E-learning 11 -02-55

1

1. ( Cartesian Product)

A B A B

6

E-learning 11 -02-55

2 2. (Relations)

30 2 3

(Subset) ( ) ( ) ()

() ()

()

A A Ar rA A

(x, y) r x y r

E-learning 11 -02-55

3 ()

()

E-learning 11 -02-55

4

()

()

E-learning 11 -02-55

5 ( )

{(,25), (,30), (,30), (,10)}

( )

E-learning 11 -02-55

6 3.

E-learning 11 -02-55

7

r ()

y = x x = y ()

-

- - -

() ()

() () () ( )

E-learning 11 -02-55

8

( y 1 ) #

() #

()

#

()

E-learning 11 -02-55

9

#

-

E-learning 11 -02-55

10 4. ( )

5. ( Function ) 1.

(1) (2) 2

() ()

E-learning 11 -02-55

11

2. (1) (2)

()

()

()

()

z x y

E-learning 11 -02-55

12

3.

(1) (2) ( )

( ) ( )

x

1

2

E-learning 11 -02-55

13 4. y

(1) y 1 x y

(2) y 1 x y 1 (

)

( ) ()

() ()

x

E-learning 11 -02-55

14 ()

y () x = 2 y 1 () y 1 ( )

E-learning 11 -02-55

15 () y 1

E-learning 11 -02-55

16

1-1

6. (1) A B :f A B fD A fR B (2) A (onto) B :f A B fD A fR B (3) (one to one) A B :f A B fD A y B x A A {1, 2, 3} B {3, 4, 5, 6, 7, 8} A B 1. f = {(x,y) A B| y = x + 3} 2. 2g = {(x,y) A B| y = x } 1. f y = x + 3 x A y B x = 1 y = 4 x A y B x = 2 y = 5 x A y B x = 3 y = 6 x A y B f = {(1, 4), (2, 5), (3, 6)} D = {1, 2, 3} = Af R = {4, 5, 6} Bf f A B 2. g 2y = x x A y B x = 1 y = 1 x A y B x = 2 y = 4 x A y B x = 3 y = 9 x A y B g = {(2, 4)} D = {2} Ag g A B

E-learning 11 -02-55

17 (3.1) 1,x y f 2 ,x y f 1 2x x 1 2( ) ( )f x f x 1 2x x f x y (3.2) x () () () x 1x 2x () x

1 1 1. 2.

y 1x

2x

0y

1x

2x

y

x

0y

1x 2x 0

y

x 0

x1 2 33 2 112

y

x

y

0

3

E-learning 11 -02-55

18

2. 4.

1.

3y x 1 1 1 2.

x ( 2)y 1 1 1 3.

x 1 1 1

4.

x 1 1-1

x

y

0x

y

0

x

y

3

0

x1 2 33 2 112

y

x

y

012 21

x

y

0

E-learning 11 -02-55

19

1-1

(4) A B :f A B 1 1

:f A B

:f A B 7. f ( 1f ) f f 1 ffD R 1 ffR D

1( )f w v ( )f v w 1. f 1f 2. f A B 1f B

A 3. f 1f f f 8. f g f g ( f g ) f gD D f x g x x

1 {(1, 4), (2, 6), (3, 8)}f {(1, 6), (2, 4), (3, 8)}g f g f g {1,2,3}f gD D (1) 4f (1) 6g (1) (1)f g f g # 2 {(1, 1), (2, 4)}f :{1, 2}g 2( )g x x f g {1, 2}f gD D (1) (1) 1f g (2) (2) 4f g f g #

E-learning 11 -02-55

20 9.

2{( , ) } f x y y x 2( ) f x x

( )f x ( y ) x () ( )f x f x (1) 1 ( ) 3 f x x () ( 1)f () (0)f () (2 1)f x () ( )f () ( 1) ( 1) 3 4 f () (0) 0 3 3 f () (2 1) (2 1) 3 2 2 f x x x () ( ) 3 f # 2 2( ) 1, ( ) 1 f x x g x x 3( ) h x x () ( (3))f g () ( (1))g f () ( ( (0)))h g f () ( ( (3)))g h f () 2( (3)) (3 1) (10) 10 1 11 f g f f () 2( (1)) ( 1 1) ( 2) ( 2) 1 2 1 3 g f g g () 2 3( ( (0))) ( ( 0 1)) ( (1)) (1 1) (2) 2 8 h g f h g h g h h () 3 2( ( (3))) ( ( 3 1)) ( (2)) (2 ) (8) 8 1 65 g h f g h g h g g # 3

2

2

3 2 , 1

( ) 5 , 1 4

, 4

x x

f x x x

x x

( ( (3)))f f f

E-learning 11 -02-55

21 3 1 2( ) 3 2 , 1 f x x x 2 ( ) 5 , 1 4 f x x x 3 2( ) , 4 f x x x

1 (3)f 3x 1 4 x ( ) 5 f x x (3) 3 5 2 f

2 ( 2)f 2 x 1 x 2( ) 3 2 f x x 2( 2) 3( 2) 2 10 f

3 (10)f 10x 4x 2( ) f x x 2(10) (10) 100 f ( ( (3))) 100 f f f # 4

2

2

3 , 1

( ) ,1 2

5 , 2

x x

f x x x

x x x

( ) 7f x 2

8

x

1x 1 2 x x ( ) 7f x 2x

2

2

( ) 7

5 7

12 0

( 3)( 4) 0

3, 4

f x

x x

x x

x x

x

4x

2 24 162

8 8 8

x #

E-learning 11 -02-55

22 (2) 1 (3 1) 6 5 f x x () ( )f x () (5)f ()

1 k 3 1 k x 1

3

k

x

2 1( ) 6 5 2 33

kf k k

3 k x ( )f x ( ) 2 3 f x x

() 1 () ( ) 2 3 f x x (5) 2(5) 3 7 f 2 1 3 1 5 x 2x 2 (5) 6(2) 5 7 f # 2 ( ) 2 3 f x x (4 1)f x ( )f x 1 (4 1)f x ( ) 2 3 f x x (4 1) 2(4 1) 3 8 1 f x x x 2 x ( )f x ( ) 2 3 f x x ( ) 3

2

f xx

3 ( ) 32

f xx (4 1) 8 1 f x x

( ) 3(4 1) 8 12

4 ( ) 11

f xf x

f x

#

E-learning 11 -02-55

23 3 ( ) 2 3 f x x () 1( )f x () 1(5)f () 1 1 ( ) 2 3 f x x 1(2 3) f x x 2 2 3 k x 3

2

k

x

3 1(2 3) f x x 1 3( )

2

kf k

4 k x 1 3( )

2

xf x #

2 () x y ( ) 2 3 f x y x 2 3 x y 3

2

x

y

1 3( )2

x

f x #

() 1 () 1 3( )

2

xf x

1 5 3 8(5) 42 2

f

2 ( ) 2 3 f x x 1(2 3) f x x 2 3 5 x 8 4

2 x

4x 1(2 3) f x x 1(5) 4 f #

1( ) f w v ( ) f v w

E-learning 11 -02-55

24

4 ( 1) 4 3 f x x () 1( )f x () 1(5)f () ( 1) 4 3 f x x 1(4 3) 1 f x x 4 3 k x 3

4

k

x

1(4 3) 1 f x x 1 3( ) 1

4

kf k

k x 1 3( ) 1

4

xf x #

() 1 () 1 3( ) 1

4

xf x

1 5 3(5) 1 2 1 14

f

2 ( 1) 4 3 f x x 1(4 3) 1 f x x 4 3 5 x 8 2

4 x

2x 1(4 3) 1 f x x 1(5) 2 1 1 f #

E-learning 11 -02-55

25 10. 1 2,x x [ , ]a b f [ , ]a b 1 2x x 1 2( ) ( )f x f x f [ , ]a b 1 2x x

[ , ]a b [ , ]a b

2( )f x

1( )f x

0a 1x 2x b

y

x

2( )f x1( )f x

y

a 2xx

1x b0

E-learning 11 -02-55

26

1-1

1-1

1-1

1-1

11. f g f g

( )( ) ( ) ( ) f g x f x g x f gx D f g f gD D D , , , ( ( ) 0g x )

1 1( )2

xf x

x

2

( 1)( )

9

x xg x

x

, ,f g f g fgD D D f

g

D

{2}fD { 3,3}gD {2, 3,3}f gD D {2, 3,3}f g f g fgD D D f

g

D ( ) 0g x 0,1x

{0,1, 2, 3,3}fg

D #

12. (Composite Function) f g f gR D f g ( og f )

( )( ) ( ( ))g f x g f x fx D ( ) f gf x R D :f A B :g C D g f f x g { | ( ) } g f f f gD x D f x R D 1. :f A B :g B C : g f A C 1 1 1( ) g f f g

2. :f A B 1 : f f A A ( )f x x x A 1 : f f B B ( )f x x x B 3. ,f g h ( ) f g h ( ) f g h ( ) ( ) f g h f g h (( ) )( ) ( )( ( )) ( ( ( ))) f g h x f g h x f g h x

( )y f x

A B C D f g

g f

x ( ) ( ( )) z g y g f x

E-learning 11 -02-55

27 1 f g g f () f gR D g f (1) , (2) , (3) , (4)f a f c f b f d ( )g a , ( ) , ( ) , ( ) , ( )g b x g c z g d y g e z f g ( (1)) ( )g f g a ( (2)) ( )g f g c z ( (3)) ( )g f g b x ( (4)) ( )g f g d y g f ( )(2) ( (2)) g f g f z (2, ) z g f ( )(3) ( (3)) g f g f x (3, ) x g f ( )(4) ( (4)) g f g f y (4, ) y g f

{(2, ), (3, ), (4, )}g f z x y {2,3, 4} g f fD D g f fD D #

1

2

3

4

a

b

c

d

e

x

y

z

fD fR

gR gD

2

3

4

x

y

z

g f

g fD g fR

f

g

E-learning 11 -02-55

28 2 2( ) 3f x x 2( ) 6g x x g f g f ( )( )g f x

2

22 2 2

( )( ) ( ( )) ( 3 )

3 6 3 6 9

g f x g f x g x

x x x

2{( , ) | 9 } g f x y y x g fD 29 0x [ 3,3]x 2 [ 3,3] (2)f 2 g fD { | [ 3,3], 2} g fD x x x # 3 f g 3( ) 1 f x x

3 2( )( ) 3 3 2 f g x x x x 1( )( 7) g f 1 1( )( 7) ( ( 7)) g f g f 1( 7) f 1( ) f w v ( ) f v w 3( ) 1 f x x 1 3( 1) f x x

3 1 7 x 2 x 1( 7) 2 f ( 2)g 3 2

3 2

3 2

( )( ) 3 3 2

( ( )) 3 3 2

( ( 2)) ( 2) 3( 2) 3( 2) 2 0

f g x x x xf g x x x x

f g

1( 2) (0) g f 1(0)f ( 1(0) f x ( ) 0f x ) 3 1 0 x 1 x 1(0) 1 f 1 1( )( 7) ( 2) (0) 1 g f g f # 4 f g

: f : g ( )( ) 3 ( ) 1 g f x f x 2( ) 4 g x x ( )f x

E-learning 11 -02-55

29 ( )( ) ( ( )) 3 ( ) 1 g f x g f x f x ......(1) 2( ) 4 g x x ......(2) (2) 2( ( )) 4( ( )) g f x f x ......(3) (1) = (3) 2

2

3 ( ) 1 4( ( ))

4( ( )) 3 ( ) 1 0

(4 ( ) 1)( ( ) 1) 0

1( ) 1,

4

f x f x

f x f x

f x f x

f x

: f ( ) 1 f x #

E-learning 11 -02-55

30

1. f(x) x

f fD , R

2. x , x 0

f(x) x x , x 0

f fD , R 0

3. 2f(x) x

2f(x) x 2f(x) x

f fD , R 0 f fD , R 0

y

x0

y

x0

y

x0

y

x0

()

E-learning 11 -02-55

31

y

x0

4. 2f(x) ax bx c x () b

2a x f(x)

a 0 a 0

5. f(x) x

f fD 0 , R 0

6. 1f(x)x

f fD 0 , R 0 7. f(x) ln(x)

f fD , R

y

x0

y

x0

E-learning 11 -02-55

32

8. xf(x) e ; e 2.7182183...

f fD , R

( x ) 0c

- ( )y f x c ( )y f x c - ( )y f x c ( )y f x c ( ) | 1|f x x ( ) | |f x x 1

( y ) 0d

- ( )y f x d ( )y f x d - ( )y f x d ( )y f x d

( ) | | 2f x x ( ) | |f x x 2

y

x0

y

x0

( ) | | 2f x x

( ) | |f x x

y

x0 1 212

( ) | |f x x

( ) | 1|f x x

E-learning 11 -02-55

33 ( )y f x ( )y f x c d 1 ( )y f x c ( )y f x c ( x ) 0c 0c ( )y f x | |c 2 ( )y f x c ( y ) d 0d 0d ( )y f x c | |d 1 2y x

1 1y x y x 1 2 1 2y x 1y x 2

y

x0 1 2

1

2

y x

1y x

1 2y x