thesis.swu.ac.ththesis.swu.ac.th/swuebook/h377210.pdf µ¦ ´ µ¦Á¦¸¥ ¦¼oÁ¦ºÉ°...
TRANSCRIPT
⌦⌦⌫
⌦ ⌫⌦
. 17 /2551
1 2551
1,000
99/20 10300
. 0-2668-7974 0-2668-7123 2530
. 0-2243-1129, 0-2668-7329
Web site: http:// www.onec.go.th www.thaigifted.org
580 8 . 34 1
. 10230
0-2943-8373-4 0-2510-7753
371.95
691
: 2550
81
ISBN 978-974-559-127-1
1. -
2. - 3.
. .2542 ( 2) . .2545
10
28
( 2547)
(Acceleration Program)
5
6
-
( )
. . 2542 ( 2) . .2545
10 ( 4)
28
2543
School in school Program 2547
Gifted Education
(Acceleration Program)
(Enrichment
Program)
( 3
6 5 6 )
(Extension Program)
(mentor)
AP Program (Advanced
Placement Program)
18
(Acceleration
Program) 5 ( 6 )
( )
Pre-test Post-test
1 1.
2.
3.
4.
10
6
24
38
.
4
2 5.
6.
7.
8.
38
30
48
6
200
1 9.
10.
11. 2 3
12.
27
20
36
24
5
2 13.
14.
15.
15
38
40
.
200
6
1
16.
17.
18.
(12 )
(25 )
(13 )
30
20
50
100
1
1
1 3
1 5
2
6
2 8
2 12
3
13
3 15
3 19
4
20
4 22
4 26
5
27
5 29
5 36
6
39
6 41
6 43
7
44
7 47
7 51
6
8
52
8 55
8 56
9
57
9 60
9 64
10
66
10 69
10 72
73
⌦⌦ 1
1
4
2
1.
1.
2.
2.
1. A B )b,a( Aa Bb A B
2. A B A B
3.
3.
(Ordered pairs)
b,a b,a )b,a(
a b
)b,a()y,x( ax by
A B )b,a(
Aa Bb
A B A B
⌫ ⌫ ⌦⌫ 2
4.
1.
(....) ,
2. 1
3.
4. 1
5.
1. 1
2. 1
3.
6.
1.
2.
7.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
8.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
⌦⌦ 3
1
2
“ ” 2
5
( )
1
2
3
4
5
6
5
10
15
20
25
30
(1,5), (2,10), (3,15),
(4,20), (5,25), (6,30)
(1,5) 1 5
(4,20)
4 20 ( 4,20 ) ( 20,4)
)y,x( x y
ba )b,a( )a,b(
)b,a()y,x( ax by
⌫ ⌫ ⌦⌫ 4
3,2,1A 4,2B
A B
6
A B
2 (1,2)
4 (1,4)
2 (2,2)
4 (2,4)
2 (3,2)
4 (3,4)
)4,3(),2,3(),4,2(),2,2(),4,1(),2,1(
A B
A B
A B )4,3(),2,3(),4,2(),2,2(),4,1(),2,1(
5,3,1A 4,2B
A B , B A , A A B B
A B }BbAa/)b,a{(
A B )4,5(),2,5(),4,3(),2,3(),4,1(),2,1(
B A }AbBa/)b,a{(
B A )5,4(),3,4(),1,4(),5,2(),3,2(),1,2(
A A }Ab,a/)b,a{(
A A )5,5(),3,5(),1,5(),5,3(),3,3(),1,3(),5,1(),3,1(),1,1(
B B }Bb,a/)b,a{(
B B )4,4(),2,4(),4,2(),2,2(
1
2
3
⌦⌦ 5
1
1. 2,1A , 20,10B BA AB BA
AB BA AB
2. 3,2,1A , 5,4,3B 4,2C
2.1 )CB(A
2.2 )CA()BA(
2.3 )CB(A
2.4 )CA()BA(
2.5 )CB(A
2.6 )CA()BA(
3.
4 2
⌫ ⌫ ⌦⌫ 6
2
4
2
1.
1.
2. A B
3. A
2.
2
3.
1. 2
2. r A B r A B
3. r A r A A
4.
1.
“ ” “ ” “ ”
4 2
“4 2” “4 2 2” “ 2
4”
⌦⌦ 7
2.
3. x y x > y , y = x + 1 , y = x 2
( x, y )
4. A B A
A B A
5. 2
6. 2
5.
1. 2
2. 2
3.
6.
1.
2.
7.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
8.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
⌫ ⌫ ⌦⌫ 8
2
2 600
50
( ) ( )
50
60
70
80
30,000
36,000
42,000
48,000
( 50, 30,000 ), ( 60 , 36,000 ), ( 70, 42,000 ), ( 80, 48,000 )
x
y
x600y 50x
2
r A B r A B
⌦⌦ 9
⌫ ⌫ ⌦⌫ 10
⌦⌦ 11
⌫ ⌫ ⌦⌫ 12
2
1. 4,3,2,1A
A
1.1 1xy/AA)y,x(r1
1.2 1xy/AA)y,x(r2
1.3 1xy/AA)y,x(r3
1.4 xy/AA)y,x(r4
1.5 x5y/AA)y,x(r5
1.6 54 rr
2.
A ,...3,2,1,0A
1)
2)
3)
2.1
X 0 1 2 3 8 20 100
Y 0 7 14 21 28
2.2
X 0 1 2 3 4 8 20 100
Y 6 7 8 9
2.3
X 0 1 2 3 4 8 20 100
Y 1 3 5 7
2.4
X 0 1 2 3 4 6 20 100
Y 1 2 5 10 17
3.
3.1 1xy/II)y,x(r 2
1
3.2 3x2y/II)y,x(r2
3.3 1xy/II)y,x(r3
⌦⌦ 13
3
4
2
1.
1.
2.
2.
3.
r A B
r rD r
D1 = {x / X A (x , y) r }
r rR r
R1 = {y / X B (x , y) r}
4.
1.
2.
3. Dr
Rr
⌫ ⌫ ⌦⌫ 14
4.
5.
3
6. 3
5.
1. 3
2. 3
3.
4. Internet
6.
1.
2.
7.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
8.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
⌦⌦ 15
3
}6,5,4,3,2,1{A }4,3,2,1{B
}x2
1y/BA)y,x{(r
A B r
)}3,6(),2,4(),1,2{(r
r }6,4,2{
( Domain ) r
r }3,2,1{
( Range ) r
r A B r A
r B
r A B
r rD r
D1 = {x / X A (x , y) r}
r rR r
R1 = {y / y B (x , y) r}
1 r “ ” }4,3,2,1{A
}25,20,15,10{B r
r
r = { /),( BAyx By,Ax (x , y) r}
x y x y By,Ax
r
r )}20,4(),15,3(),20,2(),10,2(),25,1(),20,1)(15,1(),10,1{(
r }4,3,2,1{Dr
r }25,20,15,10{Rr
⌫ ⌫ ⌦⌫ 16
⌦⌦ 17
r x
y 1x2
1y
r D1 = {x I / x A (x , y) }r
r R1 = {y / y y B (x , y) }r
3 r
02y2x/RR)y,x(r
r r
r r
x y
y = 0 x – 2 y – 2 = 0 x = 2
x ( 2, 0 )
x = 0 x – 2 y – 2 = 0 y = - 1
y ( 0, - 1 )
r
r D1 = {x / x A (x , y) }r
r R1 = {y B (x , y) }r
02y2x 1x2
1y
}1x2
1y/RR)y,x{(r
(0,-1)
(2,0) x
y
⌫ ⌫ ⌦⌫ 18
r r 1x2
1y
x y
x = 2 y = 0
x > 2 y > 0
x < 2 y < 0
r
4 2x4y/RR)y,x(r r
r 2x4y
x Rx4 2
}Rx/x{Dr
2x4y y4x2
0x2
0y4
y4
4y
}4y/Ry{Rr
5 2xy/RR)y,x(r r
r 2xy
x R2x
02x
0y
r }Rx/x{Dr
r }0y/Ry{Rr
⌦⌦ 19
3
1. 2x2/IxA , 4y4/IyB
1.1 xy/BA)y,x(r1
1.2 xy/BA)y,x(r2
1.3 xy/BA)y,x(r3
2. I
2.1 2yx/II)y,x(r1
2.2 3x2y/II)y,x(r 2
2
2.3 2
3 yx/II)y,x(r
2.4 2xy/II)y,x(r4
2.5 yx/II)y,x(r5
3. 36 d w
r
( d, w ) r r
4.
4.1 2x3
2y/RR)y,x(r1
4.2 1xy/RR)y,x(r2
4.3 2x/RR)y,x(r3
4.4 x2
1y/RR)y,x(r4
4.5 y2x/RR)y,x(r5
4.6 1xy/RR)y,x(r6
4.7 2
7 x9y/RR)y,x(r
4.8 3xy/RR)y,x(r 2
8
4.9 xy/RR)y,x(r 2
9
4.10 1xy/RR)y,x(r 3
10
⌫ ⌫ ⌦⌫ 20
4
4
2
1.
1.
2.
2.
3.
1. r 1r
r
2.
4.
1.
1r
2.
3.
⌦⌦ 21
4.
5. 4
6. 4
5.
1. 4
2. 4
3.
4. Internet
6.
1.
2.
7.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
8.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
⌫ ⌫ ⌦⌫ 22
4
}5,4,3,2,1,0{A , }9,8,7,6,5,4,3,2,1,0{B
r A B
}x2y/BA)y,x{(r
)}8,4(),6,3(),4,2(),2,1(),0,0{(r
r }4,3,2,1,0{ r }8,6,4,2,0{
r
r 1r
)}4,8(),3,6(),2,4(),1,2(),0,0{(r 1
1r }8,6,4,2,0{ 1r }4,3,2,1,0{
r B A 1r
}2
xy/AB)y,x{(r 1
r A B r
1r B A )x,y(
r)y,x(
}r)y,x/()x,y{(r 1
r
}xy/RR)y,x{(r 2 r
x y r
1) r }/),{( 21 yxRRyx 2) r }/),{(1 xyRRyx
3) r }/),{( 21 xyRRxy 4) r }/),{(1 yxRRxy
⌦⌦ 23
1 }3x2y/RR)y,x{(r 1r
x y 1r
x y r
r 1r
3x2y 3y2x
1r
3y2x
y23x
y2
3x
2
1
/RR)y,x{(r 1}2
3x
2
1y
}3x2y/RR)y,x{(r x
)0,2
3( y )3,0(
1r r )2
3,0( )0,3(
1r
r 1r xy
xy r 1r
y=2x-3y=x
2
3x
2
1y
(0,-3)
)0,2
3(
(-3,0)
)2
3,0(
y
x
⌫ ⌫ ⌦⌫ 24
2 2xy/RR)y,x(r 1r
1r x y 2xy
1r 2yx
/RR)y,x{(r 1 }2yx
2yx
y2x
y 0y
02x
2x
1r }2x/Rx{
1r }Ry/y{
1r
2yx
02 yy
02 yy
1r
y
x = y – 2 y 0
x
x = - y – 2 y< 0
X =
(0,2)
(0,-2)
(-2,0)
⌦⌦ 25
3 }1xy/RR)y,x{(r 2 1r
1r x y 1xy 2
1r 1yx 2
2y1x
/RR)y,x{(r 1 }1xy2
y 0y2
01x 1x
1r }1x/Rx{
1r }Ry/y{
4 }1xy/RR)y,x{(r 3 1r
1r x y 1xy 3
1r 1yx3
y1x3
/RR)y,x{(r 1 }y1x3
1r
x -3 -2 -1 0 1 2 3
1xy 3 -26 -7 0 1 2 9 28
1r }Rx/x{
1r }Ry/y{
⌫ ⌫ ⌦⌫ 26
⌦⌦ 27
5
4
2
1.
2.
(x, y)
x y
R r R x R
r r
3.
1)
2)
3)
4)
5)
4.
1.
2. r R x R
R
⌫ ⌫ ⌦⌫ 28
3.
4. 5
5. 5
6.
5.
1. 5
2. 5
3.
4. Internet
6.
1.
2.
7.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
8.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
⌦⌦ 29
5
1)
2)
3)
y = ax + b , a > 0
y = ax + b
(0 , 0)
y = ax2 , a > 0
y = ax3 , a > 0
y
x
(0 , 0)
⌫ ⌫ ⌦⌫ 30
4) Hyperbola
5)
6) 2
xy = c , c > 0
(0 , 0)
x
y = x
y
(0 , 0) x
y =
(0 , 0)
y
x
x
y
⌦⌦ 31
1)
1)
2)
2)
3)
3)
y = x
y
(0 , 0) x
y = x
y > x
y
x
y x
y
(0 , 0) x
y x
⌫ ⌫ ⌦⌫ 32
4)
4) y > x
1) r = { (x , y) RxR y = x2 }
2) r = { (x , y) RxR y > x2 }
y
(0 , 0) x
(1 , 1)
(1 , 1)
(0 , 0)
y
x
y = x2
(0 , 0)
y
x
⌦⌦ 33
3) r = { (x , y) RxR y x2 }
4) r = { (x , y) RxR y < x2 }
5) r = { (x , y) IxI y = | x | }
(0 , 0)
y
x
y = x2
(0 , 0)
y
x
y
(0 , 0) x
y = | x |
X
X
X
X
X
X
X
X
X
⌫ ⌫ ⌦⌫ 34
6) r = { (x , y) RxR | y > | x | }
7) r = { (x , y) RxR | y | x | }
8) r = { (x , y) IxI | y = - | x | }
y
(0 , 0) x
y
x
y = | x |
y
xX
X
X
X
X
X
X
X
X
⌦⌦ 35
9) r = { (x , y) IxI | y = - | x | + 2 }
10) r = { (x , y) IxI y = x2 + 2 }
y
(0 , 0) x
X
X
X
X
X
X
X
X
X
(0 , 2)
y
x
y = x2
+ 2
(0 , 2)
⌫ ⌫ ⌦⌫ 36
5
⌦⌦ 37
4. r ={(x,y) RxR|y=x+1}
x -3 -2 -1 0 1 2 3
y 9
5. r ={(x,y) RxR|1 x 4}
6.
6.1 r ={(x,y) AxA|y=x} A = {1,2,3,4,5}
6.2 r ={(x,y) AxA|y=x-1} A = {3,4,5,6,7}
⌫ ⌫ ⌦⌫ 38
6.3 r ={(x,y) RxR|y=4x}
6.4 r ={(x,y) RxR|y=x+3}
7. r r 1
r = {(-3,-2),(-2,-1),(2,3),(3,4),(4,5)}
r 1=……………………………………
8. r r 1
r = {(x,y) RxR|y=x+3}
r 1= {(x,y) RxR|y=………}
⌦⌦ 39
6
4
2
1.
2.
3.
r f ( x , y ) f ( x , z ) f
y = z
4.
1.
2. 5
1r = {(1,2),(2,5),(4,2),(3,4)}
2r = {(2,1),(5,3),(6,5),(7,3)}
3r = {(1,2),(1,3),(2,4),(3,5)}
4r = {(2,1),(3,1),(4,2),(5,3)}
5r = {(2,3),(3,4),(3,5),(2,5)}
1r , 2r , 4r
3r , 5r
3.
⌫ ⌫ ⌦⌫ 40
4.
5.
1. 2
1 xy/)y,x(r
2. 1x2y/)y,x(r2
3. 3
3 xy/)y,x(r
4. xy/)y,x(r4
5. 2
5 yx/)y,x(r
6. 1yx/)y,x(r 22
6
2
2
1 , 2 , 3 , 4 5 , 6
6.
7. 6
8. 6
5.
1. 6
2. 6
3.
6.
1.
2.
7.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
8.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
⌦⌦ 41
6
1r
)}200,4(),180,4(),150,3(),135,3(),100,2(),90,2(),50,1(),45,1{(r1
2r ( )
1 2 3 4 5 6 7 8 9 10 11 12 … 100
( ) 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 … 50.00
)}00.50,100(),...,00.2,4(),50.1,3(),00.1,2(),50.0,1{(r2
}x50.0y/RI)y,x{(r2
1r 45
50 ( 1, 45) ( 1 , 50 ) 1r
1 45 50
1r
1r 45
50
1 90
2 100
3 135
4 150
180
200
⌫ ⌫ ⌦⌫ 42
1r
2r 2r
2r
2r
1 0.50
2 1.00
3 1.50
. .
. .
100 50.00
2r 1r
⌦⌦ 43
6
1.
1. )6,5(),4,3(),3,2(),2,1( ………………….
2. )3,4(),7,3(),4,3(),2,1( ………………….
3. )6,5(),4,3(),4,3(),4,2( ………………….
4. )8,7(),8,6(),6,4(),4,2( ………………….
5. )1,1(),1,5(),1,3(),1,2( ………………….
2.
1. x
1y/)y,x( ………………….
2. 1yx/)y,x(2
………………….
3. 5xy/)y,x(2
………………….
4. 2
yx/)y,x( ………………….
5. 1x
1y/)y,x( ………………….
6. x2y/)y,x( ………………….
7. x2yx/)y,x(2
………………….
3.
…………………………………………………………………………………………………………..
…………………………………………………………………………………………………………..
…………………………………………………………………………………………………………..
…………………………………………………………………………………………………………..
…………………………………………………………………………………………………………..
…………………………………………………………………………………………………………..
…………………………………………………………………………………………………………..
…………………………………………………………………………………………………………..
…………………………………………………………………………………………………………..
…………………………………………………………………………………………………………..
…………………………………………………………………………………………………………..
⌫ ⌫ ⌦⌫ 44
⌦⌦ 45
4.
1.
2.
3.
A = {1 , 2 , 3}
A {a,b}
B = {a , b , c , d}
A = {1 , 2 , 3 , 4 , 5}
A B
B = {a , b , c , d}
A = {1 , 2 , 3 , 4}
A {a,b,c,d}
B = {a , b , c , d , e}
A = {1 , 2 , 3 , 4}
A B
B = {a , b , c , d }
4. A B
6. A
B A B f : A B
7. 7
8. 7
5.
1. 7
2. 7
3.
4. Internet
⌫ ⌫ ⌦⌫ 46
6.
1.
2.
7.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
8.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
⌦⌦ 47
⌫ ⌫ ⌦⌫ 48
⌦⌦ 49
)}9,t(),8,n(),7,m{(f
)}7,t(),8,n(),7,m{(f
f g f
fD f fR
m 7
n 8
t 9
g 7 m t
gD g gR
m 7
n 8
t 9
f g
f
f A B f A B
1x 2x A )x(f)x(f 21 21 xx
f ; A B
1-1
⌫ ⌫ ⌦⌫ 50
3 )}8,4(),9,3(),8,2(),7,1{(f
f 1 – 1
)}8,4(),9,3(),8,2(),7,1{(f
f
1 7
2 8
3 9
4
f 1 – 1 f)8,2( f)8,4( 8
42
4 1x)x(f f 1 – 1
f 1 – 1 1x 2x 1x 2x
)x(f 1 )x(f 2
1x 2x 2x1 2x2
2x1 31212)2(f)x(f 1
2x2 31212)2(f)x(f 2
)3,2( )3,2( f 22
f 1 – 1
5 f 1 – 1 4x3)x(f
)x(f)x(f 21
4x34x3 21
- 4
)4(4x3)4(4x3 21
21 x3x3
3
1
21 xx
)x(f)x(f 21 21 xx
f 1 – 1
⌦⌦ 51
7
1. R R
1. 4x9)x(f
2. 2x3)x(f
3. 14)( 2xxf
4. 1x7)x(f
5. 3x)x(f
6. 5x2x)x(f 2
7. 3)x(f
8. 0x,x)x(f 2
9. x
1)x(f
10. x)x(f
2.
}3,2,1{A , }5,4{B }6,5,4{C
………….1) {(1,4),(2,5),(3,5)} A B
………….2) {(1,4),(2,5),(3,5)} A C
………….3) {(1,4),(2,4),(3,5)} A B
………….4) {(1,4),(2,5),(3,6)} A C
3. 1 – 1
1) x)x(f
2) 1x)x(f 2
3) 5x)x(f
4) 2x3)x(f
5) 1x)x(f
6) x
1)x(f
7) 4x)x(f
8) 2x)x(f
9) 3x)x(f
10) x)x(f
⌫ ⌫ ⌦⌫ 52
8
4
4
1.
1.
2.
3.
4.
2.
gof(x) A C A
C
3.
f g gf DR f g
gof (gof)(x) = g(f(x)) x f(x) gD
4.
1. )x(f
2. f g
A B C
1 a p
2 b q
8 3 c r
3. f(1) = a , f(2) = c , f(3) = b
g(a) = p , g(b) = p , g(c) = q
g(a) = p g(b) = p g(c) = q
A B C f g
f g
⌦⌦ 53
f g
g(f(1)) = g(a) = p
g(f(2)) = g(c) = q
g(f(3)) = g(b) = p
4. gof ( ) A C
(gof)(1) = g(f(1))
(gof)(2) = g(f(2))
(gof)(3) = g(f(3))
gof = {(1,p),(2,q),(3,p)}
5. f g gof
A B C
4 7 9
5 8 10
6
gof = {(4,9),(5,10),(6,9)}
6.
7.
8. 2 5 f g
gof
9.
10. 8
11. 8
f g
⌫ ⌫ ⌦⌫ 54
5.
1. 8
2. 8
3.
4. Internet
6.
1.
2.
7.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
8.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
⌦⌦ 55
8
f g f g
}D)x(f/Dx{ gf g
gof)z,x( y f)y,x( g)z,y(
1 )}5,c(),2,b(),1,a{(f
)}9,3(),8,2(),7,1{(g f)1,a( g)7,1( gof)7,a(
f)2,b( g)8,2( gof)8,b(
f)5,c( 5 g
)}8,b(),7,a{(gof
2 1x2)x(f 2x)x(g 2
gofD , fogD , )x)(gof( )x(fog
gofD fogD
}D)x(f/Dx{D gfgof
RDf RDg
}R1x2/Rx{Dgof
R }D)x(g/Dx{D ggfog
}R2x/Rx{ 2
R
gofD fogD gof fog
))x(f(g)x)(gof(
)1x2(g 1x2)x(f
2)1x2( 2 2x)x(g 2
2A)A(g 2 1x2A
2)1x4x4( 2
1x4x4 2
))x(g(f)x)(fog(
)2x(f 2 2x)x(g 2
1)2x(2 2 1x2)x(f
1A2)A(f 2xA 2
)14x2( 2
3x2 2
⌫ ⌫ ⌦⌫ 56
⌦⌦ 57
9
4
2
1.
1.
2.
3.
4.
5.
2.
( Inverse Function )
3.
( Inverse Function )
f 1f f 1-1
4.
1.
2.
f = {(1,2),(2,3),(3,4)}
1f = {(2,1),(3,2),(4,3)}
⌫ ⌫ ⌦⌫ 58
g = {(1,2),(2,3),(3,2)}
1g = {(2,1),(3,2),(2,3)}
h = {(1,2),(3,2),(4,1)}
1h = {(2,1),(2,3),(1,4)}
3. f , g h
4. 1f
5.
6.
f = {(3,2),(4,3),(5,1)}
g = {(4,1),(5,3),(6,2)}
h = {(2,3),(3,5),(4,1)}
1. 1-1
2.
7.
8.
9.
f = {(3,2),(4,3),(5,1)}
g = {(4,1),(5,3),(6,2)}
1f = {(2,3),(3,4),(1,5)}
1g = {(4,1),(5,3),(6,2)}
10.
fD = {3,4,5} 1fD = {1,2,3}
fR = {1,2,3} 1fR = {3,4,5}
11. 10
fD = 1fR fR = 1f
D
12
f = {(1,2),(2,3),(3,5)}
1f
1f = {(2,1),(3,2),(5,3)}
⌦⌦ 59
f 1f
y
5 f
4
3 1f
2
1
0 1 2 3 4 5 x
13. 9
14. 9
5.
1. 9
2. 9
3.
4. Internet
6.
1.
2.
7.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
8.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
y = x
⌫ ⌫ ⌦⌫ 60
9
⌦⌦ 61
⌫ ⌫ ⌦⌫ 62
⌦⌦ 63
⌫ ⌫ ⌦⌫ 64
9
1. f
1. 1x5)x(f
2. 3)x(f
3. 1x)x(f2
4. 2
)2x()x(f
5. x34)x(f
6. 1x)x(f
7. x
1x)x(f
2
8. 1x0,x)x(f2
9. x
3)x(f
10. 4x0,x16)x(f2
2. 1
…………………………………………………………………………………..
⌦⌦ 65
3. f f
1f
ff
1D,R,D,f 1
fR
1. 3x2)x(f
2. x3)x(f
3. 9x)x(f2
4. 2
)1x()x(f
5. 2
x3)x(f
6. 2x)x(f
7. 1x
1x)x(f
8. 0x1,x)x(f2
9. x
1)x(f
10. 3x0,x9)x(f2
⌫ ⌫ ⌦⌫ 66
10
4
4
1.
1. 2
2.
3.
4.
2.
3.
f g gf , gf , gf g
f
1. )x(g)x(f)x)(gf(
2. )x(g)x(f)x)(gf(
3. )x(g)x(f)x)(gf(
4. g
f(x) =
)(
)(
xg
xf g(x) 0
x 1 – 3 f
g gf DDx
⌦⌦ 67
4. )x(g
)x(f)x)(
g
f( 0)x(g
x g
f f g
0)x(g gf DDx }0)x(g/x{
4.
1. f x
2.
f = {(1,2),(2,4),(3,6)}
g = {(1,1),(2,2),(3,3)}
f(1) = 2 f(2) = 4 f(3) = 6
g(1) = 1 g(2) = 2 g(3) = 3
3. 2 (f+g)(1) = f(1) + g(1) = 2+1 =3
(f+g)(2) (f +g)(3)
f+g = {(1,3),(2,6),(3,9)
4.
5.
6. 2 f – g
f – g = {(1,1),(2,2),(3,3)}
7. f+g f – g
1. f = {(2,4),(3,6),(4,8)}
g = {(2,3),(3,5),(4,6)}
2. f = {(1,3),(2,5),(3,7)}
g = {(1,2),(2,4),(3,2)}
8.
9. 2 f g g
f
f g = {(1,2),(2,8),(3,18)}
g
f = {(1,2),(2,2),(3,2)}
10.
11.
⌫ ⌫ ⌦⌫ 68
f = {(1,2),(2,4),(3,6),(4,7)}
g = {(2,3),(3,1),(4,2),(5,3)}
f + g
f + g = {(2,7),(3,7),(4,9)}
f + g
gfD = {2,3,4} , gfR = {7,9}
12. 10
13. 10
5.
1. 10
2. 10
3.
4. Internet
6.
1.
2.
7.
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
8.
……………………………….………………………………………………………………………………
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
⌦⌦ 69
10
f g gf , gf , gf g
f
1. )x(g)x(f)x)(gf(
2. )x(g)x(f)x)(gf(
3. )x(g)x(f)x)(gf(
x 1 – 3
f g gf DDx
4. )x(g
)x(f)x)(
g
f( 0)x(g
x g
f
f g 0)x(g gf DDx }0)x(g/x{
1 2x)x(f 3 1x)x(g
)x)(gf( , )x)(gf( , )x)(gf( )x)(g
f(
RDf }1x/Rx{Dg
RDD gf }1x/Rx{
}1x/Rx{
}1x/Rx{
⌫ ⌫ ⌦⌫ 70
)x(g)x(f)x)(gf(
)2x( 3 1x
2x3 1x
)x(g)x(f)x)(gf(
)2x( 3 1x
2x3 1x
)x(g)x(f)x)(gf(
)2x( 3 1x
)x(g
)x(f)x)(
g
f( 0)x(g
1x
2x3 01x 1x
g
f }1x/Rx{
2 10x
1)x(f x2)x(g )x)(gf(
}10x/Rx{Df
}2x/Rx{Dg
gf DD
fD gD )x)(gf(
3 3x2)x(f 2x)x(g
)1)(gf( , )0)(gf( , )1)(gf( )3)(g
f(
RDf RDg RDD gf
)1)(gf( )1(g)1(f
15
4
)0)(gf( )0(g)0(f
03
3
⌦⌦ 71
)1)(gf( )1(g)1(f
1)1( 1
}0{Rg
f
)3(g
)3(f)3)(
g
f(
9
3
3
1
4 )}3,4(),4,2(),3,1(),2,1{(f f5
f5
f5 gf g 5)x(g
g R
RDD gf fD}4,2,1,1{}4,2,1,1{
f5 )}35,4(),45,2(),35,1(),25,1{(
)}15,4(),20,2(),15,1(),10,1{(
⌫ ⌫ ⌦⌫ 72
10
1. )}4,5(),0,2(),3,1(),1,0{(f
)}0,5(),3,2(),1,0(),2,1{(g
gf , gf , gf , g
f f)3(
2. x35)x(f 1x)x(g 2
)x)(gf( , )x)(gf( )x(g
f
3. 5x2)x(f 2 2x4)x(g
)1)(gf( , )2)(gf( )2(g
f
4. x35)x(f 3x4 1x)x(g 5x2
)x)(gf( , )x)(gf( , )x)(gf( )x(g
f
5. 4x)x(f 2 2x)x(g
gf , gf , gf g
f
⌦⌦ 73
1. 1x x8x4)x)(fog( 2 4x)x(f 2
)4(g 1
1. 1 2. 2 3. 3 4. 4
2. 5x4xy/RR)y,x{(r 2 }03x2x2
A = r B = 1r BA
1. ]1,1( 2. )10,3[ 3. )3,1( 4. )10,1(
3. 3x2)x(f 4x2 )x(f
1. 4,2 2. 11,1 3. 11,3 4. 4,3
4. )x(gx)x(f )x(fx)x(g )x(g
f
1. x 2. x
1 3.
x1
x 4.
2x
x1
5. 1x2)x(f 4x2
3x2)x)(ogf( 1 )4)(ofg( 1
1. 4
13 2.
10
13 3.
4
15 4.
10
15
6. Axx2)x(f 3 )2,3( 1f A
1. – 15 2. 15 3. – 54 4. 54
7. 5x4x4)x)(fog( 2 2
3x)x(g 1 )x(f
1. 8x4x2 2. 10x8x2 3. 20x8x2 4. 6x4x2
8. )x2)(x3()x(f 3x
1)x(g gf
1. 2. ]2,( 3. )2,3( 4. ]2,3(
9. I 12y2x/)y,x{(f }Iy,x
fof
1. )}4,4(),5,8{( 2. )}4,4(),8,5{( 3. )}4,4(),2,2{( 4. )}4,4(),3,6{(
10. x1
x)x(f 1x)x(g 2 gofDA gDB
BA
1. }1,1{R 2. ),1( 3. ),1()1,2
1( 4. ),1()1,1(
⌫ ⌫ ⌦⌫ 74
⌦⌦ 75
เพือ่เปนการใชทรพัยากรของชาตใิหคมุคาหากทานไมใชหนงัสอืเลมนีแ้ลว
โปรดมอบใหผอูืน่นำมาใชประโยชนตอไป
กลมุพฒันาการเรยีนรขูองผเูรยีนทีม่คีวามสามารถพเิศษสำนกัมาตรฐานการศกึษาและพฒันาการเรยีนรูสำนกังานเลขาธกิารสภาการศกึษา (สกศ.)
99/20 ถนนสโุขทยั เขตดสุติ กรงุเทพฯ 10300โทรศพัท : 0-2668-7123 ตอ 2530โทรสาร : 0-2243-1129, 0-2668-7329เวบ็ไซต : http://www.onec.go.th
http://www.thaigifted.org
