4) - ) . 1) 1 , 2 ^ ;
7.18.* :
1) /\ > 0;
2) < 0;
3) ^16\ > 0;
4) \/256fe8, ft < 0;
5)
6) j o ,25b u , > < 0;
7) - 0 f > 0;
8) 70,01 aV\ a < 0, > 0;
9) -1,2 /4, < 0;
10) 4/ , 28 32
/
, > 0, b < 0.
65
-
3.
7.19.' :
1) ^62524; 5) -0,1 000 00042, > 0;
2) 0; 6) 1$3660, < 0, < ;
3) -5 yfx*, < ; 7) ab2 ^ V V 4 , > 0, < ; gm3 4 40
4) ^p 3 0q 4 0, > 0; 8) ]}25' < 0, k > 0.
7.20.* :
1) 2)
7.21.' :
1) \[* + , > ; 3) s/o^ + V4;
2) < 0; 4) -2 - /7.
7.22.* ' :
1) + 4? = + 4; 3) $J(x 2-2x-3f = 3 + 2- 2.
2) t](l-3x) 8 =(1-3) 2-,
7.23.* :
1) ^/(-2)3; 2) 3) ^ - S f i 4)
7.24.' :
1) 2) - J E f ; 3) - ) 3 ; 4) ^ / 7 - ) \
7.25.* :
1) = 7-; 4) y = !](x-2) s; ) = - + 2 ;
2) = 2 + 5) = 7) = .
3) =
7.26." :
1) =-2; 2) y = 3) =
7.27.* ' :
1) = -4; 2) 3) 2& = +3.
7.28.' ' :
1) /8 = + 8; 2) 1^ 2=6:-10.
7.29.* '
-
8. , /-
, 9 it-
,
V48:
*/8 = 4/16.3 = ^16.^/3 = 2^3.
, /48
2 /. -
- . - - 2.
:
2 ^3 = 16 . ^3 = 16 .3 = ^48.
.
1 - : 1) \250;
2) \162; 3) ; 4) tf-b"; 5) ^ /V, < 0. '
1) , , -
, :
^/250 =$/125.2 = 5^/2.
2) \/628 = /818 2 = 2.
3) 3 , & > 0.
4) 3 , b < 0.
" = ffi^bf = 5 = -.
5) 3 , > 0.
2 : 1) -2 2) 7 3) 4f7i 4) >^| .
'
1) -2 \/ = \/4 / = \/92.
2) ^ 0, W = = *; < 0,
atff -tff = -7*.
67
-
3.
3) 3 , > 0. 1 ^ / = = fc
4) 3 , b < 0.
3bfl= -Vsib 1. f l =^81 b 4 . (_|) =
: 1) ^54 + 3/60-2000; 2) ^ 4 ^4 ;
3) /4->/7 -^23+8>/7.
'
1) :
+ - ?/2000 = 27 2 + *}8 2 - / 2 =
= 3 $2 + 2 /2 -10 /2 = - 5 ^ .
2) 4 ,
:
3".
, 7.6, :
3) ]4~7 -^23+8/7 =4-\7) 2-^23+8^7 =
= ^16-8^7+7-^23 + 87 = - 8 7)(23 + 8 /7) =
= ^232-(8V7 )2 = /529-448 = /81=3.
4 : 1) 2) 3) ^ ^ ^ -. \/>+1 [2 Ja-2Vab + yJb
'
1) , -
:
+ 1 /fc+l
2-2 -2 jifeit-) Q 2 ) " w
3) -
, :
Ja-2 ifeb+yfb (^-Vb) 2 V^-Vb' *
68
-
8. , /-
5
: 1) 2) 2?/ 2-v3
-
,
- .
'. 1)
\/% :
15 _ _ 1 5 _ 1 5 _ 5
3 ~ 2-3 ~ 2 *
2) -
2 \/3, :
5 _ 5(4 + 2^/3 + /9) _ 5(4 + 2^/3 + 9) _
2-*/3 (2 - %/) (4 + 2 / + \/) 23 - (/)3
8-3
6 2 kl ,
tfn- - - Ja
ab +
'. + -
2 yfb 4
HZ- ijb-ifc
v r
tfi+
= ) W f l _ V b J = - - / + = m
7 MJab
'. ,
. .
: > 0 > 0. :
Vfa + Sfb ^ + . ^ _ + ) _ + ^
69
-
3.
: < 0, b < . :
Vjab ^-
, < 0 b < 0, .
= -X, b - -, X > 0, > 0. :
8 , 3->/2 ^49-20/ = /->/2.
'. : ^ / ~/2 ^5-2 >/ ^49-20V6 =
= ^(>13-^2)' -^5-2/-^49-20/ =
= ^5-2 -^52 V6 -^49-20 >/ =
= ^(5-2/)2 -^49-207 =^/49-20/-^49-20N/6 =
= ^(49 - 20 7 )2. 49 - 20 / = (49- 20 VfS)3 =
-
8. , /-
8.1.
1
2
8.2."
1
8.3.
1
2 8.4.
1
2
8.5.
1
8.6.
1
2
8.7.
1
2
8.8.
1
2
8.9.
1
2 8.10
1
- :
?/6; 3) 20; 5) ^40';
/2; 4) >/7290; 6) ^ V ;
:
- | ^ 4 ; 2) {/40; 3) |^686; 7
- :
4/86;
7) -54% 9;
8) /-108710.
4) -1,2^/96.
5)
3) ^432;
4) ^30 000000; 6) V243fcV8.
:
2/3; 3) -10^/0,271; 5) 5/25->/320-+ \/40;
8) * " 2 '
5) 2*3 (fi-.
5 + \/-189 - /-8In -1,5 24^ + /448m.
:
^4-3^/16+5^/128 + ^/2000;
>/25 +2^1+4 /296.
:
ffi3; 3) t / W ; 5) ^ W ;
3/3W; 4) ^/bW; 6) ^2 \f2~j2.
:
3) 5)
; 4) tf
:
( + / + \fcf) (l \/a);
6) V .
2) (l + >/^)(l + ) ( l - V e ) .
71
-
3.
8.11. :
1
2
8.12
1
2 Ja-Sfb;
3 yfm- lyfn;
8.13
1 ifx-;
2
8.14
1
2
8.15
1
2
8.16
1
2
:
'-V 4) 1 71 5) 8) i f*b-lfi?- lb;
6) \jfa -yfa; 9)
V Sab2
\[ab
3) V /; 5)
\fy* yfy*', 4) 6) V
:
+ 4)
^ / 0 2 5 + 3 2 5 ) + .
4 + 3 v2
(2 $/2 - 2 V5 + $/l) + V4);
:
, f f .
:
' 3)
1 5)
.
V i ' 7) 9)
X 2
. ' 4)
12 6) 18 . 8)
12 . 10)
a + b
a + bf
72
-
8. , -
8.17. :
'
2) 20
3)
4)
Va'
15
/25
-
3.
8.24/ & :
1) ifc=abtlab-, 2) $f*b = al!b; 3) $[* = -$1?
8.25/ - :
1) tf^rf; 5) ^/l62a W * , > 0, < 0;
2) ilabb13, > ; 6) ^/1515;
3) ^ , * 0; 7) -'^.
4) /2 ,1817. im
8.26." - :
1) 32 3, < 0; 3) < 0, < 0;
2) 1-62 5; 4) \! ' 1 9, > 0.
8.27/ :
1) 2, > 0; 4) ^;
2) ab > 0, b < 0; 5) a b
3) : 6) abSlrtf, b < . V
8.28/ :
1) , < 0; 4) a b f - ^ j , < 0; V a b
2) a $fa; 5) * .
3) -ab \/, < 0, b > 0;
8.29/ :
1) %//- *^/9+6 ; 4) ^4 + 2>/2-^/-4/2;
2) . & 5, v 2 1
3) />/15 + 4-'^31-8 >/l5;
8.30, :
1) ^7-4 / '^2+/; 2) ^2-1-^25+4
8.31/ :
^ | 1 yf + l . V
/-1 / J 1'
74
-
8. , /-
Ja + 27 ( \Ja-3 tft-9
\/-3%/+9 -Ja +27 la-*
2)
3)
4)
5)
6)
7)
IJa + 1 Ja-1
8.32.* :
6 -
zi ! - f - L + J L >
{l'za + zja*
1) ^+1 /* J"$/x+2$/*+l *
+ 2-
2) +
3) / + / -J-4b sf + ifb J+ifb yfa+^fb
+ 4^lm-4: yjjm.^4 +2 m-4 %/m-4 4) =, * = 1.
^/m-4>/m-4 ^Jtn-4-2 m _ 8
8.33." , :
1) ^7 + 5>/2+^7-5V2; 2) ^/^3+10-^//-10.
8.34." , ^20 + 14 >/2 + ^20-14 V2 = 4.
8.35." -
, / + /9 .
8.36. -
, /2+3 .
75
-
3.
8.37." , :
1) i + ; 2) $B + yf2.
8.38." , :
1) - ; 2) +
8.39." (3^2 +1) (#2 +1) $ 2 +1) +1) +1).
8.40." { 6$ + ){ 3$ + )'...-{yf+ ).
8.41." '" + 1 3 2 + 22 +... + '^11.
8.42." - 31 2 + " - 22 - . . . + " .
8.43."
J 2 + 2+.. + 2 + = + & + Wi2-S. 10
8.44,"
]12 + ^2 + 2+^2 + /2 + 44 +
-
9.
1
0
1 = )
/ 0 . 9.1 . 9.2
~ X, =, = X
(. 9.3).
= X 2 . , -
, 4, ( = -2
*2 = 2.
9.1. (), .
. , f,
. 0 (/), JC,
2 (JCJ < JC2) , f (JCJ) = f (;2) = y0. f
, 1 < 2 , f (jct) < f (2).
.
, f -
.
, ,
- ().
> 1 "
0 * ^ ^ X X . 9.3
77
-
3.
, 9.4 -
, , .
= f (),
:
' 1 \ / >
0 X
. 9.4
5 6 7
7 f .
= g (),
:
X
5 6 7
f g ' : 1) D (/) = () ( = D
-
9.
, ,
, = 2 1.
: 2 - + 1; = V 2
X.
, -
. , ,
+1 , = :
2
, g () = ^ f (JC) = 2 - 1
.
: D (f) = (g) = R, (f) = D (g) = R.
f (0) = 0, 0 ~ 2 - 1. , g (0) = xQ.
, + 2: -1 + 1 : g (0) = -^= = 0.
= X 2 .
[0; +). , f () = JC2, >(/) = [0; +=),
. , =
[0; .
: - 2, [0; +>). ^y=yfx2-\ \ = .
, -
y = Jx.
, f (je) = 2, D (/) = [0; g(x)=yfx
.
: D (f) = (g) = [0; +
-
3.
* b (. 9.5): ON = -Ja 2 +b 2, ~ Ja 2 +b 2,
MN, ,
MN. MN -
la + b g + b\
\ 2 ' 2 }'
. 9.5
= . , =
MN.
, -
(. 9.6).
' 1 * / /
^ ^ /
/ 's/ /
V / /
) )
. 9.6
9.3. f (),
g ().
. , f
g .
D (g) 2 D (), < 2
g (,) > g (2). g () = xv g (2) = 2. ,
Xj ^ 2. f , f (JCj) > f (2),
/, > 2. .
. *>
9.4.
= .
. (; )
f i g . , = .
. , -
, < .
80
-
9.
f g = , N (; )
. f :
f () < f (b). f () = b, f () = . b < ,
< . ,
> b. , = b.
. ,
9.4 '. ,
F () = -X G () = -X , -
, {1; 1) (1; -1), = .
. f i g -
, f () = g ()
[ () -- g () = .
.
2 ' yJ*Jx+5= -5. '. t. \lt + 5 = t 2 -5.
f (t) = Vf+5 g (f) = t 2 - 5, D (g) = [0;
.
9.4 , Jt + 5 = t 2 -5
5 = t, . 1 + 2
t . t> 0. 2
>[ -1+V21
= -
:
22 + 2 11 +
4 ~ 2
I I + V2T
9.1.* , 9.7,
?
J
' 1 ' >
X ( 0 * /
) )
. 9.7
81
)
\ [
0 \
)
-
3.
9.2/ , 9.8,
?
V 1 0 X
.
\ L
0 0
) ) ) ) . 9.8
9.3/ , :
l ) j / = |*|; 2 ) = - 3) = 5; 4) = [].
9.4/ , / g :
1) f (*) = + , ( * ) = 3 * - 1;
2) = ^
3 ) = g ( X ) = X2 - 2, D {g) = [0; +).
9.5/ , f g :
1 ) / ( * ) = 4* + 2, = 4 2
2) f (X) = *+1' ' 1-'
3) f (X) = (X - )2, D ( f ) = [3; +=), g(x) = ^ + 3. 9.6/ , :
1 1) = 3- 1;
2) = - \
3) = 2 + 1
4) = | + 4.
9.7" , :
4 1) = 0,2* + 3;
1 2) =
-
3) - + 2
4) = 4 - 5.
9.8/ , :
1) --
2) = /2-1;
3) = 2yc-l;
4) = \ D(y) = (-o-, 0];
82
-
9.
5) = 1 - 6) =
yJx-2, X>,
1+' ' [2-5, 1, 2) = 4) = \
yjx [2-, 0, 1) = -0,5 + 2; 2) = \[ + 1; 3) = \
[2, 0,
1) = - 1; 3) = <
2) = X 2 - 4, X > 0;
-X, 1 / / j /
1
0 1
" 1
5 --s
/ -h
-
J
yt t
\ V \ 0 X
) ) . 9.9 . 9.10
9.13." = f (), - 9.10, , f .
[xz+l, 0
-
3.
9.15.* , ,
= kx + fe * 0, .
9.16." fei b = kx + b,
fe * 0, ?
9.17." = -, ax + t>
* , ?
* 9.18." , , ,
.
9.19." g , f () = 5 + 6.
1) g (7).
2) ' g () = 1.
3) g () = -
9.20." g , f () = + -Jx-2.
1) g (28).
2) ' g () = 1.
3) , g () =
?
9.21." g f () = 3 + - 3.
' g () = X s + + 3.
9.22." g f () = 3 + + 12.
' g () = X 3 + X - 12.
9.23." g f () = 5 + - 1.
' f () = g{x).
9.24." f g () = 3 + - 8.
' f () = g ().
9.25." ' J x - - =2 +-. \ 8 8
9.26." ' / + / =-1.
9.27." g, f () = 2, D(f) = (-1; 0] U [3; 4).
9.28." g, f () = - 2,
D{f ) = [-3; -2) U [0; 1).
9.29." f ,
f (f ()) = X. , f .
84
-
9.
9.30." f g , f (/ ()) = g (X). , f - .
9.31." f , D ( / ) = N U {0}, ( f ) = N?
9.32." f , D {/) = Z, ( = N?
9.33.* / , D ( f ) Q, ( f ) = W
9.34.* / , D ( f ) = [0; 1], E ( f ) ~ [0; 1)?
9.35.* / , D ( f ) - [0; 1], ( = N?
9.36.* f i g ,
1 f () = 2 g ()?
9.37.* f i g ,
R f () - g () = X.
9.38.* f g. ,
-/- + X , 2 2
g () = 10 - 22 . 1 0,25.
9.39.* f g. , 2 - 8 < f(x) < 2 - 6 R, g () = 22 - 3 . 0,1.
9.40.* g f , D ( = [0; 1],
-
3.
9.42.* f , - R,
R
xf {f () - 2) = 9 (X - ) + yf ().
9.43.* / , - R,
R
yf (f () - 2) = (X + ) - xf ().
.
'
.
? -
.
;
,
, -
.
X X .
' -
, -
.
,
.
20-30- pp. X X .
.
, ,
, , , ,
, .
, -
, -
.
, -
.
.
86
-
5
(1892-1945)
.
.
. .
. -
. ,
-, , .
, -
... '. . (-
) ', -
.
' ,
. ,,
. '
, . -
. ' -
, . ,
, (1936 p.),
' 1972 p., .
, , -
, , '
, .
.
87
-
3.
=
6 ,
- .
X
, = 2 +[. k N
f () = 2" *[ R.
, f g () = 2*+1,
ft N.
= -
= 2* + / R.
: D (/) = (g) = ,
(f) = D {g) = . 1R 2^2+1 = .
, f (g ()) = D (g). ,
f i g .
= 2*+ 9.2, -
= 2 + (. 10.1). ,
10.2 - $ .
88
-
10. = six
1
= Vx
-1 , X
. 10.2
g (JC) = 2 1 1 , - 9.3 f (JC) =2 - .
f(x) = ' - 0,
JC < 0, f (JC) < 0; JC > 0, / (JC) > 0. ,
(-=; 0) (0; -
f.
- X f
f (-) = - - 2 k +ifx - f (). , / -
.
6 , -
- '
.
[0; +) ,
= 2yfx. f () = 2tfx, k N, -
[0; +).
, / g () = 2, k N, [0; +).
2s[x = - > 0
je = 2 - < 0 ,
f [0; +=).
: D ( / ) = (g) = [0; E(f) = D(g) = [0;
- JC [0; 2\ - .
, / (g (JC)) = JC D (g). , f i g .
89
-
3.
10.3 ,
ke N. 10.4 -.
. 10.3 . 10.4
g () = X2*, N, D {g) = [0; +), -, f(x) = 2\[x .
f = 0.
X > 0, / () > 0. , (0;
f.
f , f , .
= %/, -
.
, > 1
[0; +)
[0; +)
X = 0 X 0
. > 0
(0; +)
< 0 ( 0 ) ,
> (0;
,
/
90
-
10. = [
1 ' : 1) $
: $2 > \2.
10.1/ = the:
(2; 16); (16; 2);(-1;1); D ( ; ); (81; 3); F(0,001; 0,1);
G (10 000; 10)?
10.2/ -\[:
(-8; -2); (3; 27); D (0,64; 0,4); (-216; 6);
F (-1000; -10)?
91
-
3.
10.3. :
3) -: 1} =
2) = + 1; 4) = 2--2;
10.4. :
1) = 2 ; 3) * =
2) f/ = VT r2; . 4) - 2 -4 + 3;
10.5. :
1) = 1; 3) = [-3\
5) =
6) (-3).
5) =
6) =
2) - Jx
5) y=\tfx + 2\.
5) I \[x +11.
4) y = \ \fx-l |;
10.6. :
1) = yfx + 2; 3) ~[-2\
2) ^^-4; 4) /=) %/-2);
10.7. :
1) f () = D (f) = [-27; 8]; 3) f () = yfx, D(f) = ].
2) f(x) = tfc, 10 000 ;
10.8. yfx, :
1) 1 < * < 216; 2) -729 < < 8.
10.9 \[, :
1) 0 < < 256; 10.10. :
1) ^ ;
2) ^23 \Z-26;
3) 2
-
10. y = tfx
10.13. -
:
1) 78; 2) 739; 3) -7212?
16.14/ ,
:
1) 4 740; 2) 7^35 740.
10.15/ ,
-Vi300 70.
10.16/ :
1) 7 73; 3) 7 tfljf; 5) 7 7) 72 72;
2) 72 7; 4) 7; 6) 78; 8)
10.17/ :
1) 75 70; 3)
2) 76 ^/; 4) / ^/7.
10.18/ :
1) / 2 , 7 74; 3) 73, 75 77;
2) 7, 72 '; 4) ^125, 7 ^ 7 4 .
10.19/ :
1) 75, 74 73; 2) 7, 7 70.
10.20/ :
1) = -[; 4) =
2) = [-2\ 5) = %[-2 -2;
3) = 7 * - 2 ; 6) =
10.21/ :
1) --[; 4) = ;
2) = 7 ^ ; 5) 0 =7 ^+3 + 1;
3) / = 7 +3; 6) =
7) =
8) =
9) / = | +1-2 |.
7) =
8) = + \;
9) = J tfx + 2 2 |.
10.22/ f (*) - \
:
) [1; 2];
2) [-3; -1]; 3) [-1; 1]; 4) [-1; 2];
5) [-3; +);
6) (-; -1].
93
-
3.
5) [-1; +>;
6) (-; 2).
5) $ 2 + 2>$ 2--6.
10.23.
/*) = >/ :
1) [2; 3]; 3) [-2; 2]; 2) [-1; 0]; 4) [-2; 1];
10.24." ' :
1) 3) +
2) $3+1 $2;
10.25." ' :
1) 1\ + 2 > 1; 3)
-
11.
10.37.*
100
10.38.*
00
10.39.* ' 3 +1 = 2 \'2 -1.
10.40.* ' 3 + 2 = 3 '-2.
10.41.* ' a* + x = %Ja-x.
10.42.* f () = . * . V 1-3
f (f U ( - / (2)))). 989
10.43.* u e N , A e N , f t > l = +
Y = I + 2* + ... + \ , X + = + 1 + .
10.44.* ' , &
:
=-'...-, N, > 1;
1 = .
,
:
1. -" = + ;
2. : = ~", * 0, > ;
3. () = ; 4. (ab)" = "";
' :
95
-
3.
0 = 1, * 0;
" = , 0, N. "
: ' -
-
.
, -
, r = ,
Z, N, > 1. , -
-
.
. 2
X 2, ,
( ) = , , X 22,
= 2^ =
.
. -
, N,
> 1, \[ ,
, 5*= {/*, = 3 * = /jF, 0,4'3 = 0,4">
, , -
, , .
, a" =\fa "* -
nklmk n/_m = va =
-
11.
, "
< 0, , (-2)3 .
:< 7-2 . :
, t P t - i - t h , , .
:
7=2 = (-2)^ = (-2)^ = 7(-2) = 74.
, ' 7-2 .
, = xf, Q,
.
, Z, n e N, > 1 , -
, = " [0; +),
' , (0;
=
. 4 5. , ,
11 . .
= X2", k N, 1
= 2[. -xz*+1 ='six, k N,
. , [0; +=) -
, {-; 0) - '
11.1 - , - ,
~4.
'
1 * 4
-
3.
, -
.
11.1 ( ). - > 0 - q
9 = +
. q -
: , q = , Z, Z,
N, > 1. : k w+ft fft. h
"." = " . " = ] - = + = = "
. - > 0 -
. 11.1, : > =
= ' + = = 1. "
11.2 ( ). - > 0 - q
" :
. 11.1, : 4 4 -
= + " = ". a p' q = " : ".
11.3 ( ). - > 0 - q
( ) =
. ~ , Z, N, > 1, q = 4, s Z, k
N, > 1. :
{ - ()* = \1() - VfV")" = = = a^ =
11.4 ( ). - > 0 b > 0 -
( - a"W ( ! )
-
.
98
-
11.
/ ( * ) = (*") .
* '.
/ (0; +). -
:
/ () = X, D (/) = (0;
11.2.
. 11.2
11.1." :
1) 3) 5) (; 7) ( + ) 5 3;
1-, 4) 10 6) ab'; 8) (-/)'*
:
13*; 3) 0-2; 5) 31'50-;
.
2.5.
4)
2)
11.2."
1)
2)
11.3.*
1)
2)
11.4.
1)
2)
11.5."
1)
2)
11.6.
1) 8^;
2) 10000*;
6) (-2)16.
:
3) V61; 5) *; 7) ^ ( a - b f ;
V 7 ; 4) 6) '^49; 8) % 7 ~b 7.
:
/2; 3) (* ; 5) +
6)
3)
4)
:
4*.
25
3) 3-64 3;
4) -5-0,01
5) 0,216
6) 7) 27;
8) 32"0 2.
3) 0,0081 - 0 , 2 5 . 5) 0,125 3;
6) ("If 99
-
3.
11.7. :
1) / = *"; 3) = (X - )2 6;
2) - X 1' 4; 4) = ( 2-6-7)~.
11.8. :
1) =
2 ) = X s' 2;
11.9. :
3) = ( + ) ! 2;
4) i/ = {jez-x-3)s .
1) 2 3; 5) ( J ) ;
2) U)3;
3) 2 : 3;
ja *;
7) 3
9) 8 : 4;
-1
10) 2 3;
13) ( V I* 2 7) ;
11) (0,4)0'80,18; 15)
1 1 5 1 14) 3 ^8;
L 2 1 2
.1 :
4 3
,1.8 .
4) ~0,61,6; 8) ("2'4)"3;
11.10/ :
12) 16) () (") .
1) t>3-V4'2; 5) W , 5 7 3 2 1 1 1
9) 2 *~ ; 13) b 2b 3b l-,
- 3 2) b 7b 7;
3) b:b 3;
4) b:b 4;
7 5
,\0,5_0.4. 6)fc"2:Vb; 10) 14) (bT'ab
*
7 1 1 ) (oM) ;
8) 12) b5
4 ;
15) (b
16)( 4 0- 7:-*\
11.11. :
1 )3 1 8 . 3 26.32'8; 4) 7 ) 4 2
2) (5 Y-54 '8; 14.5
3)
(25 )4;
5)
6)
1,24'5;
UO \700/
8) 360,4-61,;
9) (4-) . 0
100
-
11.
11.12. : ! , \-0,2S
1) 53-4-5-l'e-5 2'e;
2) (7 '7)8 : 7 7 , 6;
42
3) ) 3;
4) '
5) ! ( 2 ? )
6) 81 .
6) ' 3
0,4 1,8,
2,5
7) 8'
8) (6 18) -2160'2?
11.13. , . :
1) ; 3) ;
2) ; 4) .
11.14. , . -
:
1 )4; 3) 3; 5) 3 ; 7) " ; 1 5 2
2) 6; 4) 2; ) ; 8) 7.
11.15. , .
:
1)&6; 3) 2; 5) t>3; 7) '
2)*15; 4) (3; 6 ) ^ ; 8 ) 6 " .
11.16/ :
1) ((-2)) = -2; 2)
11.17.* :
({ 2) = -2?
1) = (*3) ; 2) = ( (*-2) ;
11.18.* :
1) 12*
-
3.
9)
10)
(l2 3 18 3 - 3'5) -5* 4 -258; 11)
( .5 v e
.7
U r ^ j 11.19." :
1) 3432
2) 104 404 >5*;
3 1 2 3) 0,0016 4-0,04 2+0,216";
( 3 - ( 12828 * 27 9 162-81s
* 3 -168 , t 93-4 J
5) 32'24 '4'7
640,* 160,25 ' 5
12 3*7
6) 73-8
7)
( V 1* 5 3-3 3
8 )
( ^
4) 625"15-251,5*125;
11.20." ' :
1) ^ =0,04; 2) (-2)^ =32;
11.21." ' :
8) 8 1 3 8 9
?
{ ^ 29 -275
18 7 3 -128";
3) ( 2-2) 4=-1
1) "1,6 = 27; 2) (jc-l)~s =100; 3) ( - 5)7 =0.
, , -
, .
1 : 1) (0-3 + &-2) (0-3 - 4&0,2) - (0'3 + 20 2) (0'3 - 2b02);
2) .
102
-
12. ,
'
1) ,
, , -
':
(0,3 + 0' 2) (0'3 - 4- 2) - (0 3 + 22) (0-3 - 2 0- 2) =
= 0, - 12a"'V'2 + ' - 4 - 0"6 + 4 0 = 20 6 - lla0'3b0'2.
2) : (a12-&12~)(* + a12b'~2 + ) + ( + fts) =(12) -(12)
/ \2 1 / \2 I I + U ej +2 1 +\* = a* ~b l +a l +2** +* = 24 +2**.
2 2 - , - : 1) ; 2) .
'
. - . - W U
5 11
3 : 1) 2) '+ 3* ' ' 4 3 ) 3 2 * ~ X f . 2 ~
3-9~ 2 4-23
' 1) ,
, :
43 43 4 - 11 1 \ 1 -
2- 3 3 vae a e - l .
2) ,
: 5 1
1 + 2 4 6*1 + 3 *) 1 ^
3 -92 1 [3-4) (b3 +3*) /
, , 23 -163 16 23 - ) 163 /\3 -3) : = ~1 = V =8 3 =2 .
43 -2 2
+ 2 ** " 2 1 6
1 1 2 3-2 X +2 X s-
103
4
-
3.
'. 3=. -
:
+ 2 - 2 16 -2 + 2 2 -4
. ' -
.
: 8
3 +2
12.1. :
1) + 2\
2) 22 {' -4 + 8 2;
3) (0 5 - 0-3) (20'5 + 0'3);
4) (~*-*)(* +J);
5) (&*-)(
6) U + b O ;
7) (4n"e+3nJ ;
8) ( - - S J ;
9) (b0'4 + ) 2 - 6b0-4;
10) - l ) ( J + c 3 + l ) ;
11) y + j ) U - J + a ) ;
12) a* (a +lo)-(a^ +) ;
/ 3 1 \2 1 /23 !\
13) lb5 -2b 3) +4b 6 lb30 -b 2 j;
14) (a^+6) (o - ft) (a^ - );
15) (x-l)(*+* + l ) ( * '+ l ) .
12.2. :
1) (5a0'4 + ') (*4 - 4 ); 6) (* +)(-2*+4);
2) (m0'5 + nua) (mu'& - ", ' 7) {y l b - 4y"*f + 8y*; -0,5 0 ,5
3) (a3 -5b 0 (a3 +5b 4);
4) (m^-J) ;
5) (&L&-I) ;
8) + 3c) - ^ (* +2J ) ;
9) (e*-l)(a* + l)(a*+l);
10) (+(*-+*)(*-).
104
-
12. ,
12.3."
( ' ):
1) - ; 3) }-&; 5) * -
2) 3 - 3; 4) 2 -3; 6) 4 0 - 9y 0 J .
12.4. ,
( ' ):
1) 5 - 5; 3) 5) 5-;
2 ) / 3 - 3 ; 4 ) ^ - 2 ; 6) 16jc0'3-2*.
12.5.
( ' ):
1) + ; 3)
2) *+3;
5) + 2; 2 2
4) 2 +27; 6)8+>3 .
12.6. ,
( ' ):
1 ) - ; 2 ) a w - b 1 5 ; 3) m0'6 - 818; 4) '-6.
12.7. :
1) + 2; 3)ab 3-a 3b-, 5) - ;
2) 4) ^ - 6 ) 6 *-
12.8. : 2
1) -5 ;
2) a2+6a3 ;
3) -;
12.9. :
-5 2
4) - 0 ;
5)
6) m ' n 4 m n + m 2 n a ;
1)
2)
3)
a 2 -5
2
fl3
,
4)
5) ,3 " +*
11 * + 2*
\ * +
-Ab
0 , 5 + 2 0 , 5'
- 1 '
ab 2 +a 2b
6) + 22(>2 +
2 + 2
105
7)
8)
9)
7) 2-2*;
5 8) 6 +* 3 .
7) 8 ^ - 4 ^ ; 5
8) 10+10"8.
? 1 1 ? 43 -12c3cf3 + 9d3
23 - 3d3
+
3+8
' -*
-
3.
)
X -6-3
12.10. :
+ 2 3
. . . 4 +72,
J !
- 492 12)
5 - 5
1' 105 - 2 5
)
2)
3)
2 3 +2
5 1
4)
5 --
5 5
-
2
5)
6)
- 2 11 2 '
3+33+3
0 , 5 . 0 . 5 -
-
+2-*+'
7)
8)
9) - 2
12.11.' :
1} (^^1)-'-1);
-125. 2 '
3-25 1
-36 , '
*-6*
244 -4
' 4 -24
2)
f 1 52.33 +53.32
1 1 5 +3
^ 34-22 -32 '24
24 - 4
12.12. :
1) -
0,5 1.0,5 -
1.5 1,1,6 ~ -
0,5 , 0 , 5 0,5 2\ z +
+
+ 0.5
0,5 .0,6
4)
5)
1 I i i 2 +24&4 + 2
2 ~* 1
1 S 5 7
* 6-%* ! 1'
4> 4 + 2
2 1 / 2+ 2 3(-)
2 * (,-) 3 2- 2
3)
12.13. :
1) +
3 +
2) (-36) ( 1+38 + )+ ). 1
4- 6
3)
4)
(^-):
( 5 4
- +
2- 12 ffl 2 +
^*
\
106
-
12. ,
12.14." :
1)
2)
3)
/ 2 2 + +
\ + 2 2+ 2
- 2
( *+ 2
V + 20'5 +1 / .
'*
1 ft-1
-5-2 gc,s _ 2 .
-1 )'- 5+1 -
3-
2 2 ~3+> 3
4) ^ + -2 2
12.15.* :
33
? _1 1 3 + 3b a + 3 3 ->3
= 2^-2&2.
1)
/2 t
^jty'+je2^ ~' 2,
. 2 + 2
4*
x R +3 6 -3
' + 3-2 6 6+ 3 3-* J
12.16." :
> . ., X +3 3
X - 0
1)
2)
' 2 2 \ 2 2 + ") -4 '
' 1 2 1+1
' -") +4 1
1 5 2 4 * -2 3 3+ 3
3) [ 3 + 2%[ + 4 3)
5 1 2 2 3 a&3 + a3b
12.17/ :
- D i l i
X4 + je2y4
2/4 + 4 2 * *
2)
3)
X' + 4 1
3-27 3
X 2 -2 4 4 +
2
2
3 +3 +
1-3 -'iLv,2.
-9
+ 2 +9 V- s-27
,0,5
1
107
-
3.
12.18.' 1,2 1'3 1 1'5 . . . 8'8, = ^2.
12.19." 2 -4 ~8 ...6*, = 2
12.20." (0'125 + t>'75) (0,25 + 1'5) (0,5 + 3) ( + ft6).
12.21." 0-2 + 0,5 + 0-8 + + ... + 7'1. 12.22." b12J - 1 + 12's - 12'4 + ... + 3'3.
12.23."
12.24."
4 -
3'8 + 3-% + a 8' 4b 2 + 3' 2 3 + ... + a'V8 + 1 9 '
g _ * + .4,8 0,6 _ .3.60,9 + 2, _ + ^
^+
= X. , , -. = 3 . : \ = \ ,
, = 2. , = 2 , 3 = 2 ,
, , , ,
1 ' /2 + 1 = -3.
'. ,
, . :
( ^+ ) 3 = (-3 ) 3 ;
2 + 1 = -27;
= -14.
; 14,
- x 2 k + 1, k N, , ,
' 1,
.
13.1.
, ,
.
. ,
/ (X) = g () (1)
108
-
13.
(*)) * _ 1 = *(*))~\ * N (2)
.
(1).
f () = g (). :
if (a)) 2 k l = (g(a)) 2 h-\
, (2).
(2). ,
(/ ())2* 1 = (g ())2*"1- = 2 - k N, -
, f () = g (). , (1).
, (1) -
(2) , (2) -
(1). , (1) (2) . Ja
2 ' '.
. :
(VT--2 )7 = {yfx)7; X 2 - 2 = ; X 2 - X - 2 = 0;
= -1, 2 = 2. -. 1; 2. , 1 2,
. .
:
-3 = 2; Jx-2$[x + l = Q; J3-x = ^J2+x.
' 1 2
( ( ) } , . -
, .
3 ' ( + 4F = {4X^2F. (3)
'. :
+ 4 = X - 2. (4)
X -3.
, -3
. , (3) .
,
(Ja) -
. (4) (3).
'
= 2 , k N.
109
-
3.
, 2 = 2 ' ,
jCj = 2. , (-2)4 = 24, -2 # 2.
2 2 1 - 2 , = | .
,
.
13.2.
.
. ,
if U)f k = (g ())2\ k N, (5)
f (X) = g (). (6)
(6), f () = g (). (/ (,))2* = {g ())2*. , (5).
, (6) -
(5). , (5) (6).
, (5), -
{f ())2* = ( ())2* ' , f () = g (). f () = g () (f ())* = (g ())2* ' , .
4 ' /4 + = . '. -
, , :
4 + = X 2 -,
X 2 - - 4 = 0; 1 = -1, 2 = 4.
, -1 , 4
.
: 4.
5 ' $2-3 + *j4x + l = 4. ' , .
:
2-3+2>/2-3 /4+ + 4 + 1 = 16.
>/2-3^/4+1 = 9-.
-, :
82 - - 3 = 81 - 54 + 92;
X 2 - 44 + 84 = 0; , = 42, 2 = 2.
, 42 , 2
.
: 2.
110
-
13.
, :
/-2 + 1 = 0; 3) /-4 + / = 5; 5) -J26+Jx + 1 =5.
4) ^ + / 7 * ^ ;
' :
2-2 = 2; 3) Vx-6=-3; 5) 7=3 ;
7-4 = 2; 4) 7 3 -2+3= ; 6) +15 = 27+ .
' :
>/-3 = 4; 2) %/2 --15 = 3; 3) ^25+7x^43=3.
' :
/ 2 - 1 = / - 2 ;
' :
7+3 = 72-3;
3) /2-1 = >/-3;
4) /2-1 = Vx2 +4-16.
3) 7 2-25 = 72+;
4) /2~36 = /2-1.
5) 2 -s/xT = X + 2;
6) -/15- -1= ;
7) ~2 2 + -21 = 3;
8) + 2+/8--2 =0.
V4x-5 = >/-;
' :
>/2- = ;
Vx + l = x- l ;
/-2 = ;
/22-3~10 = ;
' :
Vl0-3x = -x;
x = Vx+5 + l ;
V2x2 +5+4 = 2 + 2;
' :
(2+3)(-4) = -4;
V(x-2){2x-5) + 2 = x;
' :
/(3-1)(4+3) = -1; 2) (-1)/2-- = 5-5.
4) 3/+-11 = 2;
5) -/ 2--20 = 5.
3) ( + 2)/2 --20 =6 + 12;
4) (+1) /2 -5+5 = +1.
111
-
3.
13.10." ' :
1) ^jl + x 4x^24= +1;
13.11." ' :
1) /22--/-=2;
2) $ + 2-$2-3 = 1;
13.12." ' :
1) /2: + 5 /-5 = 2;
13.13." ' :
1) N/X-5 + V10-jc =3;
2) -7 +$-1=4;
13.14." ' :
1) \j4-x + VJC + 5 =3; 2) VSjc+I + V?-^ = 6;
13.15." ' :
1) $2x + l + yjx-3 = 2jx;
2) 1-1~13-2 = 4 :;
13.16." ' :
1) Jx+2 + y/3x + 7 = s/8-x;
13.17." ' :
1) yjx- 1-24^2+^+7-6!-2 = 6 ;
2 ) YJX+3-4YJXPL +}+8-6-1=1;
3) ^]++24-+-44 + = 4.
2)
-
14. '
13.21."
-1 = \8~ 2 -15 '? 13.22."
0.
. .
yjf () = yjg (). f () > 0.
.
f () = g (). , ' .
' ,
j /() = S(),
l / ()>0.
, g () 5s 0. , '
f () g () , , ^/() = yjg (). ,
.
, ' '
, , ' ' -
. , '
.
113
-
3.
. , J f () - Jg ( )
[f(x) = g(x),
\g{x)>0.
, , ' ,
, f () > 0 g () > 0, ' .
1 ' --Jx-1. '.
= 2+>/, = 2 + >/3.
>1;
[ - = -1,
[>1.
: 2 + .
14.2. Jf()-g()
if(x) = (g(x)f,
\g(x)>0.
14.1,
.
2 ' \! + 7 = -3. '.
+ 7 = (-3),
{-3 >0.
7 + V41
X2-7+2 = 0,
X >3;
: 7 + V41
X = -
X
2 '
7-41 X
7 +
>3;
14.1 14.2 , -
: 3* 0 ? > 0, 2 = 2 ,
k N, , - .
14.3. - - f () > 0 ig () > 0, f () = g () (/ ())2* = = (g ())2*, k N, .
114
-
14. '
14.1,
.
3 ' >2-3 + -J4x + 1 = 4. . '. -
' . -
(j2x-3 + \l4x + lf = 42.
2*-3 + 2 /2-/4 + 1 + 4*+1 = 16; yj2x-3 4 + = 9-3*.
- [1-
' . , 9 - ,
'. 9 - > 0; < 3.
'-[I*] , \j2x - 3 \[4 +1 = 9 ' . , 14.3 -
f (2 - 3) (4 +1) = (9 - ),
.
2-44*+ 84-0,
12
-
3.
{j2x-5 + Jx + 2) = + 1) . 2j2x-5 Jx + 2 = 4-x.
14.3, :
f4(2x-5) (jc+2) = (4-)2,
12
7jc + 4JC 56 = 0,
5.
-
14. '
(>1, fx>l ,
[4(2x-1)(-1) = 2 + 14 + 49; |72-26-45 = 0;
>1,
* = = 5.
=
2)
/l JC =;
14.2. ' :
1) / + 2 / + 8 =4;
2) -1 = 2-5 yjx+;
7) /7- + - = = 2/5 + 37. V 7 - X
3) /
4) . 1 2 -/2 + 3= / + 10. /+ 10
117
7
-
3.
14.3. ' :
1) $4 + 2- 2 = -2; 3) /2 +8 = 2 + 1; 5) 4 = -1;
2) \Ig-4X-X 2 = + 4; 4) $2 2-7 + 5^1-; 6) $ 2-1=3-2.
14.4.* ' :
1) 2-4 + 13 = ^ + 2;
2) $2 2 +8 + 7-2 = ;
14.5." ' :
1) /2 + 6-/ + 1=2;
2) yjx + - / j c = 1 ;
3) Vx-5-V9~x = l;
4) /2 + 5 = 8-7x^1 ;
14.6." ' :
1) >/2-4 / + 5 =1;
2) V* + l l -V2x + l = 2 ;
14.7." ' :
1) /+4 + sjx-4 = 2jx;
14.8.* ' :
1) $ + 3-2-1--2 = 0; 2) Jx + 1 + Jx-l =$3-1.
14.9." ' :
1) $ 2 -4 + yJx 2 +2-8^$ 2 -6 + 8;
2) YJIX 2 +5X+2-4X 2 + -2 = >/+6.
14.10." ' :
1) $2- + 2 + $ 2-6 + 8 = 2-11 + 18;
2) 2 - -10 + /2 + +2 = /2 + 8 +12.
118
3) / + 2 =1-.
5) / + 5 + /5- = 4;
6) /3-+/ + 3=2;
7) $ + $ + 11+$-[ + 11=4.
3) / + 1 + Vl6-3x = 5.
2) / + 1-/9-x =/2-12.
-
15. '
'
'-
,
.
'
.
.
' 2 + -18 + 4 /2 + - 6 = 0.
'. six 2 + 3-6 = t. 2 + - 18 = t 2 - 12,
t2 - 12 + 4f = 0.
~t = - 6,
t = 2.
t > 0, t = 2. ,
:
/2 + 3-6 =2. 2 + - 6 = 4; = -5 = 2.
: -5; 2.
2 '
>/+4 + Vx-4 =2 + 2 Vx2 -16 -12.
'. Jx + 4 + Jx-4 = t. ,
,
2 + 2 /2 -16 =t 2.
t - t 2 - 12. t- 4
f = -3.
, \+4 + >/-4=3 '.
, : Vx + 4 + /-4 = 4.
,
fx >4, jx>4,
[2+2/-16 = 16; |Vx2-16 = 8-;
f 4
-
3.
3 ' 2( + 1)- Jx + 1 - 2 -0 .
'. 0 -
2 ( +1) Jx +1 , , _ 1 = 0 .
X
sjx + l = f, 2f - t - 1 = 0. t - 1 t = .
X 2
:
V H I - 1 1- X
\1 + 1 . 2*
: 2-2 72.
( > 0,
[ + 1 = \
< ,
4 + 4 = ;2;
X = 1 + 5
= 2-2^2.
' -
.
Ux + y + ijxy + 22 = 5,
\+ + +22 = 3.
'. tfx+y = , + 22 = b, > 0, > 0,
4 '
= 1,
= 2
2 +2 =5,
+ = 3.
(a+b) 2 -2ab = 5,
+ = 3;
ab = 2,
a + b = 3.
= 2,
6=1.
:
,
I $jxy + 22 = 2,"
\[ + = 2,
+ = 1,
= - 6,
+ / = 16,
[ ^ + 2 2 = 1,
' , .
: (3; -2), (-2; 3), (8 + 785; 8-785), (8-785; 8 + 785).
120
-
15. '
5 ' 7(2-xf + 7 ( 7 + ) 2 - 7 ( 7 + ) ( 2 - ) = 3-
' . 72- = , 77 + = >.
[a2 + i>2-a& = 3,
3 + 3 =9;
ab = 2,
a + b = 3;
| =
1& = 2;
a2+i>2-ai> = 3,
( + 6)(2+&2-&) = 9;
[6 =
[ = 2,
*=1.
inoeib: 1; -6.
' ( 7 l + x +1) ( 7 + +2 - 5 ) = . .
7 + -1 . -: X (7+ + 2 - ) = (7+ -1).
X = 0,
7 + + 2-5 = 7 + - 1 . ' .
2 - 5 = -1. = 2.
. ,
2 , 0 .
: 2.
15 . 1 . ' , :
1) 7 + 2 7 ? - 3 = 0;
2) 7 + 7 - 6 = 0;
3) 2-7 7 15 = 0;
4 ) 7 + 7 = 4 ;
5) 27 + 1 -5 = -T==; V X + 1
6) X2 - + 9+7 - + 9 = 12;
7) 72 -4+ 4-2 /-2-3 = 0;
8 j 1 3 _ 2.
7 - 7*+
7 - 7 7 - 1 7 2 - / + 1
121
-
3.
15.2. ' , -
:
1) -/-12 = 0;
2) 3J7 + 8 = 9&c;
sfx7==1; \
5 j 1 2 _
+ 1 + 3
6) 9-6 + 2+2^3--8 = 0;
7) 2-+%/- + 4 = 2;
4) 7 ^ 7 5 - 3 ^ ^ 7 5 + 2 = 0; 8) = 2 5 < \2x-3 \ +2
15.3.' ' , -
:
1) 2-5+16-7*~5+20=0; 4) 2-9-26 = 12+-2;
2) 2 + 4-5/ -2 = 0; 5) 2+6-3/2+3-3 = 5;
3) 72- + 5+2= + 7; 6) J x k f x = 72.
15.4.* ' , -
:
1) -4-3/2-4 + 20 + 10 = 0; 3) /22-6 + 40 = 2- + 8;
2) 27 - + 11 = 4+-2; 4) 52+10+/2+ 2-15 = 123.
15.5.* ' :
1)
2)
[+/=5,
( + +4/ = 37;
[/- = 7;
4)
5)
3 ) 6 )
[ = 8;
15.6.* ' :
1)
2)
[=27;
+ = 5.
3)
\$ + + $-=4,
[$ + -$- = 8;
2 _2 V 2 \ - 2
2 - 82 = 18 - 18/;
$4~ + +$9-2 + =7, [2-3 = 12.
($ + 2 + $- + 2 = 3, [2 + = 7.
122
-
15. '
15.7." '
Jx^i + ylx + 3 + 2j(x-l)(x + 3)=4-2x.
15.8." '
x+J{x + 6) {x-2) = 2 + Jx+6 + Jx~2.
15.9." '
42+3 + /+=3 + 2 /2 + 5 + 3 -16. 2
15.10." ' * ..+j2x + 5-2x. yJ2x + 5
15.11." ' 4 2 +12 \ll + X = 27 (1 + ).
15.12." ' 6 2-5xJx + 3 + x + 3 = 0.
15.13." '
%( + )2 +%](6-) -$]( + 3)(6-) = 3.
15.14." '
%](+4) 2 +yJ(x-5) 2 +%]( + 4)(-5) = 3.
15.15." ' + 8-yJx-S -2.
15.16." ' >/18 + 5jc +\/4-5: =4.
15.17." ' /-2 + /-1 ~5.
15.18." ' \2- -l-Jx-l,
15.19." ' J 2 - J 2 - X - .
15.20." ' JG-\I6-x = 15.21." ' :
1) 22 + +5 + 22- + 5 = -,
2) (Vx+I + l )(Vx+I-4) = x.
15.22." ' :
1) Jx 2+3x-2 + yIx 2-x + l=4x-3;
2) {/^+ + )( + + 2 + --7) = .
123
-
3.
, '
. -
14.1.
16.1. yffjx) > fgjx)
U(x)>g(x),
\g(x)>0. -
1 ' yjx 2 - +1 ^ s/ - 4.
'.
2-6+5>0, X 4 - + 1>-4,
-4>0;
>5,
-
X > 5.
: [5;
16.2. yjf () < g ()
f(x) ,
f(x)> .
2 ' ^22 --5 < -1 .
'.
2 2--50.
>1,
2,5.
-1 2,5 3
. 16.1
' -
16.1,
2,5 < X < 3.
: [2,5; 3),
124
-
16,
16,3. yjf(x) >g(x)
~\g(x) ,
\gix)>Q,
\f(x)>{g(x)f.
3 ' yjx 2 +7 + 12 >6-. '.
.
>6, X 6.
>~3;
(6-*)2 ;
: ( f f ; *- ) .
jc> 19 19 *
4 ' (* -'. :
(-3){^ 2 + 4--)3,
1) * - 3 > 0 , >3,
5 * > 3. [/2 + 4 < Jc+3; * 3 +4 + 3.
+ 30,
2+4>( + 3) 2.
-
3.
4 ' , -
. , '
( - 3) (/2 + 4 - - ) = 0,
X = , X = . '
16 2
16.2,
' -
.
16.4. - - f(x)>Oig(x)> 0, f(x)>g () (/ ())2* >
> (g ())3*, k N, .
5 ' $2 + + $-3 4; 2) ^ < 4 ; 3) >-4 ; 4) ^ < - 4 .
126
-
16,
4) sJx 2-3x + l>yj2x-3;
5) V8-5x>Vx2-16;
6) yjx 2- + 2 < 72X 2~3X + 1.
3) 7* + -10\ 5) sjx 2+x-2>;
2) ! + 7 > X +1; 4) six 2-2 >4-; 6) 7-2 + 6-5 >8-2.
16.7.* ' :
1) 7+2> ;
2) 72 + 14> + 3;
16.8.* ' :
1) ( + 10)-Jx~4
-
3.
16.11." ' :
1) (x+l)Jx2 + l>x2-l; 3) 2-1 -5
2) & 0 ; .. / 2 +X-6 + 3X + 13 4) 4..
16.15." ' yJx-2+Jx3 +8 < 4. 16.16." '
Jl-(x + 2) 2 > ~ 5
16.17." '
-2 >(-) 5
2
-
, J
.
. .
.
, , -
.
17.1 .,
. : Z = 1 .
, AB .
: AB = 1 .
() . -
, R
(R > ) (. 17.2). D
. , AB -
R, , , .
129
-
4.
17.3 R MN, -
-R. MON ( MN)
^ . , -
R -
,
, -
.
.
,
,
180. '
, :
= 180. (1)
i p ^ f i f )
180 3,14 (, ~ 3,14), -: 1 = 57.
(1) ,
80 (2)
, , ,
15 = 15 = , 90 = 90 ^ = | ,
135 = 1 3 5 - = .
. , 135 =
4
,
:
0 30 45 60 90 120 135 150 180
0 2 5
0
6 4 3 2 3 4 6
130
-
17.
,
. -
1 ,
R, , . , , ,
= aR
-
. -
.
, 0(1; 0), -
.
,
ZP0OP = ~- = 120 (. 17.4).
, 0 -
- ( 120). 2
: = 03(
,
2 ZPOP0 = = 120 (. 17.5). , -
0
~ ( -120). : = 03(0) .
, -
, , -
'.
N V
> '
, \/ J / > . 17.4 . 17.5
131
-
4.
-
>
N J 11
. 17.6 ( 270)
. 17.6.
, 0
( 90)