a
A = (0, 0), B = (3, 4) A = (2, 4), B = (6, 8)
U
U = (-1, 5) U = (k, -1k)
2
x + 2y - 1 = 0 3x - 3y + 27 = 0
f(x) = x + 5 y = 3 f(x) = 4x - 5 f(x) = 2x + 8
Å„Å‚
òÅ‚ x+3
, x < 2
5
f(x) =
ół
2x - 3, x e" 2
Å„Å‚ Å„Å‚
x+16
ôÅ‚ ôÅ‚
, x < -1 x + 4, x d" 1
ôÅ‚ ôÅ‚
3
òÅ‚ òÅ‚
f(x) = - 3x, -1 d" x < 2
f(x) = -2x + 6, 1 < x < 5
2
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
ół ół
x-14
, x > 2 x - 6, x e" 5
3
10x - 2 > 5x + 4 3 + 2(x - 4) e" 2(2x + 3) - 3(3x - 1)
1 1 3 1
5 - (2x - 3) + (4 - x) = (5x - 2) - (x - 2)
4 2 4 8
|x + 1| = 3 |2x + 6| > 4
|2x| - |x + 5| < 8 |x - 1| > |2x + 1|
|3x - 2| + x = 11
|x| - |x - 2| = 2 |x - 2| = -|x| + 2
2|x + 1| > x + 4 |x + 2| - |x| < 1 |3x - 2| + |x| < 10
 f(x) = -36 + x2
f(x) = 1 - 3x2 f(x) = -32x + 64x2 + 4
f(x) = -(x + 2)(2x - 3) f(x) = 5 - 3x2 f(x) = x2 - 5 f(x) = (1 + x)2
f(x) = 2x2 + (x - 1)2 f(x) = -(5x + 1)2
x2 + 3x = -2 x2 + 4 = -5x x2 = -4
1 5
2 - x2 = 5x2 - x 3x2 - x + 2 = 7 - 3x x(1 - 2x) + 3 = (x - 1)2
3 6
5x2 = 0
1
x2 - x = 4 - 2x2
4
x2 + 4 > -5x x2 > -4
(1 + x)(x - 3) e" -2x + 1 x2 > 0
3
7x2 + x + 2 < -3x + 2 5x2 + 2x e" 0
4
1 1
- 3x2 e" -2(4 + 3x)2 x2 d" -2(x - 2) + x -7x2 e" 0
3 7
f(x) = -3 + x2 - x 2
1
f(x) = 10 - 2x + x 4
5
x
Å„Å‚
òÅ‚
x2 + 1, x < 1
f(x) =
ół
-x2 + 1, x e" 1
Å„Å‚
òÅ‚
(x - 2)(x + 3), x d" 2
f(x) = f(x) = |3x + 1| - x2
ół
-(x + 1)2, x > 2
" " "
7·322-5·320 15
3 5 15
318 3-10·94
1 1 1 1
2 2 2 2
(8 - 18 + 50 )2 (2)50(1, 5)50
3
" "
1 1
2 2
(2 5 - 2) (2 5 + 2)
ab a " N, b " Q
" "
5
5·5-2 7 64
" "
(125)-1 49· 3 7 2
" "
" "
3
8 27
"
2 2 8 32
3
2
9
1
a-1 b4
a · b3(a-3 + ) (4x4y6)- 2
(a4b)(1a-3b-1)
a2·a-5 a 9 2
3 1 1 1
1
8 8 2 2
a · b2(a · b + ) (a b ) : (a- b )
b2
"
f(x) = x-1 f(x) = x
1
1
3
f(x) = x- g(x) = x8 f(x) = x6 f(x) = x-6 f(x) =
x-4
"
3 4
3
2 3
(3x )-1 = 24-1 x = 3 3
x-1 > x-2 x-3 e" x-2
m, n
(3)m < (3)n 1.3m < 1.3n (0.2)m > (0.2)n (4)m < (0.8)n
4 4 5
32x+2 + 32x = 30 7 · 5x - 5x+2 + 450 = 0 8x-3 = 9x-3
152x+4 = 33x · 54x-4 62x+4 = 33x · 2x+8
x
1
> 4
2
2
22x+4 - 4x > 15 5x -7x+12 > 1
2
1 1 1
5x+3 < 25 · 7-x-1 · 4x+1 < 0.5x · 22x+2 <
64 64
2x2
x
log2 x = 5 log3 x = -3
1 1
1
log x = -2 log0,1 x = -1 logx 8 = logx 81 = 4
2
2
3 3
logx 4 = -2 logx 1 = logx 625 =
8 2 4
 f(x) = log2 (1 - x) f(x) = log3(-x)
f(x) = log3 (x2) f(x) = log3 (x2 + 2x + 3)
f(x) = log3(-x)
f(x) = log0,5 (1 - x) f(x) = | log3 x
f(x) = log2 |x| f(x) = 1 - log3 x
a b
log1 a > log1 b log3 a < log3 b log3 a > log3 b
2 2 4 4
log"2 a < log"2 b loga 2 > logb 2 loga 2 < logb 2
3 3
1
log2(x2 + 6x + 17) = 3 log(3x + 4) + log(x + 8) = 2 3log x =
27
log1 |x - 3| < -2
4
1
logx 4 < 2 logx 27 > -3
45ć% 60ć% 105ć% 150ć% 210ć% 270ć%
1 1 3 5 11 19 7
Ä„ Ä„ Ä„ Ä„ Ä„ Ä„ Ä„
6 3 4 18 6 16 4
a, b, c
Ä… ² sin Ä… cos Ä… tgÄ… ctgÄ… sin ² cos ² tg² ctg²
a = 8, b = 12.7, c = 15.03 a = 10.9, b = 14.3, c = 18
a = 10cm, ² = 30ć%
a = 6cm, c = 12cm c = 16cm, ² = 60ć%
f(x)
f(x) = tgx f(x) = ctg3x
3
f(x) = sin 3x f(x) = cos x · cos x f(x) = sin3 x
f(x) = x sin x f(x) = x cos x
2ctgx
2(sin6 x + cos6 x) - 3(sin4 x + cos4 x) + 1
1+ctg2x
1-cos 2x+sin 2x
1+cos 2x+sin 2x
1 sin x 1
(tgx + ctgx)2 = ctgx + =
sin2 x cos2 x 1+cos x sin x
1 1
(sin x - )(sin x + cos x) = ctgx - tgx
cos x
4
x = x = 1
3
f(x) = -5x f(x) = -1x
2
y = -1x + 1 y = x + 9
2
5
x0 = -5 x0 x0 = x0 = -4
4
13 9
x " (6, ") x " [21, ") x = x = x e" 2
5 7 4 2
x " {0, 1, 2}
x " (-", -2) *" (2, ") x " (-1, ") x " (-2, 3)
2
" "
3 3 1
(x - 6)(x + 6) 3( - x)( + x) 64(x - )2
3 3 4
"
3 1 49 15
x1 = -2, x2 = , W = , x1,2 = Ä… , W = (0, 5)
2 4 8 3
1 2
x1 = -1, W = (-1, 0) W = , x1 = -1, W = -1, 0
3 3 5 5
3
x = -1 x = -2 x = -1 x = -4 x =
2
" " "
2 -4+ 151 -4- 151 17- 1057
x = -5 x = -1 x = x = x = x =
2 3 9 9 32
" " "
17+ 1057 2+2 37 2-2 37
x = x = 0 x = x =
32 9 9
x " (-", -4) *" (-1, +") x " R x " (-", -2] *" [2, +")
"
91
x " R \ {0} x " (-15, 0) x " (-", -2] *" [0, +") x " -", -8 - *"
28 5 5 15
" " "
91 -7- 161 -7+ 161
-8 + , +" x " ,
5 15 2 2
" "
1- 21 1+ 21 10
x " , x " (-", ) 522 135 15
2 2 3
8 1 4
3 32 11 1 3 14 5
2 15 2 2 4 3 6
5 7- 2 2 2 2 3
7 3
b3 3 1
8
a2 + b7 a a2b3 + a a b- 8
a2 2x2y3 2
4 3 x " (1, +") x " (0, 1]
m > n m < n m < n m > n
1 3
x = x = 2 x = x = 3 x = 4 x = 4
2 2
x " (-", -2) x " (0, ") x " (-", 3) *" (4, ") x " (-", -1)
x " (-", -2) *" (4, ")
16
1 1 1 1
3
32 4 10 64 5
27 3 2 4
x " (-", 1) x " (-", 0) x " R \ {0} x " (-", -1) *" (1, ")
x " R x " (-3, 2)
a < b a > b a > b a < b a < b a < b
-3 2 10-3
x " (-", -13) x " (0, 1) *" (2, +") x " (0, 1) *" (3, +")
Ä„ Ä„ 7 5 7 3
Ä„ Ä„ Ä„ Ä„
4 3 12 6 6 2
30ć% 60ć% 135ć% 50ć% 330ć% 570ć% 315ć%
sin Ä… = 0.532 cos Ä… = 0.845 tgÄ… = 0.63 ctgÄ… = 1.5875 sin ² = 0.845
cos ² = 0.532 tg² = 1.5875 ctg² = 0.63 sin Ä… = 0.6055 cos Ä… = 0.794 tgÄ… = 0.762
ctgÄ… = 1.312 sin ² = 0.794 cos ² = 0.6055 tg² = 0.762 ctg² = 1.312
" "
"
10 3 20 3
b = , c = , Ä… = 60ć% b = 6 3, Ä… = 300, ² = 600 a = 8, b =
3 3
"
8 3, ą = 30ć%
Ä„
T = 3Ä„ T =
3