relations-function
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The document discusses mathematical relations and functions. It defines Cartesian products and relations. Cartesian products combine elements from two sets into ordered pairs. Relations are subsets of Cartesian products that satisfy a given condition. The document provides examples of calculating Cartesian products and defining relations for given sets. It also establishes properties of Cartesian products, such as A × (B ∪ C) = (A × B) ∪ (A × C) and discusses domains and ranges of relations.
![F F 7
F x = y
y - 1
F F x y - 1 = y
กก F F x2
(y 1) = y2
=> y2
x2
y + x2
= 0
กF ก y F y =
2 2 2
x x (x - 4)
2
±
F 2 2
x (x - 4) ≥ 0 กF ก F ( ∞, 2] ∪ [2, ∞)
F x
x - 1
> 0 ก F x {( ∞, 2] ∪ [2, ∞)} ∩ [0, ∞) = [2, ∞)
F F Rr = [2, ∞)
1.3 ก F
F 1.7 ก F r = {(x, y) | y = 5 - 2x
3 } r 1
ก 1.4 F F x ˈ y y ˈ x ก F ก
F F ก r 1
= {(x, y) | x = 5 - 2y
3 }
F F x ก F
=> ก x = 5 - 2y
3 F F 3x = 5 2y
2y = 5 3x y = 5 - 3x
2
r 1
= {(x, y) | y = 5 - 3x
2 }
1.4
ก F r ˈ F ก A B F r 1
ˈ F ก B A
r 1
= {(y, x) | (x, y) ∈ r}](https://image.slidesharecdn.com/lecture005-relationsfunction-140713074949-phpapp02/85/relations-function-15-320.jpg)
![F F 11
ก F ˆ กF ก ก F
F ˆ กF F 2 ˆ กF f : A → B ก g : A → B F F F F
f = g F F F
F ก F f : A → B g : A → B
(fl) F f = g F f(x) = g(x) ก x ∈ A
F F y ∈ B (x, y) ∈ f
F f = g F F (x, y) ∈ g
y = f(x) y = g(x) f(x) = g(x)
(›) F f(x) = g(x)
(⊆) F (x, y) ∈ f F x ∈ A y ∈ B y = f(x)
F ก F F F y = g(x) F
(x, y) ∈ g f ⊆ g
(⊇) F ก ⊆
ก F F f = g
ก F F ˈ
2.2
F f ˈ F ก A B ก F F f ˈ ˆ กF ก A B (function from A to B)
F ก F f : A → B ก F
1) Df = A
2) ∀x ∈ A, ∀y, z ∈ B, [(x, y) ∈ f - (x, z) ∈ f fl y = z]
2.1
ก F f : A → B g : A → B F f = g ก F f(x) = g(x) ก x ∈ A](https://image.slidesharecdn.com/lecture005-relationsfunction-140713074949-phpapp02/85/relations-function-19-320.jpg)
![14 F ˆ กF
2.2 ˆ กF ˆ กF
F 2.3 ก F f(x) = x2
2 x ∈ [0, 3] F f(x) ก F ˈ ˆ กF
ˆ กF
ก F x1, x2 ∈ [0, 3] x1 < x2 F F
x1
2
< x2
2
3x1
2
< 3x2
2
3x1
2
2 < 3x2
2
2
f(x1) < f(x2)
F f ˈ ˆ กF F [0, 3]
ก F ˆ กF ก F ˈ ˆ กF ˆ กF ก ก
2.4 F F กก ก ˆ กF ( F ก F F ) F
F
F 2.4 ก ˆ กF f ก F F 2.3 ก y = f(x) F F
F [ 3, 3] ˈ ˆ กF ˆ กF
ก y = f(x) = x2
2 ก F
2.4
ก F f ˈ ˆ กF ก R R F A ⊆ Df
1) f ˈ ˆ กF (increasing function) A ก F x1, x2 ∈ A F x1 < x2 F
f(x1) < f(x2)
2) f ˈ ˆ กF (decreasing function) A ก F x1, x2 ∈ A F x1 < x2 F
f(x1) > f(x2)](https://image.slidesharecdn.com/lecture005-relationsfunction-140713074949-phpapp02/85/relations-function-22-320.jpg)
![F F 15
ก F ˈ ก y = x2
2 ก x ∈ [ 3, 3] F F ก ˈ 2 F
[ 3, 0] F ก F ก F [0, 3] F ก F ก F [0, 3]
x F f(x) F F F ˆ กF f F [0, 3] ˈ ˆ กF
ก F [ 3, 0] F x F f(x)
ก ก ก F x F f(x) F F ก
F F f(x) = x2
2 ˈ ˆ กF F [ 3, 0]
ʿก 2.2
1. ก F f(x) = 2
1
x + 1
F f(x) ˈ ˆ กF ก x ∈ [0, 2]
2. ก F f(x) = 4x2
3 F F ก x ∈ [0, 1] F f ˈ ˆ กF
3. F F F 1 F 2 ˈ Fก
2.3 ˆ กF ก (Composite function)
F ก
1. ก 2.5 F F Dgof = A
2. ˆ กF ก f g F F F F F
F Rf ∩ Dg ≠ φ F ˆ กF ก f g F F ก ก ˆ กF ก
g f F F F F Rf ∩ Dg ≠ φ F ˆ กF ก
g f F F F F F F F
F 2.5 ก F f(x) = 2x + 1 g(x) = x2
+ 3 F gof F F F Fก
gof ก F
ก Rf = R Dg = R Rf ∩ Dg ≠ φ F gof
2.5 F F gof(x) = g(f(x))
= g(2x + 1)
= (2x + 1)2
+ 3
2.5
ก F f : A → B g : B → C F gof : A → C](https://image.slidesharecdn.com/lecture005-relationsfunction-140713074949-phpapp02/85/relations-function-23-320.jpg)
![16 F ˆ กF
= 4x2
+ 4x + 1 + 3
= 4x2
+ 4x + 4
F 2.6 ก ˆ กF f g F 2.5 F fog F F F Fก F fog
ก F
ก Rg = [3, ∞) Df = R Rg ∩ Df ≠ φ F fog
2.5 F F fog(x) = f(g(x))
= f(x2
+ 3)
= 2(x2
+ 3) + 1
= 2x2
+ 6 + 1
= 2x2
+ 7
F 2.7 ก F f(x) = 2
1
x - 2x - 3
g(x) = x 2 F ˆ กF ก g f
F
ก Rg ∩ Df = R ∩ [R { 1, 3}] = R { 1, 3} ≠ φ F fog
ก fog(x) = f(g(x)) = f(x 2)
= 2
1
(x - 2) - 2(x - 2) - 3
= 2
1
x - 6x + 5
F 2.8 ก F 2.7 F fog(0)
ก fog(x) = 2
1
x - 6x + 5
F F fog(0) = 2
1
0 - 6(0) + 5
= 1
5](https://image.slidesharecdn.com/lecture005-relationsfunction-140713074949-phpapp02/85/relations-function-24-320.jpg)
![F F 17
F ˈ ก F F ก F F ˆ กF ก ˈ
ˆ กF F
F 1) F ∀x1, x2 ∈ A, [gof(x1) = gof(x2) fl x1 = x2]
F x1, x2 ∈ A F F y ∈ C (x1, y) ∈ gof (x2, y) ∈ gof
y = gof(x1) y = gof(x2)
F gof(x1) = gof(x2)
ก g ˈ ˆ กF F f(x1) = f(x2)
ก f ˈ ˆ กF F F F x1 = x2 F ก
2) F Rgof = C
(⊆) F y ∈ Rgof F F x ∈ A z ∈ B (x, z) ∈ f (z, y) ∈ g
y = g(z)
y ∈ C
(⊇) F y ∈ C F F x ∈ A (x, y) ∈ gof
y = gof(x)
y ∈ Rgof
F F Rgof = C F ก
ʿก 2.3
1. ก F A, B, C ˈ F ˈ F f : A → B g : B → C F
gof : A → C Dgof = A
2. F f : A → B B F g : B → C, h : B → C F F gof = hof
F g = h
2.2
F f : A → B g : B → C F F F F gof : A → C ˈ
ˆ กF F C](https://image.slidesharecdn.com/lecture005-relationsfunction-140713074949-phpapp02/85/relations-function-25-320.jpg)
![F F 29
3.2.2 F ˆ กF ก
2 F ก ก F ˆ กF F F
F ˆ กF ก F F
ˆ กF F
y = sin x R [ 1, 1]
y = cos x R [ 1, 1]
y = tan x R { n
2
π | n ˈ } R
y = cosec x R {nπ | n ˈ } R ( 1, 1)
y = sec x R { n
2
π | n ˈ } R ( 1, 1)
y = cot x R {nπ | n ˈ } R
3.2.3 ก ˆ กF ก
ก ก ก ˆ กF ก F F F F ก ก F
ก F ˆ กF ก F ก F F
3.2 ก y = sin x F [ 2ππππ, 2ππππ]
3.3 ก y = cos x F [ 2ππππ, 2ππππ]](https://image.slidesharecdn.com/lecture005-relationsfunction-140713074949-phpapp02/85/relations-function-37-320.jpg)
![30 F ˆ กF
3.4 ก y = tan x F ( ∞∞∞∞, ∞∞∞∞)
ʿก 3.2
1. ก F y = cos x + sin x ก x ∈ [0, π] F F F
ก ˆ กF ก F F
2. ก F f(x) = sin x F ˆ กF f ก F ˈ ˆ กF F ก x ∈ [0, 2
π ]](https://image.slidesharecdn.com/lecture005-relationsfunction-140713074949-phpapp02/85/relations-function-38-320.jpg)
![32 F ˆ กF
F 3.7 F cos2 5
12
π + sin2 5
6
π + sin2 5
12
π + cos2 7
6
π
ก. 2 . 3
. 11
4 . 3
2
. F F ก F
ก F cos2 5
12
π + sin2 5
6
π + sin2 5
12
π + cos2 7
6
π ก F F F
(cos2 5
12
π + sin2 5
12
π ) + (sin2 5
6
π + cos2 7
6
π )
= 1 + (sin2 5
6
π + cos2 7
6
π )
= 1 + [sin2
(π 6
π ) + cos2
(π + 6
π )]
= 1 + (sin2
6
π + cos2
6
π )
= 1 + 1
= 2
ʿก 3.3
1. F ก ก F ก ก F F ˈ
1) cos2
θ cot θ + sin2
θ tan θ + 2 sin θ cos θ = tan θ + cot θ
2) tan2
θ sin2
θ = tan2
θ sin2
θ
3)
2
2
cosec
1 + tan
θ
θ
= cot2
θ
2. F 3.1 F 4) F 5) ˈ](https://image.slidesharecdn.com/lecture005-relationsfunction-140713074949-phpapp02/85/relations-function-40-320.jpg)
![F F 35
( )sin (3A - A)
sin A cos A ⋅( )sin (3A + A)
sin A cos A = 2
( )sin2A
sinAcosA ⋅( )sin4A
sinAcosA = 2
( )sin2A
sinAcosA ⋅( )sin2(2A)
sinAcosA = 2
( )2sinAcosA
sinAcosA ⋅( )2sin2Acos2A
sinAcosA = 2
( )2sinAcosA
sinAcosA ⋅( )4sinAcosAcos2A
sinAcosA = 2
2 ⋅ 4 cos 2A = 2
cos 2A = 1
4
ก F F ˈ ก ก F ก ˈ
ก
F F ก F F 2) F 3) F F ก F ก F F ก
F F F F F ˈ ʿก
2) ก 2 sin( )x + y
2 cos( )x - y
2 = 2 sin( )yx
2 2+ cos( )yx
2 2-
ก F A = x
2 B = y
2 F F F F F F
= 2 sin (A + B) cos (A B)
= 2 (sin A cos B + cos A sin B)(cos A cos B + sin A sin B)
= 2 (sin A cos B ⋅ cos A cos B + cos A sin B ⋅ cos A cos B
+ sin A cos B ⋅ sin A sin B + cos A sin B ⋅ sin A sin B)
= 2 (sin A cos A cos2
B + cos2
A sin B cos B + sin2
A cos B sin B + cos A sin A sin2
B)
= 2 [sin A cos A (cos2
B + sin2
B) + sin B cos B (cos2
A + sin2
A)]
= 2 (sin A cos A + sin B cos B) ( ก cos2
B + sin2
B = 1 cos2
A + sin2
A = 1)
= 2 sin A cos A + 2 sin B cos B
= sin 2A + sin 2B
3.4 ก ก F ˈ
1) cos x + cos y = 2 cos( )x + y
2 cos( )x - y
2
2) sin x + sin y = 2 sin( )x + y
2 cos( )x - y
2
3) cos x cos y = 2 sin( )x + y
2 sin( )x - y
2
4) sin x sin y = 2 cos( )x + y
2 sin( )x - y
2](https://image.slidesharecdn.com/lecture005-relationsfunction-140713074949-phpapp02/85/relations-function-43-320.jpg)
![36 F ˆ กF
F A = x
2 B = y
2
F F 2 sin( )x + y
2 cos( )x - y
2 = sin x + sin y
3) ก 2 sin( )x + y
2 sin( )x - y
2 = 2 sin( )yx
2 2+ sin( )yx
2 2-
ก F A = x
2 B = y
2 F F F F F F
= 2 sin (A + B) sin (A B)
= 2 (sin A cos B + cos A sin B)(sin A cos B cos A sin B)
= 2 [(sin A cos B)2
(cos A sin B)2
]
= 2 sin2
A cos2
B + 2 cos2
A sin2
B
= 2 sin2
A cos2
B + 2 (1 sin2
A)(1 cos2
B)
= 2 sin2
A cos2
B + 2 (1 sin2
A cos2
B + sin2
A cos2
B)
= 2 sin2
A cos2
B + 2 (1 sin2
A) 2 cos2
B + 2 sin2
A cos2
B
= 2 (1 sin2
A) 2 cos2
B
= 2 cos2
A 2 cos2
B
= (2 cos2
A 1) (2 cos2
B 1)
= cos 2A cos 2B
F A = x
2 B = y
2
F F 2 sin( )x + y
2 sin( )x - y
2 = cos x cos y F ก
ʽ F F F ก ก ก ก ˈ ก F
ก
F 1) ก cos(x + y) + cos(x - y)
2
= (cos x cos y - sin x sin y) + (cos x cos y + sin x sin y)
2
= 2 cos x cos y
2 = cos x cos y
2) ก cos (x - y) - cos (x + y)
2
= (cos x cos y + sin x sin y) - (cos x cos y - sin x sin y)
2
= 2 sin x sin y
2 = sin x sin y
3.5 ก ˈ ก F
1) cos x cos y = cos (x + y) + cos (x - y)
2
2) sin x sin y = cos (x - y) - cos (x + y)
2
3) sin x cos y = sin (x - y) + sin (x + y)
2](https://image.slidesharecdn.com/lecture005-relationsfunction-140713074949-phpapp02/85/relations-function-44-320.jpg)
![58 F ˆ กF
F F
F 4.8 F log93, log9(3x
2), log9(3x
+ 16) ˈ F ก ก ก S ˈ
ก F ก ก F 3S
F F ก F
ก log93, log9(3x
2), log9(3x
+ 16) ˈ F ก ก ก
F F d1 = log9(3x
2) log93
= log9( )x
3 - 2
3
d2 = log9(3x
+ 16) log9(3x
2)
= log9( )x
x
3 + 16
3 - 2
F d1 = d2 ( ก ˈ )
log9( )x
3 - 2
3 = log9( )x
x
3 + 16
3 - 2
x
3 - 2
3 =
x
x
3 + 16
3 - 2
(3x
2)2
= 3(3x
+ 16)
(3x
)2
4(3x
) + 4 = 3(3x
) + 48
(3x
)2
4(3x
) + 4 3(3x
) 48= 0
(3x
)2
4(3x
) 3(3x
) 44 = 0
(3x
)2
7(3x
) 44 = 0
(3x
11)(3x
+ 4) = 0
F F 3x
11 = 0 (3x
+ 4 F F F)
x = log311
3 F Fก F (log93) = 1
2 , 1, 3
2
ก ก Sn = n
2 [2a1 + (n 1)d]
F n = 4, a1 = 1
2 F S4 = 4
2 [2( 1
2 ) + (4 1) 1
2 ] = 2[1 + 3
2 ] = 5
3S
= 35
= 243
F 4.9 x F ก ก
log 2x
log 3 + log3(x 12) = ( )3log x x + 5 - x - 5
F F ก F
กF F ก F ก F F ก ก F F ก กF](https://image.slidesharecdn.com/lecture005-relationsfunction-140713074949-phpapp02/85/relations-function-66-320.jpg)
![F F 59
log3 2x + log3(x 12) = 2 log3 ( )x x + 5 - x - 5
log3 [2x(x 12)] = log3 ( )
2
x x + 5 - x - 5
2x(x 12) = ( )
2
x x + 5 - x - 5
2x(x 12) = x[(x + 5) + (x 5) 2( x + 5 )( x - 5 )]
2x(x 12) = x[2x 2( x + 5 )( x - 5 )]
2x2
24x = 2x2
2x( x + 5 )( x - 5 )
24x = 2x( x + 5 )( x - 5 )
กก F ก F F
576x2
= 4x2
(x2
25)
= 4x4
100x2
4x4
676x2
= 0
4x2
(x2
169) = 0
กF ก F x = 0 x2
= 169
x = 0 x = 13 x = 13
F F x = 0 x = 13 F ก ˈ
ก x = 13
4.6.2 ก ก
ก กF ก ก กก F ก ก กF ก ก ก F ก F
ก F ก ก ก ก กก F F ˆ กF ก
ก F ˈ ˆ กF ก F ˈ F ก ก ก F F ˈ ˆ กF F
ก ก F ก F F
F 4.10 ก F A ˈ ก log4log3log2(x2
+ 2x) ≤ 0 ˈ ก
A ก
ก ก log4log3log2(x2
+ 2x) ≤ 0
F F log3log2(x2
+ 2x) ≤ 1 (4 > 0 ˈ ˆ กF )
log2(x2
+ 2x) ≤ 3 (3 > 0 ˈ ˆ กF )
x2
+ 2x ≤ 8 (2 > 0 ˈ ˆ กF )
x2
+ 2x 8 ≤ 0
(x 2)(x + 4) 0
F F 4 ≤ x ≤ 2 ˈ F F F [ 4, 2]](https://image.slidesharecdn.com/lecture005-relationsfunction-140713074949-phpapp02/85/relations-function-67-320.jpg)
![60 F ˆ กF
F ก x2
+ 2x > 0 F F x(x + 2) > 0
( ∞, 2) ∪ (0, ∞)
ก A = [ 4, 2] ∩ [( ∞, 2) ∪ (0, ∞)] = [ 4, 2) ∪ (0, 2]
F F ˈ ก A F กF { 4, 3, 1, 2} 4](https://image.slidesharecdn.com/lecture005-relationsfunction-140713074949-phpapp02/85/relations-function-68-320.jpg)
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relations-function
- 1. ความสะมพะนธแลัฟงกชะนความสะมพะนธแลัฟงกชะนความสะมพะนธแลัฟงกชะนความสะมพะนธแลัฟงกชะนความสะมพะนธแลัฟงกชะนความสะมพะนธแลัฟงกชะนความสะมพะนธแลัฟงกชะนความสะมพะนธแลัฟงกชะน ((((((((MMMMMMMMaaaaaaaatttttttthhhhhhhheeeeeeeemmmmmmmmaaaaaaaattttttttiiiiiiiiccccccccaaaaaaaallllllll RRRRRRRReeeeeeeellllllllaaaaaaaattttttttiiiiiiiioooooooonnnnnnnnssssssss aaaaaaaannnnnnnndddddddd FFFFFFFFuuuuuuuunnnnnnnnccccccccttttttttiiiiiiiioooooooonnnnnnnnssssssss)))))))) F ก ““““ F F”””” F 5 F F F F F . . 2537 www.thai-mathpaper.net
- 3. ก ก ก ˈ ก ก ก F F F F F ˈ ก 3 F ก ˆ กF ก ก ก F F ʿก F F F F F F F ก F F F ก Fก F F F ก F F ก F ก F F F F F F F F F 19 ก . . 2552
- 5. F F ˈ F 5 ก 15 F F ก F F ˆ กF F ก F F F ก F F ก ก F ก 1 ˈ ก F F กก F F ก F F F ก F F F ก F ก F 2 F กF ˆ กF F F ˆ กF ˆ กF ˆ กF ˆ กF ก ก ˆ กF ʽ F F ˆ กF 3 ˈ F ก ˆ กF ก F ก ก กกF F F F ก F ˆ กF ก F ก ก F ก ˆ กF ก ก ก F ก F F ʽ F F ก ˆ กF ก 4 ˈ F F F ก F ˆ กF ก F ˆ กF ก F ก ก ˆ กF ก F ก ก ก ˆ กF ก ˈ ก ˆ กF ก F F ก ก ˆ กF ก F ก ก ˆ กF ก ก ก ก ก F F ʽ F F ก ก ก F F F F F F กF ก F F F กก F ก F F F ก ก F ก F F F F F F F F F 14 ก F . . 2549
- 7. 1 F 1 8 1.1 F 1 1.2 F 4 1.3 ก F 7 2 ˆ กF 9 22 2.1 ˆ กF 9 2.2 ˆ กF ˆ กF 14 2.3 ˆ กF ก 15 2.4 ก ˆ กF 18 2.5 ˆ กF 22 3 F ก ˆ กF ก 23 45 3.1 F ก 23 3.2 ˆ กF ก 26 3.3 ก ก F ก 31 3.4 ˆ กF ก ก 33 3.5 ก F ก F 37 3.6 40 3.7 ก ˆ กF ก 41 4 ˆ กF ก F ˆ กF ก 47 60 4.1 ˆ กF ก F 47 4.2 ˆ กF ก 49 4.3 ˆ กF ก 50 4.4 ก 54 4.5 ก ก ก F 55 4.6 ก ก ก 57 ก 61 ก 63
- 9. 1 F 1.1 F (Cartesian Product) F 1.1 ก F A = {1, 2, 3}, B = {0, 1} F A × B, B × A, A × A, B × B 1.1 F F A × B = {(1, 0), (1, 1), (2, 0), (2, 1), (3, 0), (3, 1)} B × A = {(0, 1), (0, 2), (0, 3), (1, 1), (1, 2), (1, 3)} A × A = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)} B × B = {(0, 0), (0, 1), (1, 0), (1, 1)} ก F 1.1 F F F ก ก F A × B ≠ B × A ˈ F ก ˈ F 1.1 ก F A, B ˈ F F (a, b) a ∈ A b ∈ B F ก F A × B 1.1 ก F A, B, C ˈ n(A), n(B) ˈ ก A B F F F 1) A × B ≠ B × A 2) A × B = B × A ก F A = B A = φ B = φ 3) A × φ = φ× A = φ 4) n(A × B) = n(A) ⋅ n(B) 5) A × (B ∪ C) = (A × B) ∪ (A × C) 6) A × (B ∩ C) = (A × B) ∩ (A × C) 7) A × (B C) = (A × B) (A × C)
- 10. 2 F ˆ กF F F ก F F 2), 3), 4), 5), 6) 7) 2) (fl) F (x, y) ∈ A × B F A × B = B × A ก F F x ∈ A fl x ∈ B F A ⊆ B ก ก ก F F B ⊆ A A = B ก F φ ˈ ก F F F φ = A F φ = B (›) F ก fl F F A × B = B × A ก F A = B A = φ B = φ 3) A × φ ‹ x ∈ A x ∈ φ ‹ x ∈ φ x ∈ A ‹ x ∈ φ ‹ φ ‹ φ× A 4) ก F A, B ก n(A), n(B) F F F x ∈ A Fก ก y ∈ B F F ก n(A) ⋅ n(B) n(A × B) = n(A)⋅n(B) 5) (⊆) ก F (x, y) ∈ A × (B ∪ C) (x, y) ∈ A × (B ∪ C) ‹ x ∈ A - y ∈ (B ∪ C) ‹ x ∈ A - (y ∈ B / y ∈ C) ‹ (x ∈ A - y ∈ B) / (x ∈ A - y ∈ C) ‹ ((x, y) ∈ A × B) / ((x, y) ∈ A × C) ‹ (x, y) ∈ (A × B) ∪ (A × C) A × (B ∪ C) ⊆ (A × B) ∪ (A × C) (⊇) ก F (x, y) ∈ (A × B) ∪ (A × C) (x, y) ∈ (A × B) ∪ (A × C) ‹ ((x, y) ∈ A × B) /((x, y) ∈ A × C) ‹ (x ∈ A - y ∈ B) / (x ∈ A - y ∈ C) ‹ (x ∈ A) - (y ∈ B / y ∈ C) ‹ (x ∈ A) - (y ∈ B ∪ C) ‹ (x, y) ∈ A × (B ∪ C) (A × B) ∪ (A × C) ⊆ A × (B ∪ C) F F A × (B ∪ C) = (A × B) ∪ (A × C) F ก 6) (⊆) ก F (x, y) ∈ A × (B ∩ C) (x, y) ∈ A × (B ∩ C) ‹ (x ∈ A) - (y ∈ B ∩ C) ‹ (x ∈ A) - (y ∈ B - y ∈ C) ‹ (x ∈ A - x ∈ A) - (y ∈ B - y ∈ C) (Idempotent)
- 11. F F 3 ‹ (x ∈ A - y ∈ B) - (x ∈ A - y ∈ C) ‹ ((x, y) ∈ A × B) - ((x, y) ∈ A × C) ‹ (x, y) ∈ (A × B) ∩ (A × C) A × (B ∩ C) ⊆ (A × B) ∩ (A × C) (⊇) ก F (x, y) ∈ (A × B) ∩ (A × C) (x, y) ∈ (A × B) ∩ (A × C) ‹ ((x, y) ∈ A × B) - ((x, y) ∈ A × C) ‹ (x ∈ A - y ∈ B) - (x ∈ A - y ∈ C) ‹ (x ∈ A - x ∈ A) - (y ∈ B - y ∈ C) ‹ (x ∈ A) - (y ∈ B ∩ C) ‹ (x, y) ∈ A × (B ∩ C) (A × B) ∩ (A × C) ⊆ A × (B ∩ C) F F A × (B ∩ C) = (A × B) ∩ (A × C) 7) (⊆) ก F (x, y) ∈ A × (B C) (x, y) ∈ A × (B C) ‹ (x ∈ A) - y ∈ (B C) ‹ (x ∈ A) - (y ∈ (B ∩ C′)) ‹ (x ∈ A) - (y ∈ B - y – C) ‹ (x ∈ A - x ∈ A) - (y ∈ B - y – C) (Idempotent) ‹ (x ∈ A - y ∈ B) - (x ∈ A - y – C) ‹ ((x, y) ∈ A × B) - ((x, y) – A × C) ‹ ((x, y) ∈ A × B) - ((x, y) ∈ (A × C)′) ‹ (x, y) ∈ (A × B) ∩ (A × C)′ ‹ (x, y) ∈ (A × B) (A × C) A × (B C) ⊆ (A × B) (A × C) (⊇) ก F (x, y) ∈ (A × B) (A × C) (x, y) ∈ (A × B) (A × C) ‹ ((x, y) ∈ A × B) ∩ ((x, y) – A × C) ‹ (x ∈ A - y ∈ B) ∩ (x ∈ A - y – C) ‹ x ∈ A - (y ∈ B - y – C) ‹ x ∈ A - (y ∈ B - y ∈ C′) ‹ x ∈ A - (y ∈ B ∩ C′) ‹ x ∈ A - y ∈ (B C) ‹ (x, y) ∈ A × (B C) (A × B) (A × C) ⊆ A × (B C)
- 12. 4 F ˆ กF F F A × (B C) = (A × B) (A × C) ʾก 1.1 1. ก F A = {0, 1, 3, 4}, B = {2, 3, 4, 5} A × B 2. ก F A = {x | x ∈ N x ≤ 10}, B = {y | y ∈ N y ≤ 0} n(A × B) 3. ก F A, B, C ˈ F F F F (A × A) ∩ (B × C) = (A ∩ B) × (A ∩ C) 1.2 F (Relation) F 1.2 ก F 1.1 F A × B = {(1, 0), (1, 1), (2, 0), (2, 1), (3, 0), (3, 1)} F ก F r1 = {(x, y) ∈ A × B | x > y} r1 ก ก F {(1, 0), (2, 0), (2, 1), (3, 0), (3, 1)} F F r1 ⊆ A × B F 1.2 F ก ก F ก F 2 ก ก ก ก ก ก Fก ˈ F ก ก ก 1.2 ก F A, B ˈ F r ˈ F ก A B r ⊆ A × B
- 13. F F 5 กF ก F ก ก F F F ก 2 กF F 1.3 ก F 1.2 F F r1 = {(1, 0), (2, 0), (2, 1), (3, 0), (3, 1)} Dr = {1, 2, 3} Rr = {0, 1} F 1.4 ก F A = {x ∈ I+ | 3|x x ≤ 10}, B = {y ∈ I+ | 2|y y ≤ 10} F r = {(x, y) ∈ A × B | x + y < 12} Dr, Rr ก Fก A, B F ก ก F F ก ก F A = {3, 6, 9}, B = {2, 4, 6, 8, 10} F F A × B = {(3, 2), (3, 4), (3, 6), (3, 8), (3, 10), (6, 2) (6, 4), (6, 6), (6, 8) (6, 10), (9, 2), (9, 4) (9, 6), (9, 8), (9, 10)} r = {(3, 2), (3, 4), (3, 6), (3, 8), (6, 2), (6, 4)} Dr = {3, 6}, Rr = {2, 4, 6, 8} ก Dr = {x | 3|x x ≤ 6}, Rr = {y | 2|y y ≤ 8} F ก ก F 1.3 F 1.4 F F F r ⊆ A × B Dr ⊆ A Rr ⊆ B F F r ⊆ R × R F ก F Fก ˈ F F ˈ R × R ก F F ก F ก y x x y ก ก ก ก F F F y x F 1.3 1) (Domain) F r F Dr ก F F (x, y) ˈ ก r Dr = {x | ∃y ∈ B (x, y) ∈ r} 2) F (Range) F r F Rr ก F (x, y) ˈ ก r Rr = {y | ∃x ∈ B (x, y) ∈ r}
- 14. 6 F ˆ กF F 1.5 ก F r = {(x, y) | x + 2y = 1} ก F r ก F F ก F F F ก F r ก F ˈ F F (linear relation) F ก ˈ F F F F fl Dr; x + 2y = 1 2y = 1 x y = 1 - x 2 F ก F x ∈ R F y F Dr = R fl Rr; x + 2y = 1 x = 1 2y F F y ∈R F x ∈ R F Rr = R F 1.6 ก F r = {(x, y) | y = x x - 1 } F r ก x x - 1 F x 1 > 0 => x > 1 F Dr = (1, ∞) y = x x - 1 F ก F r x y
- 15. F F 7 F x = y y - 1 F F x y - 1 = y กก F F x2 (y 1) = y2 => y2 x2 y + x2 = 0 กF ก y F y = 2 2 2 x x (x - 4) 2 ± F 2 2 x (x - 4) ≥ 0 กF ก F ( ∞, 2] ∪ [2, ∞) F x x - 1 > 0 ก F x {( ∞, 2] ∪ [2, ∞)} ∩ [0, ∞) = [2, ∞) F F Rr = [2, ∞) 1.3 ก F F 1.7 ก F r = {(x, y) | y = 5 - 2x 3 } r 1 ก 1.4 F F x ˈ y y ˈ x ก F ก F F ก r 1 = {(x, y) | x = 5 - 2y 3 } F F x ก F => ก x = 5 - 2y 3 F F 3x = 5 2y 2y = 5 3x y = 5 - 3x 2 r 1 = {(x, y) | y = 5 - 3x 2 } 1.4 ก F r ˈ F ก A B F r 1 ˈ F ก B A r 1 = {(y, x) | (x, y) ∈ r}
- 16. 8 F ˆ กF F 1.8 ก F r = {(x, y) | y = x2 3x +2 ก x > 0} r 1 x ก y = x2 3x +2 F F x = y2 3y + 2 x 2 = y2 3y = (y 3 2 )2 9 4 => x + 1 4 = (y 3 2 )2 => y 3 2 = ± 1 4x + => y = 3 2 ± 1 4x + F F r 1 = {(x, y) | y = 3 2 ± 1 4x + } F ก ก F F กก F ก F F F F F F F ก ก F F ก Fก ˈ ก ก F F F F F 1.9 F r 1 F 1.8 ก r 1 = {(x, y) | y = 3 2 ± 1 4x + } F F -1 rD = {x | x ≥ 1 4 } -1 rR = {y | y ≥ 3 2 } 1.5 ก F r ˈ F ก A B Dr, Rr F r F F F 1) -1 rD = Rr 2) -1 rR = Dr
- 17. 2 ˆ กF 2.1 ˆ กF 2.1.1 ˆ กF ก 2.1 ก F F ˈ ˆ กF ก ก Fก F F 1) F F ก F ˈ F F F F F F ก F ก F ก F ก F ก F F ก F F F F ˈ ˆ กF 2) F F ก F ˈ ก F ก F F F ก F F ก ก Y ก ก F F F ก ก F F ก ก F F F ˈ ˆ กF F (x, y) ∈ f F y = f(x) ก F F F 2.1 ก F r1 = {(x, y) | y = 2x 1}, r2 = {(x, y) | x = y2 + 3}, r3 = {(x, y) | y = ( )x1 2 } F 1.1 F F ก F ˈ ˆ กF F ก r1 = {(x, y) | y = 2x 1} F ก ˈ ก ก F 2.1 1) ˆ กF (Function) F F F ˈ ก F F ก F ก F ก F ก F 2) ˆ กF (Domain) ก F F ˈ ก ˆ กF 3) F ˆ กF (Range) ก F ˈ ก ˆ กF
- 18. 10 F ˆ กF ก F F กF F y = 2x 1 ก F F F ก ก Y F F F ก ก F y = 2x 1 r1 ˈ ˆ กF ก r2 = {(x, y) | x = y2 + 3} x = y2 + 3 ก F ก F F ก x = y2 + 3 ก F F F ก ก Y F ก F F F ก กก F 1 F F F ก F F ˈ ˆ กF ก r3 = {(x, y) | y = ( )x1 2 } y = ( )x1 2 ก F ก F ก y = ( )x1 2 ก F x = 0 F F ก F F ก F ˈ ˆ กF
- 19. F F 11 ก F ˆ กF ก ก F F ˆ กF F 2 ˆ กF f : A → B ก g : A → B F F F F f = g F F F F ก F f : A → B g : A → B (fl) F f = g F f(x) = g(x) ก x ∈ A F F y ∈ B (x, y) ∈ f F f = g F F (x, y) ∈ g y = f(x) y = g(x) f(x) = g(x) (›) F f(x) = g(x) (⊆) F (x, y) ∈ f F x ∈ A y ∈ B y = f(x) F ก F F F y = g(x) F (x, y) ∈ g f ⊆ g (⊇) F ก ⊆ ก F F f = g ก F F ˈ 2.2 F f ˈ F ก A B ก F F f ˈ ˆ กF ก A B (function from A to B) F ก F f : A → B ก F 1) Df = A 2) ∀x ∈ A, ∀y, z ∈ B, [(x, y) ∈ f - (x, z) ∈ f fl y = z] 2.1 ก F f : A → B g : A → B F f = g ก F f(x) = g(x) ก x ∈ A
- 20. 12 F ˆ กF 2.1.2 ˆ กF F ˆ กF ก F ˆ กF ก F ˈ ˆ กF F F F F F 1) F ก F Fก ก F F F 2.3 2) F ก F Fก ก ˆ กF F ก F F ก ก X F F ก F ก กก F 1 F ˆ กF F F ˆ กF F F 2.2 ก F f(x) = x x - 1 F f ก F ˈ ˆ กF F F 1) F F x1, x2 ∈ Df F f(x1) = f(x2) F F 1 1 x x - 1 = 2 2 x x - 1 x1 2x - 1 = x2 1x - 1 กก F ก F F x1 2 (x2 1) = x2 2 (x1 1) x1 2 x2 x1 2 = x2 2 x1 x2 2 x1 2 x2 + x2 2 = x2 2 x1 + x1 2 x2(x1 2 + x2) = x1(x2 2 + x1) F F x1 ≠ x2 F F ˆ กF f ก F F ˈ ˆ กF F 2) F ก f(x) = x x - 1 ก F 2.3 1) ˆ กF F ก A B (injective function from A to B or 1 1 function) ˆ กF ก A B x1, x2 ∈ Df F (x1, y) ∈ f (x2, y) ∈ f F x1 = x2 F ก F f : A 1-1 → B 2) ˆ กF ก A B (surjective function from A to B or onto function) ˆ กF ก A B Df = A Rf = B 3) ˆ กF F ก A B (bijective function) ˆ กF ก A B F ˈ ˆ กF F ˆ กF
- 21. F F 13 ก F ก y = f(x) F F ก F F ก ก X F F F ก กก F 1 F F ˆ กF ก F F ˈ ˆ กF F ʿก 2.1 1. ก F r = {(x, y) | y = x2 + 2x + 3} F F r ก F ˈ ˆ กF F F r ก F ˈ ˆ กF F F ˆ กF
- 22. 14 F ˆ กF 2.2 ˆ กF ˆ กF F 2.3 ก F f(x) = x2 2 x ∈ [0, 3] F f(x) ก F ˈ ˆ กF ˆ กF ก F x1, x2 ∈ [0, 3] x1 < x2 F F x1 2 < x2 2 3x1 2 < 3x2 2 3x1 2 2 < 3x2 2 2 f(x1) < f(x2) F f ˈ ˆ กF F [0, 3] ก F ˆ กF ก F ˈ ˆ กF ˆ กF ก ก 2.4 F F กก ก ˆ กF ( F ก F F ) F F F 2.4 ก ˆ กF f ก F F 2.3 ก y = f(x) F F F [ 3, 3] ˈ ˆ กF ˆ กF ก y = f(x) = x2 2 ก F 2.4 ก F f ˈ ˆ กF ก R R F A ⊆ Df 1) f ˈ ˆ กF (increasing function) A ก F x1, x2 ∈ A F x1 < x2 F f(x1) < f(x2) 2) f ˈ ˆ กF (decreasing function) A ก F x1, x2 ∈ A F x1 < x2 F f(x1) > f(x2)
- 23. F F 15 ก F ˈ ก y = x2 2 ก x ∈ [ 3, 3] F F ก ˈ 2 F [ 3, 0] F ก F ก F [0, 3] F ก F ก F [0, 3] x F f(x) F F F ˆ กF f F [0, 3] ˈ ˆ กF ก F [ 3, 0] F x F f(x) ก ก ก F x F f(x) F F ก F F f(x) = x2 2 ˈ ˆ กF F [ 3, 0] ʿก 2.2 1. ก F f(x) = 2 1 x + 1 F f(x) ˈ ˆ กF ก x ∈ [0, 2] 2. ก F f(x) = 4x2 3 F F ก x ∈ [0, 1] F f ˈ ˆ กF 3. F F F 1 F 2 ˈ Fก 2.3 ˆ กF ก (Composite function) F ก 1. ก 2.5 F F Dgof = A 2. ˆ กF ก f g F F F F F F Rf ∩ Dg ≠ φ F ˆ กF ก f g F F ก ก ˆ กF ก g f F F F F Rf ∩ Dg ≠ φ F ˆ กF ก g f F F F F F F F F 2.5 ก F f(x) = 2x + 1 g(x) = x2 + 3 F gof F F F Fก gof ก F ก Rf = R Dg = R Rf ∩ Dg ≠ φ F gof 2.5 F F gof(x) = g(f(x)) = g(2x + 1) = (2x + 1)2 + 3 2.5 ก F f : A → B g : B → C F gof : A → C
- 24. 16 F ˆ กF = 4x2 + 4x + 1 + 3 = 4x2 + 4x + 4 F 2.6 ก ˆ กF f g F 2.5 F fog F F F Fก F fog ก F ก Rg = [3, ∞) Df = R Rg ∩ Df ≠ φ F fog 2.5 F F fog(x) = f(g(x)) = f(x2 + 3) = 2(x2 + 3) + 1 = 2x2 + 6 + 1 = 2x2 + 7 F 2.7 ก F f(x) = 2 1 x - 2x - 3 g(x) = x 2 F ˆ กF ก g f F ก Rg ∩ Df = R ∩ [R { 1, 3}] = R { 1, 3} ≠ φ F fog ก fog(x) = f(g(x)) = f(x 2) = 2 1 (x - 2) - 2(x - 2) - 3 = 2 1 x - 6x + 5 F 2.8 ก F 2.7 F fog(0) ก fog(x) = 2 1 x - 6x + 5 F F fog(0) = 2 1 0 - 6(0) + 5 = 1 5
- 25. F F 17 F ˈ ก F F ก F F ˆ กF ก ˈ ˆ กF F F 1) F ∀x1, x2 ∈ A, [gof(x1) = gof(x2) fl x1 = x2] F x1, x2 ∈ A F F y ∈ C (x1, y) ∈ gof (x2, y) ∈ gof y = gof(x1) y = gof(x2) F gof(x1) = gof(x2) ก g ˈ ˆ กF F f(x1) = f(x2) ก f ˈ ˆ กF F F F x1 = x2 F ก 2) F Rgof = C (⊆) F y ∈ Rgof F F x ∈ A z ∈ B (x, z) ∈ f (z, y) ∈ g y = g(z) y ∈ C (⊇) F y ∈ C F F x ∈ A (x, y) ∈ gof y = gof(x) y ∈ Rgof F F Rgof = C F ก ʿก 2.3 1. ก F A, B, C ˈ F ˈ F f : A → B g : B → C F gof : A → C Dgof = A 2. F f : A → B B F g : B → C, h : B → C F F gof = hof F g = h 2.2 F f : A → B g : B → C F F F F gof : A → C ˈ ˆ กF F C
- 26. 18 F ˆ กF 2.4 ก ˆ กF 2.4.1 ก ˆ กF F F ˆ กF ˈ F ก ˆ กF F F ก F F ˈ F ก ˆ กF ก ก F ก F F F 1) F F f 1 : B → A F f ˈ ˆ กF F B 1.1) F f ˈ ˆ กF F F y1, y2 ∈ A F F (y1, x) ∈ f 1 (y2, x) ∈ f 1 (x, y1) ∈ f (x, y2) ∈ f y1 = f(x) y2 = f(x) y1 = y2 f ˈ ˆ กF F F ก 1.2) F F F F ˈ ʿก 2) F F f ˈ ˆ กF F B F f 1 : B → A F F F F ˈ ʿก F 2.9 ก F f(x) = x2 F ˆ กF f ก F ก F ก ก ˆ กF y = f(x) F ˆ กF ก F F ˈ ˆ กF F 2.2 F ˆ กF f ก F F ก F 2.10 ก F g ˈ ˆ กF ก A B g(x) = x3 + 1 F g ก F 1) F ˈ ˆ กF F F F x1, x2 ∈ Dg F g(x1) = g(x2) F F x1 3 + 1 = x2 3 + 1 x1 3 = x2 3 x1 = x2 ˆ กF g ก F ˈ ˆ กF F 2.3 F f : A → B F f 1 : B → A ก F f ˈ ˆ กF F B
- 27. F F 19 2) F ˈ ˆ กF F ก. Dg = A F fl ˈ ˆ กF . Rg = B F fl Rg ⊆ B ˈ ˆ กF fl B ⊆ Rg F y ∈ B F F x ∈ A (x, y) ∈ g y = g(x) y ∈ Rg F Rg = B F ก ก F 1) F 2) F F g ก ʿก 2.4 ก 1. F 2.3 F 1.2) F 2) ˈ 2. F ˆ กF f F 2.9 F ˈ ˆ กF F Fก 3. F ˆ กF g F 2.10 ˈ ˆ กF F Fก 4. ก F 2.10 F F F g ก ก g 5. ก F f : A → B F B F F (f 1 ) 1 = f 2.4.2 ก ˆ กF ก F 2.3 F ก ก ก ˆ กF ก F F F f : A→ B g : B → C F gof : A → C F ก ˆ กF ก ก ก F gof : A → C ˈ ˆ กF F C F f : A→ B g : B → C F 2.3 F F gof ก F F F (fl) F F (gof) 1 : C → A F gof F C 2.4 ก F gof : A → C f : A→ B g : B → C F ก F F (gof) 1 : C → A ก F gof F C
- 28. 20 F ˆ กF 1) F (gof) 1 C F x ∈ C F F y ∈ A (x, y) ∈ (gof) 1 (y, x) ∈ gof F F z ∈ B (z, x) ∈g (y, z) ∈ f z = f(y) x = g(z) (gof) 1 C 2) F (gof) 1 F C F x1, x2 ∈ C F F y ∈ A (x1, y) ∈ (gof) 1 (x2, y) ∈ (gof) 1 F F y = (gof) 1 (x1) y = (gof) 1 (x2) (gof) 1 (x1) = (gof) 1 (x2) g(f 1 (x1)) = g(f 1 (x2)) ก g ˈ ˆ กF F f 1 (x1) = f 1 (x2) ก f 1 ˈ ˆ กF F F F F x1 = x2 (gof) 1 F C F ก (›) F F gof F C F (gof) 1 : C → A ก gof F C 2.3 F F (gof) 1 : C → A F F gof ก F F gof F 2 ก ˈ ˆ กF F ˈ ˆ กF C กF ก F F ก ก ก ˆ กF ก F กF 2.6 ก F A ⊆ R F ˆ กF ก ก F (Identity function) ˆ กF ก A A ˈ ก F IA : A → A
- 29. F F 21 ก 2.6 F F IA(x) = x ก x ∈ A F ˈ F ก ก ก ˆ กF ก F F ก F F 2) F F 1) F F F F ˈ ʿก 2.1) (⊆) F (gof) 1 ⊆ f 1 og 1 F (x, y) ∈ (gof) 1 F F (y, x) ∈ gof x = gof(y) = g(f(y)) g 1 (x) = f(y) f 1 (g 1 (x)) = y (f 1 og 1 )(x) = y => (x, y) ∈ f 1 og 1 (gof) 1 ⊆ f 1 og 1 F ก (⊇) F f 1 og 1 ⊆ (gof) 1 F (x, y) ∈ f 1 og 1 F F y = f 1 og 1 (x) f(y) = g 1 (x) g(f(y)) = x (y, x) ∈ gof (x, y) ∈ (gof) 1 f 1 og 1 ⊆ (gof) 1 F F (gof) 1 = f 1 og 1 2.2) F ก F 2.1) ʿก 2.4 1. F 2.5 F 1) ˈ 2.5 ก F f : A → B g : B → C f g F ก F F F F ˈ 1) f 1 of ˈ ˆ กF ก ก F A fof 1 ˈ ˆ กF ก ก F B 2) (gof) 1 = f 1 og 1 (fog) 1 = g 1 of 1
- 30. 22 F ˆ กF 2.5 ˆ กF F 2.11 ก F f(x) = 1 x ก x > 0 g(x) = x2 + 3 ก x ∈ R F h(x) = f(x) + g(x) ก x > 0 ก h(x) ˈ ก กก ˆ กF F Dh = Df ∩ Dg ก Fก F Df = (0, ∞) Dg = ( ∞, ∞) F F Dh = (0, ∞) ∩ ( ∞, ∞) = (0, ∞) h(x) = f(x) + g(x) F F F ก F 2.12 ˆ กF h = f ⋅ g = {(x, y) | y = f(x) ⋅ g(x) ก x > 1} F f(x) = 2 x x - 1 h(x) = 2 1 x - 1 F g(5) ก h(x) = f(x) ⋅ g(x) F F g(x) = h(x) f(x) F F g(5) = h(5) f(5) ก h(5) = 2 1 5 - 1 = 1 24 f(5) = 2 5 5 - 1 = 5 24 g(5) = h(5) f(5) = 1 24 5 24 = 1 24 ⋅ 24 5 = 24 120 2.7 ก F f g ˈ ˆ กF ก R R x ∈ Df ∩ Dg F 1) f + g = {(x, y) | y = f(x) + g(x)} 2) f g = {(x, y) | y = f(x) g(x)} 3) f ⋅g = {(x, y) | y = f(x) ⋅ g(x)} 4) f g = {(x, y) | y = f(x) g(x) g(x) ≠ 0}
- 31. 3 F ก ˆ กF ก 3.1 F ก 3.1.1 F ก ก F F ก ก F F ก F F F F F F ก F ก F F F F F ˈ F F θθθθ sin θθθθ cos θθθθ tan θθθθ 30° 1 2 3 2 1 3 45° 2 2 2 2 1 60° 3 2 1 2 3 3.1 F ก F ก F F F ก F F ก 2 F F กF sin 45° ก cos 45° sin 30° ก cos 60° ก F F θ ก sin θ = cos(90° θ) cos θ = sin(90° θ) 3.1 ก F θ ˈ ก F F ก 1) F F (sine : sin) F F F F θ ก F F ก ˈ ก sin θ = F ก 2) F F (cosine : cos) F F F θ ก F F ก ˈ ก cos θ = ก 3) F F (tangent : tan) F F F Fก F F ˈ ก tan θ = F ก ก = F
- 32. 24 F ˆ กF F 3.1 ก ABC θ = 30° x ก F tan 30° = 12 x F tan 30° = 1 3 F F 1 3 = 12 x x = 12 3 F 3.2 ก F cos 15° = 0.9659 F sin 75° ก ก sin θ = cos (90° θ) F F sin 75° = cos (90° 75° ) = cos 15° = 0.9659 F 3.3 θ ก F cos2 θ + sin2 θ = 1 ก ก cos2 θ + sin2 θ = ( )2 ก + ( ) 2F ก = 2 2ก + 2F 2ก = 2 2+ F 2ก = 2ก 2ก = 1 12 x θ
- 33. F F 25 3.1.2 F ก ก (inverse trigonometric ratio) ก F F F ก ก F ก F F ก ก F ก ก F F 3.4 F θ ก F F cot θ = tan (90° θ) ก tan (90° θ) = osin (90 - ) ocos (90 - ) θ θ = cos sin θ θ = cot θ ʿก 3.1 1. ก F cos 25° = 0.906 F F ก F ก ก F 1) sin 25° 2) cos 65° 3) cos 65° + sec 65° 4) sec2 65° tan2 65° 2. F (sin2 1° + sin2 2° + sin2 3° + + sin2 89° ) + (cos2 89° + cos2 88° + + cos2 1° ) 3.2 ก F θ ˈ ก F F ก 1) F F (cosecant : cosec) F F F F ก ก F F θ ˈ ก cosec θ = ก F = 1 sinθ 2) F F (secant : sec) F F F F กก F θ ˈ ก sec θ = ก = 1 cosθ 3) F F (cotangent : cot) F F F Fก F F ˈ ก cot θ = F = 1 tanθ
- 34. 26 F ˆ กF 3.2 ˆ กF ก (Trigonometric Function) 3.2.1 ˆ กF ก ก F F 3.1 ก F Fก F ก O(0, 0) ก F F OA A(x, y) ˈ F ก F ก F OA θ ก ก X A ก F กก ก X ก X B F F ก OAB ก OAB ก F ก F F cos θ = OB OA sin θ = AB OA F OA = 1 ก ˈ cos θ = OB sin θ = AB ก F F F OB = 2 2 (0 - x) + (0 - 0) = x AB = 2 2 (x - x) + (y - 0) = y 3.3 ˆ กF ก ˆ กF y = f(x) f(x) ˈ ˆ กF F ก ก F π ≤ x ≤ π θ
- 35. F F 27 F ก A(x, y) θ A(cos θ, sin θ) F F cos θ sin θ ก (1, 0), ( 1, 0), (0, 1), (0, 1) 1) (1, 0) F OA ก ก F ˈ F ก 0 F F cos 0 = 1 sin 0 = 1 2) ( 1, 0) F OA ก ก F ˈ 180° F ก π F F cos π = 1 sin π = 0 3) (0, 1) F OA ก ก F ˈ 90° F ก 2 π F F cos 2 π = 1 sin π = 0 4) (0, 1) F OA ก ก F ˈ 270° F ก 3 2 π F F cos 3 2 π = 0 sin 3 2 π = 1 F F F F ˈ ก ก F F ก F F F F F ก OA ก ก กก F 1 ก F ˆ กF F ˆ กF F F F ก ก F F cos (nπ + 2 π ) = 0 n ≥ 0 sin (nπ + 2 π ) = 1 n ≥ 0 cos nπ = 1 n sin nπ = 0 n cos nπ = 1 F n sin nπ = 0 F n ก F ˆ กF F ˆ กF F ก F F F F ˆ กF F ˆ กF F F F ก F F F F F 3.5 F ˆ กF F F F ก F 15.70 (ก F π ≈ 3.14) ก sin (15.70) = sin (3.14 × 5) = sin 5π = 0 cos (15.70) = cos (3.14 × 5) = cos 5π = 1
- 36. 28 F ˆ กF F 3.6 F F F F F 31.4 ก F F F ก F F ก F ก กF F F F กF ก ก k = 2kπr F r = 1 ก k F ก 31.4 F F 31.4 = 2 × k × π k = 31.4 2 × 3.14 = 5 F ก 5 F F cos 31.4 = cos (5 × 6.28) = cos (5 × 2π) = cos 10π = 1 ก sin 31.4 = sin (5 × 6.28) = sin (5 × 2π) = sin 10π = 0 ʿก 3.2 ก 1. θ F cos ( θ) = cos θ sin ( θ) = sin θ 2. ก F θ F cos2 θ + sin2 θ = 1 3. ก F F ก 10 F
- 37. F F 29 3.2.2 F ˆ กF ก 2 F ก ก F ˆ กF F F F ˆ กF ก F F ˆ กF F y = sin x R [ 1, 1] y = cos x R [ 1, 1] y = tan x R { n 2 π | n ˈ } R y = cosec x R {nπ | n ˈ } R ( 1, 1) y = sec x R { n 2 π | n ˈ } R ( 1, 1) y = cot x R {nπ | n ˈ } R 3.2.3 ก ˆ กF ก ก ก ก ˆ กF ก F F F F ก ก F ก F ˆ กF ก F ก F F 3.2 ก y = sin x F [ 2ππππ, 2ππππ] 3.3 ก y = cos x F [ 2ππππ, 2ππππ]
- 38. 30 F ˆ กF 3.4 ก y = tan x F ( ∞∞∞∞, ∞∞∞∞) ʿก 3.2 1. ก F y = cos x + sin x ก x ∈ [0, π] F F F ก ˆ กF ก F F 2. ก F f(x) = sin x F ˆ กF f ก F ˈ ˆ กF F ก x ∈ [0, 2 π ]
- 39. F F 31 3.3 ก ก F ก ก ก F ก F ก F กF ˆ F F F F F F ก F F 2), 3) 2) ก cosec2 θ cot2 θ = 1 F cosec2 θ = 2 1 sin θ cot2 θ = 2 2 cos sin θ θ F F cosec2 θ cot2 θ = 2 1 sin θ 2 2 cos sin θ θ = 2 2 1 - cos sin θ θ = 2 2 sin sin θ θ = 1 3) ก sec2 θ tan2 θ = 1 F sec2 θ = 2 1 cos θ tan2 θ = 2 2 sin cos θ θ F F sec2 θ tan2 θ = 2 1 cos θ 2 2 sin cos θ θ = 2 2 1 - sin cos θ θ = 2 2 cos cos θ θ = 1 3.1 ก F θ ˈ F F 1) sin2 θ + cos2 θ = 1 2) cosec2 θ cot2 θ = 1 3) sec2 θ tan2 θ = 1 4) cos ( θ) = cos θ 5) sin ( θ) = sin θ 3.3 ก ก F ก (Trigonometric Identities) ก ก F ก ˆ กF ก ˈ ก F
- 40. 32 F ˆ กF F 3.7 F cos2 5 12 π + sin2 5 6 π + sin2 5 12 π + cos2 7 6 π ก. 2 . 3 . 11 4 . 3 2 . F F ก F ก F cos2 5 12 π + sin2 5 6 π + sin2 5 12 π + cos2 7 6 π ก F F F (cos2 5 12 π + sin2 5 12 π ) + (sin2 5 6 π + cos2 7 6 π ) = 1 + (sin2 5 6 π + cos2 7 6 π ) = 1 + [sin2 (π 6 π ) + cos2 (π + 6 π )] = 1 + (sin2 6 π + cos2 6 π ) = 1 + 1 = 2 ʿก 3.3 1. F ก ก F ก ก F F ˈ 1) cos2 θ cot θ + sin2 θ tan θ + 2 sin θ cos θ = tan θ + cot θ 2) tan2 θ sin2 θ = tan2 θ sin2 θ 3) 2 2 cosec 1 + tan θ θ = cot2 θ 2. F 3.1 F 4) F 5) ˈ
- 41. F F 33 3.4 ˆ กF ก ก 3.2 F ก F ˆ กF ก ก F ˆ กF ก F ก ก F ก F F F ก F F 3.8 ก F A, B ˈ ก A + B = 2 π F 1) sin (A + B) = 1 2) cos (A + B) = 0 1) ก sin (A + B) = sin A cos B + cos A sin B A + B = 2 π F F sin (A + B)= sin A cos ( 2 π A) + cos A sin ( 2 π A) = sin A sin A + cos A cos A = sin2 A + cos2 A = 1 2) ก cos (A + B) = cos A cos B sin A sin B A + B = 2 π F F cos (A + B) = cos A cos ( 2 π A) sin A sin ( 2 π A) = cos A sin A sin A cos A = 0 F 3.9 ก F A, B, C ˈ ก F 1) sin (A + B) = sin C 2) cos (A + B) = cos C ก A, B, C ˈ ก F F A + B + C = π 1) A + B = π C sin (A + B)= sin (π C) = sin C 2) cos (A + B) = cos (π C)= cos C 3.2 ก F A, B ˈ ก ABC F ˈ ก F F F 1) sin (A + B) = sin A cos B + cos A sin B 2) cos (A + B) = cos A cos B sin A sin B 3) tan (A + B) = tan A + tan B 1 - tan A tan B
- 42. 34 F ˆ กF 3.3 F F F F ก F B = A 3.2 F F ก F F 3.10 ก 3.3 cos 2A cos A F sin A F cos 2A = cos2 A sin2 A = cos2 A (1 cos2 A) = cos2 A 1 + cos2 A = 2 cos2 A 1 cos 2A = cos2 A sin2 A = (1 sin2 A) sin2 A = 1 2 sin2 A F 3.11 ก 3.3 tan 2A sin A cos A tan 2A = sin2A cos2A = 2 2sinAcosA 2cos A - 1 tan 2A = 2 2sin A cos A 1 - 2cos A F 3.12 F 2 2 sin 3A sin A 2 2 cos 3A cos A = 2 F cos 2A F F ก F F (Ent . . 2548) ก F 2 2 sin 3A sin A 2 2 cos 3A cos A F F ( )2sin3A sinA ( )2cos3A cosA = 2 (sin3A sinA )cos3A cosA ⋅(sin3A sinA + )cos3A cosA = 2 ( )sin 3A cos A - cos 3A sin A sin A cos A ⋅( )sin 3A cos A + cos 3A sin A sin A cos A = 2 3.3 ก F A ˈ ก F F 1) sin 2A = 2 sin A cos A 2) cos 2A = cos2 A sin2 A 3) tan 2A = 2 2 tan A 1 - tan A
- 43. F F 35 ( )sin (3A - A) sin A cos A ⋅( )sin (3A + A) sin A cos A = 2 ( )sin2A sinAcosA ⋅( )sin4A sinAcosA = 2 ( )sin2A sinAcosA ⋅( )sin2(2A) sinAcosA = 2 ( )2sinAcosA sinAcosA ⋅( )2sin2Acos2A sinAcosA = 2 ( )2sinAcosA sinAcosA ⋅( )4sinAcosAcos2A sinAcosA = 2 2 ⋅ 4 cos 2A = 2 cos 2A = 1 4 ก F F ˈ ก ก F ก ˈ ก F F ก F F 2) F 3) F F ก F ก F F ก F F F F F ˈ ʿก 2) ก 2 sin( )x + y 2 cos( )x - y 2 = 2 sin( )yx 2 2+ cos( )yx 2 2- ก F A = x 2 B = y 2 F F F F F F = 2 sin (A + B) cos (A B) = 2 (sin A cos B + cos A sin B)(cos A cos B + sin A sin B) = 2 (sin A cos B ⋅ cos A cos B + cos A sin B ⋅ cos A cos B + sin A cos B ⋅ sin A sin B + cos A sin B ⋅ sin A sin B) = 2 (sin A cos A cos2 B + cos2 A sin B cos B + sin2 A cos B sin B + cos A sin A sin2 B) = 2 [sin A cos A (cos2 B + sin2 B) + sin B cos B (cos2 A + sin2 A)] = 2 (sin A cos A + sin B cos B) ( ก cos2 B + sin2 B = 1 cos2 A + sin2 A = 1) = 2 sin A cos A + 2 sin B cos B = sin 2A + sin 2B 3.4 ก ก F ˈ 1) cos x + cos y = 2 cos( )x + y 2 cos( )x - y 2 2) sin x + sin y = 2 sin( )x + y 2 cos( )x - y 2 3) cos x cos y = 2 sin( )x + y 2 sin( )x - y 2 4) sin x sin y = 2 cos( )x + y 2 sin( )x - y 2
- 44. 36 F ˆ กF F A = x 2 B = y 2 F F 2 sin( )x + y 2 cos( )x - y 2 = sin x + sin y 3) ก 2 sin( )x + y 2 sin( )x - y 2 = 2 sin( )yx 2 2+ sin( )yx 2 2- ก F A = x 2 B = y 2 F F F F F F = 2 sin (A + B) sin (A B) = 2 (sin A cos B + cos A sin B)(sin A cos B cos A sin B) = 2 [(sin A cos B)2 (cos A sin B)2 ] = 2 sin2 A cos2 B + 2 cos2 A sin2 B = 2 sin2 A cos2 B + 2 (1 sin2 A)(1 cos2 B) = 2 sin2 A cos2 B + 2 (1 sin2 A cos2 B + sin2 A cos2 B) = 2 sin2 A cos2 B + 2 (1 sin2 A) 2 cos2 B + 2 sin2 A cos2 B = 2 (1 sin2 A) 2 cos2 B = 2 cos2 A 2 cos2 B = (2 cos2 A 1) (2 cos2 B 1) = cos 2A cos 2B F A = x 2 B = y 2 F F 2 sin( )x + y 2 sin( )x - y 2 = cos x cos y F ก ʽ F F F ก ก ก ก ˈ ก F ก F 1) ก cos(x + y) + cos(x - y) 2 = (cos x cos y - sin x sin y) + (cos x cos y + sin x sin y) 2 = 2 cos x cos y 2 = cos x cos y 2) ก cos (x - y) - cos (x + y) 2 = (cos x cos y + sin x sin y) - (cos x cos y - sin x sin y) 2 = 2 sin x sin y 2 = sin x sin y 3.5 ก ˈ ก F 1) cos x cos y = cos (x + y) + cos (x - y) 2 2) sin x sin y = cos (x - y) - cos (x + y) 2 3) sin x cos y = sin (x - y) + sin (x + y) 2
- 45. F F 37 ʿก 3.4 1. ก F A ˈ ก F F F 1) sin A 2 = ± 1 - cos A 2 2) cos A 2 = ± 1 + cos A 2 2. F ก F 1 F tan A 2 3. F F 3.3 ˈ 4. F F 3.4 F F F F ˈ 5. F F 3.5 F F F F ˈ 3.5 ก F ก F 3.5.1 ก F ก F (Laws of Cosine) ˈ ก F F F ก ก F ˈ F F 1 F F 3.4 F ก กF ˆ ก ก กF ก F ก F ก F F 9 ก F F F ก F F Fก F F F F F F F F ก F F ก ก 3.4 F ก (2) F a2 = c2 b2 2ac cos B F กก ก (1) ก F F 0 = (b2 + c2 2bc cos A) + (c2 b2 2ac cos B) = 2c2 2bc cos A 2ac cos B 2c2 = 2bc cos A + 2ac cos B = 2c (b cos A + a cos B) 3.4 ก F A, B, C ˈ F a, b, c ˈ F F F A, B, C F F F 1) a2 = b2 + c2 2bc cos A 2) b2 = c2 + a2 2ac cos B 3) c2 = a2 + b2 2ab cos C
- 46. 38 F ˆ กF c = b cos A + a cos B ก ก ก ก F F F a = b cos C + c cos B b = c cos A + a cos C F F ก 3.1 F Fก F F F F ก F ก F ก ก Fก F ก ก ก F ก F F F ˈ ก F F ก F F ʿก 3.5 ก 1. F ก 3.1 F 1) F 2) ˈ 3.5.2 ก F ก F (Laws of Sine) ˈ ก F F F F F ก F F F F F F F 3.5 F F F ก 3.1 ก F A, B, C ˈ F a, b, c ˈ F F A, B, C F F 1) a = b cos C + c cos B 2) b = a cos C + c cos A 3) c = a cos B + b cos A 3.5 ก F A, B, C ˈ F a, b, c ˈ F F F A, B, C F F sinA a = sinB b = sinC c = k k ˈ
- 47. F F 39 F 3.13 ก F ABC ˈ F AC ˈ F a, b, c ˈ F F A, B, C ก F F F sinB b = 3 c cos A + a cos C = 1 2 3 F |cos(A + C)| ก A + B + C = π fl A + C = π B |cos(A + C)| = |cos (π B)| = | cos B| = cos B F Fก F c cos A + a cos C = 1 2 3 ก ก 3.1 F 2) F F b = 1 2 3 ก sinB b = 3 fl sin B = 3b = 3( )1 2 3 = 3 2 cos B = 2 1 - sin B = ( ) 2 3 21 - = 1 2 ʿก 3.5 1. ABC F k 3.5 F ก F F Fก ก ABC F F
- 48. 40 F ˆ กF 3.6 ABC F 3.5 ABC F h F c F ก ABC F F F F sin A = h b sin B = h a F h h = b sin A h = a sin B ก ก (A) = 1 2 × F × = 1 2 hc F h F F F F F ก F F A = 1 2 bc sin A A = 1 2 ac sin B กก F F sinA a = sinC c ก F F c sin A = a sin C ABC F ก A = 1 2 ab sin C F 3.14 ก F ABC ˈ F F a = 2, b = 4 C F ก 6 π ABC ก F ก A = 1 2 ab sin C F a = 2, b = 4 C = 6 π F F F F A = 1 2 (2)(4) sin 6 π = (4)( 1 2 ) = 2 F A B C h ab c
- 49. F F 41 F 3.15 ก F ก F C ˈ ก F ก ก F 1 F ก A = 1 2 ab sin C F a = 1, b = 1, C = 2 π F F F F A = 1 2 (1)(1) sin 2 π = 1 2 F ʿก 3.6 ก F ABC ˈ F F F F a F F ABC ก F F F ก 23 4 a 3.7 ก ˆ กF ก ก ˆ กF ก F F ˆ กF F ก ˆ กF ก F F ˆ กF F ˈ F F F ˈ ก ˆ กF ก F ˆ กF A B C 1 1
- 50. 42 F ˆ กF ˆ กF ก ก ˆ กF ก y = sin x y = arcsin x y = cos x y = arccos x y = tan x y = arctan x y = cosec x y = arccosec x y = sec x y = arcsec x y = cot x y = arccot x F ก ˆ กF ก ก F ก ก F F ก F F Df = -1 fR Rf = -1 fD F ก F ก F F ก ˆ กF ก ˈ ˆ กF F ˈ F ˆ ก ก ก ˆ กF ก F 3.16 F arctan x = arctan 1 4 2 arctan 1 2 F sin (180° + arctan x) F F ก F F A = arctan 1 4 fl tan A = 1 4 F B = arctan 1 2 fl tan B = 1 2 ก arctan x = arctan 1 4 2 arctan 1 2 F x = tan (arctan 1 4 2 arctan 1 2 ) = tan (A 2B) = tan A - tan 2B 1 + tan A tan 2B ก tan 2B = 2 2 tan B 1 - tan B = ( ) ( ) 1 2 21 2 2 1 - = 1 4 1 1 - = 4 3 x = tan A - tan 2B 1 + tan A tan 2B = ( )( ) 1 4 4 3 1 4 4 3 - 1 + = 3 - 16 12 4 3 = 13 16 sin (180° + arctan x) = sin (arctan x) = sin (arctan ( 13 16 ))
- 51. F F 43 = sin (arctan13 16 ) = sin (arcsin 13 425 ) = 13 425 F 3.17 F tan (arccos x) = 3 F F x ⋅ sin (2 arccos x) F ก F => F A = arccos x tan A = tan (arccos x) = 3 = tan (π 3 π ) = tan 2 3 π F F A = 2 3 π => ก A = arccos x F F cos A = cos 2 3 π = 1 2 = x x ⋅ sin (2 arccos x) = x ⋅ sin (2A) = ( 1 2 ) sin 4 3 π = ( 1 2 ) sin( )3+ ππ = 1 2 sin 3 π = 31 2 2⋅ = 3 4 F 3.18 ก F 2 arcsin a + arcsin (2a 2 1 - a ) = 3 π F arcsin a F F F 1) ( 2 π , 4 π ) 2) ( 4 π , 0) 3) (0, 4 π ) 4) ( 4 π , 2 π ) ก F A = arcsin a F F sin A = a cos A = 2 1 - a ก F B = arcsin (2a 2 1 - a ) F F sin B = 2a 2 1 - a = 2 sin A cos A = sin 2A F B = 2A 2 arcsin a + arcsin (2a 2 1 - a ) = 3 π F F ˈ 2A + B = 3 π cos (2A + B) = cos 3 π
- 52. 44 F ˆ กF F F ก F cos (2A + B) = cos 2A cos B sin 2A sin B = cos2 2A sin2 2A = cos 4A cos 4A = cos 3 π A = 12 π = arcsin a F 12 π ∈ (0, 4 π ) F 3) F 3.19 ก ก arcsin x + arcsin (1 x) = arccos x F F ก ก ก F arccos x + arcsin x = 2 π F F arccos x = 2 π arcsin x -----(1) (1) ก F arcsin x + arcsin (1 x) = 2 π arcsin x 2 arcsin x + arcsin (1 x) = 2 π F F sin (2 arcsin x + arcsin (1 x)) = sin 2 π = 1 -----(2) กก F A = arcsin x B = arcsin (1 x) sin (2A + B) = sin 2 π F F ก (2) sin (2A + B) = sin 2A cos B + cos 2A sin B = (2 sin A cos A) cos B + (1 2 sin2 A) sin B = 2x( 2 1 - x )( 2 2x - x ) + (1 2x2 )(1 x) 2x( 2 1 - x )( 2 2x - x ) + (1 2x2 )(1 x) = 1 2x 2 2 (1 - x )(2x - x ) + (1 2x2 )(1 x) = 1 2x 2 3 4 2x - x - 2x + x + (1 x 2x2 + 2x3 ) = 1 2 2 3 4 4x (2x - x - 2x + x ) = x + 2x2 2x3 3 4 5 6 8x - 4x - 8x + 4x = x + 2x2 2x3 กก F ก F 8x3 4x4 8x5 + 4x6 = x2 + 4x3 8x5 + 4x6 x2 4x3 + 4x4 = 0 x2 (1 4x + 4x2 ) = 0 x2 (1 2x)(1 2x) = 0 x = 0, 1 2 ( ก ) ก ก F F ก 0 + 1 2 = 1 2
- 53. F F 45 ʿก 3.7 1. F sin (arctan 2 + arctan 3) 2. ก F f(x) = sin x, g(x) = arcsin 2x + 2 arcsin x F F fog(1 3 ) F ก F 3. F arccos x arcsin x = 6 π F arccos x arctan 2x F F ก F 4. ก F A = {x | arccos (x x2 ) = arcsin x + arcsin (x 1)} n(A)
- 55. 4 ˆ กF ก F ˆ กF ก 4.1 ˆ กF ก F ก 4.1 F F ˆ กF ก F (R) F ˆ กF ก F ก (R+ ) ก ˆ กF ก F F 1) y = 2x 2) y = ( )x1 2 4.1 ˆ กF ก F (Exponential Function) F (x, y) y = ax a ˈ ก F F ก 1 ˈ ก F f = {(x, y) |||| y = ax , a ∈∈∈∈ R+ a ≠≠≠≠ 1}
- 56. 48 F ˆ กF F F a > 1 F y = ax ˈ ˆ กF 0 < a < 1 F y = ax ˈ ˆ กF กก F F ˆ กF ก F ˈ ˆ กF F F F ก ˆ กF ก F ˈ ˆ กF ก F ˆ กF ก (logarithmic function) ก F ʿก 4.1 1. ก F y = 3x F ˆ กF ก F ˈ ˆ กF F 1) 2) ก ก 2. ก F y = ( )x1 3 F ˆ กF ก F ˈ ˆ กF F 1) 2) ก ก 3. ก ˆ กF ก F F 1 F 2 ก F F ก ก F ก ก 4. ก F y = ax a > 1 F F ก F (2, 9) ก ก ก F
- 57. F F 49 4.2 ˆ กF ก ก 4.2 F F ˆ กF ก ก (R+ ) F ˆ กF ก (R) ก ˆ กF ก F 1) y = log2x 2) y = log1/2x F F a > 1 F ˈ ˆ กF 0 < a < 1 F ˈ ˆ กF ก F F ˆ กF ก ˈ ˆ กF F F ก ˆ กF ก F 4.2 ˆ กF ก (Logarithmic Function) F (x, y) x = ay a ˈ ก F F ก 1 ˈ ก F f = {(x, y) |||| x = ay , a ∈∈∈∈ R+ a ≠≠≠≠ 1} ก f = {(x, y) | y = logax , a ∈ R+ , a ≠ 1}
- 58. 50 F ˆ กF ʿก 4.2 1. ก F y = log1/3x F ˆ กF ก F ˈ ˆ กF F 4.3 ˆ กF ก 4.3.1 ˆ กF ก ˆ กF ก F ก F ก F F F F ก F F F F F F F F F F F ˈ ʿก ก F M = logax, N = logay x, y > 0 4.2 F F x = aM y = aN 1) xy = aM ⋅aN = aM + N loga(xy) = loga(aM + N ) = M + N = logax + logay 2) x y = M N a a = aM N 4.1 ก F x > 0, y > 0 1) loga(xy) = logax + logay 2) loga( )x y = logax logay 3) logaxk = k ⋅ logax k ˈ 4) logxx = 1 5) a blog x = 1 b logax 6) logax = b b log x log a b > 0 7) logax = x 1 log a x ≠ 1 8) alog x a = x 9) alog y x = alog x y
- 59. F F 51 loga( )x y = loga(aM N ) = M N = logax logay 3) ก x = aM F xk = (aM )k = aMk logaxk = loga(aMk ) = Mk = kM = k ⋅ logax 4) ก F M = logxx F F xM = x M = 1 = logxx 8) ก x = aM x = alog x a 9) F P = alog y x F logaP = loga( alog y x ) = logay ⋅ logax = logax ⋅ logay = loga( alog x y ) ก ก ˈ ˆ กF F F F P = alog x y alog y x = alog x y F ก F F ก (primary logarithm) log10 x ˈ ก ก log10 x F log x F 4.1 ก F log 2 = a log 3 = b F log 24 a ก b ก F log 24 = log (23 ⋅ 3) = 3 log 2 + log 3 = 3a + b
- 60. 52 F ˆ กF ʿก 4.3 ก 1. F 4.1 F F F F ˈ 2. ก F log 2 = 0.3010, log 3 = 0.4771 F F 1) log210 2) 2 log 9 3) 24 8log 2 4.3.2 ก ก ก log 24 ก F log 2 = 0.3010 log 3 = 0.4771 ก 24 = 23 ⋅ 3 F F log 24 = log(23 ⋅ 3) = 3 log 2 + log 3 = 3(0.3010) + 0.4771 = 1.3801 ก log 24 = 1.3801 = 1 + 0.3801 = (log 10) + 0.3801 ก log 10 = 1 F ก ก ก 0.3801 F ก ก F F ˈ F F 4.2 ก ก F 1) 441 2) 44.1 3) 4.41 4) 0.441 5) 0.0441 ก F log 4.41 = 0.644 4.3 ก F x ˈ x = k + log M k ˈ M ˈ ก F F ก 1 F 1) ก ก (characteristic) ก k 2) (mantissa) log M
- 61. F F 53 1) ก F N = 441 F F log N = log 441 = log (4.41 ⋅ 102 ) = log 4.41 + log 102 = 0.644 + 2 ก ก 441 2 0.644 2) ก F N = 44.1 F F log N = log 44.1 = log (4.41 ⋅ 10) = log 4.41 + log 10 = 0.644 + 1 ก ก 44.1 1 0.644 3) ก F N = 4.41 F F log N = log 4.41 = 0.644 ก ก 4.41 0 0.644 4) ก F N = 0.441 F F log N = log 0.441 = log (4.41 ⋅ 10 1 ) = log 4.41 + ( 1) ก ก 0.441 1 0.644 5) ก F N = 0.0441 F F log N = log 0.0441 = log (4.41 ⋅ 10 2 ) = log 4.41 + ( 2) ก ก 0.441 2 0.644 ก F F F F F ก F 2 ก 1) ก ก ˈ F ก ก N ก F F ก ก ˈ ก F ก N กก F ก ก F 1 ก ก F ก ก ˈ F ก N F ก F ก ก F 1 2) ก F ก F F ก F F F ก 441 F F F ก 0.644
- 62. 54 F ˆ กF ʿก 4.3 1. ก F log 3 = 0.4771 F log R R ก F F F ก ก ก R F 1) 27 2) 1 27 3) 27 81 2. ก R ก F F 1) R = 3100 2) R = 6 12 3. F F ก ก R F ก 3 R F ก log 3 F R ก F 4.4 ก F ก ก 4.4 logeN F F F F F F ก ก ก 4.1 F (5) ˈ F ก ก ก ก e F ก ˈ ก F ˆ ก logeN = logN loge F log e = log 2.718 = 0.434 F F logeN = logN 0.434 = 2.304 log N F F F ก ก ˈ ก F 4.4 ก F N ˈ e ≈ 2.718 F ก (natural logarithm) logeN ˈ ก F ก ก ก ln N
- 63. F F 55 F 4.3 ก F log 37 = 1.568 F ln 0.37 F ก ก ก ln 0.37 F ln 0.37 = 2.304 ⋅ log 0.37 = 2.304 ⋅ log (37 ⋅ 10 2 ) = 2.304 ⋅ ( 2 + log 37) = 4.608 + (2.304 ⋅ log 37) ก ก ln 0.37 F ก 4.608 2.304 ⋅ log 37 F F ln 0.37 = 4.608 + (2.304 ⋅ 1.568) = 4.608 + 3.613 = 0.995 F 4.4 F F F F ln 4 + ln 6 ln 8 + ln 3 ก F ln 4 + ln 6 ln 8 + ln 3 = ln (4 6) ln (8 3) ⋅ ⋅ = ln 24 ln 24 = 1 ʿก 4.4 F F 4.1 ˈ ก 4.5 ก ก ก F 4.5.1 ก ก F ก กF ก ก F กก F F ก F ก ก ก F F F F F ก ˈ FกF F ก ก F ก F ก ก F กก F F xy = 0 F x = 0 y = 0 ก ก F F F ก ก ก ก F F ก F ก F F F
- 64. 56 F ˆ กF F 4.5 ก ก 12x 2(3x ) 9(4x ) + 18 = 0 F F ก F ก ก ก F 12x 2(3x ) 9(4x ) + 18 = 0 F F ก F (3 ⋅ 4)x 2(3x ) 9(4x ) + 18 = 0 3x ⋅ 4x 2(3x ) 9(4x ) + 18 = 0 (3x 9)(4x 2) = 0 3x - 9 = 0 4x 2 = 0 3x = 9 4x = 2 3x = 32 22x = 2 F F x = 2 2x = 1 x = 2 x = 1 2 ก ก 2 + 1 2 = 5 2 = 2.5 F 4.6 ก F F F y = 22x 2x + 2 45 ก X A F F F A B(0, b) ก F y = (log32)x 4 F b F F ก F ก ก F F ก F y = 22x 2x + 2 45 F A ˈ ก X F y = 0 F F 22x 2x + 2 45 = 0 (2x )2 2x ⋅ 4 45 = 0 (2x 9)(2x + 5) = 0 F 2x 9 = 0 2x + 5 = 0 ( F F F) 2x = 9 x = log29 F F A ก (log29, 0) F ก F A B(0, b) ก F y = (log32)x 4 F (m1) F F A ก B (m2) F y = (log32)x 4 F F ก m1 = m2 F F 2 b - 0 0 - log 9 = log32 b = (log32)(0 log29) = log32 ⋅ log29 = 3 9 log 2 log 2 = 3 1 32 log 2 log 2⋅ = 2
- 65. F F 57 4.5.2 ก ก F ก กF ก ก F กก F ก ก กF ก ก F ก F ก F ก F ก ก ก ก กก F F ˆ กF ก F ก F ˈ ˆ กF ก F ˈ F ก ก ก F F ˈ ˆ กF F ก ก F ก F F F 4.7 ก 2 x (x - 3) 2 < ( )2 3 - x 8 ก ก ก F 2 x (x - 3) 2 < ( )2 3 - x 8 ก F ˈ ก ก F F F F ก F F F ก กF F F 2 x (x - 3) 2 < ( )2 33 - x 2 ก 2 > 0 F ˈ ˆ กF x2 (x 3) < 3( 2 3 x) x3 3x2 < 2 3x x3 3x2 + 3x 2 < 0 ก F ก ก ก F ก F ก F F ก ก F P(x) = x3 3x2 + 3x 2 ก P(2) = 8 12 + 6 2 = 0 x 2 ˈ ก x3 3x2 + 3x 2 F F x3 3x2 + 3x 2 = (x 2)(x2 x + 1) (x 2)(x2 x + 1) < 0 F x2 x + 1 > 0 F F x 2 < 0 x < 2 ก ( ∞, 2) 4.6 ก ก ก 4.6.1 ก ก ก กF ก ก กก ก ก กF ก ก ก F F F 2 ก F กF 1) กF ก ก F ก F F ก กF 2) F ก F F ˆ กF ก F กก F 1
- 66. 58 F ˆ กF F F F 4.8 F log93, log9(3x 2), log9(3x + 16) ˈ F ก ก ก S ˈ ก F ก ก F 3S F F ก F ก log93, log9(3x 2), log9(3x + 16) ˈ F ก ก ก F F d1 = log9(3x 2) log93 = log9( )x 3 - 2 3 d2 = log9(3x + 16) log9(3x 2) = log9( )x x 3 + 16 3 - 2 F d1 = d2 ( ก ˈ ) log9( )x 3 - 2 3 = log9( )x x 3 + 16 3 - 2 x 3 - 2 3 = x x 3 + 16 3 - 2 (3x 2)2 = 3(3x + 16) (3x )2 4(3x ) + 4 = 3(3x ) + 48 (3x )2 4(3x ) + 4 3(3x ) 48= 0 (3x )2 4(3x ) 3(3x ) 44 = 0 (3x )2 7(3x ) 44 = 0 (3x 11)(3x + 4) = 0 F F 3x 11 = 0 (3x + 4 F F F) x = log311 3 F Fก F (log93) = 1 2 , 1, 3 2 ก ก Sn = n 2 [2a1 + (n 1)d] F n = 4, a1 = 1 2 F S4 = 4 2 [2( 1 2 ) + (4 1) 1 2 ] = 2[1 + 3 2 ] = 5 3S = 35 = 243 F 4.9 x F ก ก log 2x log 3 + log3(x 12) = ( )3log x x + 5 - x - 5 F F ก F กF F ก F ก F F ก ก F F ก กF
- 67. F F 59 log3 2x + log3(x 12) = 2 log3 ( )x x + 5 - x - 5 log3 [2x(x 12)] = log3 ( ) 2 x x + 5 - x - 5 2x(x 12) = ( ) 2 x x + 5 - x - 5 2x(x 12) = x[(x + 5) + (x 5) 2( x + 5 )( x - 5 )] 2x(x 12) = x[2x 2( x + 5 )( x - 5 )] 2x2 24x = 2x2 2x( x + 5 )( x - 5 ) 24x = 2x( x + 5 )( x - 5 ) กก F ก F F 576x2 = 4x2 (x2 25) = 4x4 100x2 4x4 676x2 = 0 4x2 (x2 169) = 0 กF ก F x = 0 x2 = 169 x = 0 x = 13 x = 13 F F x = 0 x = 13 F ก ˈ ก x = 13 4.6.2 ก ก ก กF ก ก กก F ก ก กF ก ก ก F ก F ก F ก ก ก ก กก F F ˆ กF ก ก F ˈ ˆ กF ก F ˈ F ก ก ก F F ˈ ˆ กF F ก ก F ก F F F 4.10 ก F A ˈ ก log4log3log2(x2 + 2x) ≤ 0 ˈ ก A ก ก ก log4log3log2(x2 + 2x) ≤ 0 F F log3log2(x2 + 2x) ≤ 1 (4 > 0 ˈ ˆ กF ) log2(x2 + 2x) ≤ 3 (3 > 0 ˈ ˆ กF ) x2 + 2x ≤ 8 (2 > 0 ˈ ˆ กF ) x2 + 2x 8 ≤ 0 (x 2)(x + 4) 0 F F 4 ≤ x ≤ 2 ˈ F F F [ 4, 2]
- 68. 60 F ˆ กF F ก x2 + 2x > 0 F F x(x + 2) > 0 ( ∞, 2) ∪ (0, ∞) ก A = [ 4, 2] ∩ [( ∞, 2) ∪ (0, ∞)] = [ 4, 2) ∪ (0, 2] F F ˈ ก A F กF { 4, 3, 1, 2} 4
- 69. ก F ˆ กF ก 0°°°° 90°°°° Degrees Radians sin cos tan Degrees Radians sin cos tan 0 0.00000 0.00000 1.00000 0.00000 46 0.80285 0.71934 0.69466 1.03553 1 0.01745 0.01745 0.99985 0.01746 47 0.82030 0.73135 0.68200 1.07237 2 0.03491 0.03490 0.99939 0.03492 48 0.83776 0.74314 0.66913 1.11061 3 0.05236 0.05234 0.99863 0.05241 49 0.85521 0.75471 0.65606 1.15037 4 0.06981 0.06976 0.99756 0.06993 50 0.87266 0.76604 0.64279 1.19175 5 0.08727 0.08716 0.99619 0.08749 51 0.89012 0.77715 0.62932 1.23490 6 0.10472 0.10453 0.99452 0.10510 52 0.90757 0.78801 0.61566 1.27994 7 0.12217 0.12187 0.99255 0.12278 53 0.92502 0.79864 0.60182 1.32704 8 0.13963 0.13917 0.99027 0.14054 54 0.94248 0.80902 0.58779 1.37638 9 0.15708 0.15643 0.98769 0.15838 55 0.95993 0.81915 0.57358 1.42815 10 0.17453 0.17365 0.98481 0.17633 56 0.97738 0.82904 0.55919 1.48256 11 0.19199 0.19081 0.98163 0.19438 57 0.99484 0.83867 0.54464 1.53986 12 0.20944 0.20791 0.97815 0.21256 58 1.01229 0.84805 0.52992 1.60033 13 0.22689 0.22495 0.97437 0.23087 59 1.02974 0.85717 0.51504 1.66428 14 0.24435 0.24192 0.97030 0.24933 60 1.04720 0.86603 0.50000 1.73205 15 0.26180 0.25882 0.96593 0.26795 61 1.06465 0.87462 0.48481 1.80405 16 0.27925 0.27564 0.96126 0.28675 62 1.08210 0.88295 0.46947 1.88073 17 0.29671 0.29237 0.95630 0.30573 63 1.09956 0.89101 0.45399 1.96261 18 0.31416 0.30902 0.95106 0.32492 64 1.11701 0.89879 0.43837 2.05030 19 0.33161 0.32557 0.94552 0.34433 65 1.13446 0.90631 0.42262 2.14451 20 0.34907 0.34202 0.93969 0.36397 66 1.15192 0.91355 0.40674 2.24604 21 0.36652 0.35837 0.93358 0.38386 67 1.16937 0.92050 0.39073 2.35585 22 0.38397 0.37461 0.92718 0.40403 68 1.18682 0.92718 0.37461 2.47509 23 0.40143 0.39073 0.92050 0.42447 69 1.20428 0.93358 0.35837 2.60509 24 0.41888 0.40674 0.91355 0.44523 70 1.22173 0.93969 0.34202 2.74748 25 0.43633 0.42262 0.90631 0.46631 71 1.23918 0.94552 0.32557 2.90421 26 0.45379 0.43837 0.89879 0.48773 72 1.25664 0.95106 0.30902 3.07768 27 0.47124 0.45399 0.89101 0.50953 73 1.27409 0.95630 0.29237 3.27085 28 0.48869 0.46947 0.88295 0.53171 74 1.29154 0.96126 0.27564 3.48741 29 0.50615 0.48481 0.87462 0.55431 75 1.30900 0.96593 0.25882 3.73205 30 0.52360 0.50000 0.86603 0.57735 76 1.32645 0.97030 0.24192 4.01078 31 0.54105 0.51504 0.85717 0.60086 77 1.34390 0.97437 0.22495 4.33148 32 0.55851 0.52992 0.84805 0.62487 78 1.36136 0.97815 0.20791 4.70463 33 0.57596 0.54464 0.83867 0.64941 79 1.37881 0.98163 0.19081 5.14455 34 0.59341 0.55919 0.82904 0.67451 80 1.39626 0.98481 0.17365 5.67128 35 0.61087 0.57358 0.81915 0.70021 81 1.41372 0.98769 0.15643 6.31375 36 0.62832 0.58779 0.80902 0.72654 82 1.43117 0.99027 0.13917 7.11537 37 0.64577 0.60182 0.79864 0.75355 83 1.44862 0.99255 0.12187 8.14435 38 0.66323 0.61566 0.78801 0.78129 83 1.44862 0.99255 0.12187 8.14435 39 0.68068 0.62932 0.77715 0.80978 84 1.46608 0.99452 0.10453 9.51436 40 0.69813 0.64279 0.76604 0.83910 85 1.48353 0.99619 0.08716 11.43005 41 0.71558 0.65606 0.75471 0.86929 86 1.50098 0.99756 0.06976 14.30067 42 0.73304 0.66913 0.74314 0.90040 87 1.51844 0.99863 0.05234 19.08114 43 0.75049 0.68200 0.73135 0.93252 88 1.53589 0.99939 0.03490 28.63625 44 0.76794 0.69466 0.71934 0.96569 89 1.55334 0.99985 0.01745 57.28996 45 0.78540 0.70711 0.70711 1.00000 90 1.57079 1.00000 0.00000 Infinity value
- 71. F F 61 ก ก ก ก ก F. ก F. ก : ก F , 2542. . F ENT 44. ก : ก , 2544. . F Ent 45. ก : ก , 2545. . F Ent 46. ก : ก , 2546. . F Ent 48. ก : F F ก F, 2548. F . ก ก F 1 ก F F ˅ 2004. ก : , 2547. . F. ก : , 2533. ก กF . Ent 43 ก F. ก : ก ก , 2543. ก . ก F .4 ( 011, 012). ก : ʽ ก F F, 2539.