Function problem p

1
F ก F F ˆ กF
1. ก F { , , }A a b c= {0,1}B =
ˆ กF F F ˈ ˆ กF ก B A
[O-net ʾก ก 2548]
1. {( ,1),( ,0),( ,1)}a b c
2. {(0, ),(1, ),(1, )}b a c
3. {( ,1),( ,0)}b c
4. {(0, ),(1, )}c b

2
2. ˆ กF ( )y f x= F ก F
1. ( ) 1f x x= −
2. ( ) 1f x x= +
3. ( ) 1f x x= −
4. ( ) 1f x x= +
-1 0 1
••
(0,1)
( )y f x=
X
Y

3
3. F {(1,0),(2,1),(3,5),(4,3),(5,2)}f =
F (2) (3)f f+ F F ก F

4
4. ก F ( )n A ก A
F 1 {( 1, 2),(0, 1),(1,2),(2, 3),(3,4)}r = − − − −
2 {( , ) | 1 }r x y y x= + =
F 1 2( )n r r∩ F ก F

5
5. F {1,2,3,4}A =
{( , ) | }r m n A A m n= ∈ × ≤ F
ก F r F ก F F
1. 8
2. 10
3. 12
4. 16

6
6. ก F {( , ) | ,r a b a A b B= ∈ ∈ b F a }
F {2,3,5}A = F
F r ˈ ˆ กF B F ก F
1. {3,4,10}
2. {2,3,15}
3. {0,3,10}
4. {4,5,9}

7
7. ก ˆ กF F F ก X กก F 1
1.
2
1y x= +
2. 2y x= −
3. 1y x= −
4.
1
2
x
y
 
=  
 

8
8. ก F {1,2,3,4,5,6}A =
{1,2,3,...,11,12}B =
{( , ) | 2 }
2
a
S a b A B b a= ∈ × = +
ก S F ก F F
1. 1
2. 2
3. 3
4. 4

9
9. ก x F F ก ก
2
4 5 6y x x= − − + F ก X
1.
2 1
( , )
3 3
− −
2.
5 3
( , )
2 2
− −
3.
1 6
( , )
4 7
4.
1 3
( , )
2 2

10
10. F F 3x = ˈ F ก ˆ กF
2 2
( ) ( 5) ( 10)f x x k x k= − + + + − k ˈ
F f F F ก F F
1. -4
2. 0
3. 6
4. 14

11
11. ก F
2
( ) 2 15f x x x= − − F F
1. ( ) 17f x ≥ − ก x
2. ( 3 2 3) 0f − − − >
3. (1 3 5) (1 3 5)f f+ + = − −
4. ( 1 3 5) ( 1 3 5)f f− + + = − − −

12
12. ก F {1,2}A = { , }B a b= F F F
ˈ) ก F A B×
1. (2, )b
2. ( , )b a
3. ( ,1)a
4. (1,2)

13
13. F {1,99}A = F A F F ˈ ˆ กF
1. F ก
2. F F ก
3.
4. F

14
14. ก F r F ก
F F ก F
1. r ˈ ˆ กF (1,1),(2,2) (3,3) F F ก
2. r ˈ ˆ กF ˈ ก
3. r F ˈ ˆ กF (3,3) (3, 1)− F ก
4. r F ˈ ˆ กF (1,1) ( 1,1)− F ก
•
2
1
3
-1
-2
-3
1 2 3-1-2-3
•
•
•
•

15
15. F F ˈ ก ˆ กF
2 2
2 1
3 2 1
x x
y
x x x
−
= +
+ + −
1. 2−
2. 1−
3. 0
4. 1

16
16. F a Fก ˆ กF (2 )x
y a= F (3,16) F F
1. 2
2. 3
3. 4
4. 5

17
17. F
2
( ) 2f x x x= − + + F F ก F
1. ( ) 0f x ≥ 1 2x− ≤ ≤
2. กก ก ˆ กF f F
3. ˆ กF f F F ก 2
4. ˆ กF f F F ก 2

18
18. F F ˈ ˆ กF
1. {(1,2),(2,3),(3,2),(2,4)}
2. {(1,2),(2,3),(3,1),(3,3)}
3. {(1,3),(1,2),(1,1),(1,4)}
4. {(1,3),(2,1),(3,3),(4,1)}

19
19. F ( ) 3f x x= − ( ) 2 4g x x= − + − F f gD R∪ F
1. ( ],3−∞
2. [ )2,− ∞
3. [ ]2,3−
4. ( ),−∞ ∞

20
20. ก Fก ˆ กF f ˈ
F 11 ( 11) 3 ( 3) (3)f f f− − − F
0-5-10
5
-5
X
Y

21
21. F F ˈ ˆ กF
1. {(0,1),(0,2),(2,1),(1,3)}
2. {(0,2),(1,1),(2,2),(3,0)}
3. {(1,1),(2,0),(2,3),(3,1)}
4. {(1,2),(0,3),(1,3),(2,2)}

22
22. F F ˈ F ก ˈ
1. {( , ) | }x y y x≥
2. {( , ) | }x y y x≤
3. {( , ) | }x y y x≥
4. {( , ) | }x y y x≤
10
y x=
y x= −
Y
X

23
23. F
2
( ) 3 4f x x= − − F F F ก F
1. [ ]2,2fD = − [ ]0,3fR =
2. [ ]2,2fD = − [ ]1,3fR =
3. [ ]0,2fD = [ ]0,3fR =
4. [ ]0,2fD = [ ]1,3fR =

24
24. F ( 2) 2 1f x x− = − F
2
( )f x F F ก F F
1.
2
2 1x −
2.
2
2 1x +
3.
2
2 3x +
4.
2
2 9x +

25
25. ˈ ก ˆ กF
2
( ) 2 4 6f x x x= − −
F F
ก. ก F 1x = −
. กก F
F F ก F
1. ก. ก . ก
2. ก. ก .
3. ก. . ก
4. ก. .

26
26. F F
[Entrance ก . ʾ 2520]
ก. F x 2
( ) 4 4f x x x= − − F ( )f x x= −
. F x y ˈ 0x y+ > F x y x y+ ≤ +
. F r A B⊂ × F r ˈ F ก A B
. A B A B× ก F ก 7
. F F ˆ กF f ˈ ˆ กF g F g fo F

27
27. F {1,2,3,4}, {1,3,4,5}A B= =
{(1,1),(2,3),(3,4),(4,5)}f = F F ก
ก.
1
f f−
o ˈ ˆ กF ก A B
. f fo ˈ ˆ กF ก A A
.
1
f f−
o ˈ ˆ กF ก A A
.
1
f f −
o ˈ ˆ กF ก B A

28
28. F {1,2,3}, {2,3,4}A B= = ˆ กF 1 1− ก A B
ก. {(1,3),(2,4),(3,3)}
. {(2,2),(3,3),(4,1)}
. {(1,1),(2,2),(3,3)}
. {(1,2),(3,3),(2,3)}
. F F ก

29
29. ก F {(1, 2),(0,0)}r = − F ( )P A F
F r F F ( )P A
ก. { ,{ 2},{ 2,0},{0, 2}}∅ − − −
. { ,{1},{1,0},{0,1}}∅
. {{ 2},{ },{ 2,0}, }− ∅ − ∅
. { ,{1, 2},{0,0},{(1, 2),(0,0)}}∅ − −

30
30. ก F F
1 {( , ) | 3 }r x y R R y x= ∈ × = −
2
2
{( , ) | }
1 3
r x y R R y
x
= ∈ × =
− +
F F A B 1r 2r F F A B∩
F F
ก.
.
.
.
o o o4− 2− 3
4− 2− 3
o o •
o3
2− 3
o o

31
31. ก F ( ) 3f x x=
2 2 ; 0
( )
2 3 ; 0
x x
h x
x x
− <
= 
− ≥
2
( ) 1g x x= +
F F ( )(1)f h go o F F ก
ก. 3
. 5
. 6
. 10
. F F ก

32
32. ก F
1 1
( 1) 1
2 2
f x x+ = − F F
1
(2)f −
F F ก
ก. 6
. 4
. 2
. 0
. F F ก

33
33. F F
2 2
1 {( , ) | 2 }r x y R R y x= ∈ × ≤ −
2
2 {( , ) | }r x y R R y x= ∈ × ≥
F F 1 2r r∩ F F F
ก.
.
.
.
(0, 2)
( 2,0)
(0, 2)−
( 2,0)−
(1,1)
( 1, 1)− −
(0, 2)
( 2,0)
(0, 2)−
( 2,0)−
(1,1)
( 1, 1)− −
Y
X
Y
X
(0, 2)
( 2,0)
(0, 2)−
( 2,0)−
(1,1)
( 1, 1)− −
X
Y
(0, 2)
( 2,0)
(0, 2)−
( 2,0)−
(1,1)
( 1, 1)− −
X
Y

34
34. F ,x y ˈ ก ก x y a+ = 0a > ก F
F
ก. .
.
.
.
Y
X
a
a
a−
a−
Y
X
a
a
a−
a−
Y
X
a
a
a−
a−
Y
X
a
a
a−
a−
•
•
•
•
Y
X
a
a
a−
a−

35
35. F
2
( ) 7f x x= + x ˈ
( ) sing x x= 0 2x π≤ ≤
F F ก
ก.
2
( )( ) sin( 7)f g x x= +o 0 2x π≤ <
.
1
( ) 7f x x−
= − x ˈ
.
2
( )( ) sin 7f g x x= +o 0 2x π≤ <
.
1 1
( ) sing x x− −
= 0 2x π≤ <
.
2
( )( ) sin 7f g x x x+ = + + x ˈ

36
36. f ˈ ˆ กF 1 1− ก A B F F ก
ก. A B ก F ก
. A ก กก F B
. B ก กก F A
. A B⊂
. ก F ก F ก F F F A B

37
37. F A B= = {( , ) | 2}f x y A B y x= ∈ × = +
F ก F
ก. f ˈ F F F ˆ กF F x กก F F F F y F ก
. f ˈ ˆ กF F F F ˆ กF 1 1− ก F ก ก x F
ก ˆ กF กก F
. f ˈ ˆ กF ก A B ก A B F ก
. ก F . F . f F F ˆ กF one to one
correspondence
. F F ก

38
38. F F
(1) F
3 3
( )f x a x= − 0x > F ( )f f f f x x=o o o
(2) F
2
( )
x
f x
x
= ( )g x x= x R∈ f g ก ก
ก
(3) F ( )f x x= x R∈ 1
f −
ˈ ˆ กF
1
f f−
=
F ก F
ก. F F (1)-(3) F ก F F
. F F (1)-(3) F ก F 2 F F (1) (2)
. F F (1)-(3) F ก F 2 F F (1) (3)
. F F (1)-(3) F ก F 2 F F (2) (3)
. F F (1)-(3) ก F ก F

39
39. F {1,2,3,4}A = F r ˈ F ก A A F F
F ˈ ˆ กF F F F F ˈ ˆ กF
ก. 1 {( , ) | }r x y A A y x= ∈ × = +
.
2
2 {( , ) | }r x y A A y x= ∈ × =
. 3 {(1,1),(2,4),(4,1)}r =
. 4 {(1,1),(2,4),(3,3),(4,1)}r =
. 5 {(1,2),(2,3),(3,4),(4,1)}r =

40
40. ก F
2 2
{( , ) | 2 1x y x by x− + = , ,x y b ˈ }
F F F ก F
ก. F 2b = − ก F ˈ ก
. F 0b > ก F ˈ F
. F 0b < ก F ˈ ก
. F 0b = ก F ˈ F F ก ก x
. ก ก F

41
41. ก F
2 2
{( , ) | 0}
4 9
x y
r x y= − = F F F ก F
ก. F 1r 1
1r r−
⊂ 1r ˈ ˆ กF ก R R F
. F 2r 2r r⊂ 2r ˈ ˆ กF ก R R F
. F 3r F ˈ F r 3
[ 1,1]rD = − F
F F F
1
3r −
F ˈ ˆ กF
. ก F r ˈ F 2 F ก
. F 4
3
{( , ) | }
2
r x y y x= = ˈ F r

42
42. ก
2
log 100 ( 1)
{( , ) | 5 }x
f x y y − −
= = F F ก F
ก. { | 9 11}fD x x= − ≤ ≤ { | 0 5}fR y y= ≤ ≤
. { | 9 11}fD x x= − < ≤ { | 0 5}fR y y= ≤ ≤
. { | 9 11}fD x x= − ≤ < { | 0 5}fR y y= ≤ <
. { | 9 11}fD x x= − ≤ < { | 0 5}fR y y= < <
. { | 9 11}fD x x= − < < { | 0 5}fR y y= < ≤

43
43. ก
2
2
1
{( , ) | 1}, {( , ) | }
1
f x y y x g x y y
x
= = − = =
−
F F ก F
ก.
1
2
( ) {( , ) | }
1
y
f g x y x
y
−
= =
−
o
. 2
{( , ) | }
1
x
f g x y y
x
= =
−
o
.
1
( )( ) }f g x
x
=
−
o
.
1 2
2
1
{( , ) | 1 }g x y x
y
−
= = −
. F F ก

44
44. F
2
{( , ) | 2 2f x y y x x= = + − 3 2}x− < ≤ F
(1) { | 3 6}fR y y= − ≤ ≤
(2) { |1 6}fR y y= < ≤
(3) F h f⊂ { | 1 1}hD x x= − ≤ < ( ) ( )h x f x= F
1
h−
ˈ
ˆ กF 1 1−
F F ก F
ก. F (1) ก F
. F (2) ก F
. F (3) ก F
. F (1) (3) ก 2 F
. F (2) (3) ก 2 F

45
45. ก F {( , ) | log 0}x y R R xy∈ × < ก ( )
ก F F F (ก F F ˈ F )
ก.
.
.
.
Y
X
Y
X
Y
X
Y
X

46
46. F F ก
ก. {( , ) | ,r x y x R y R= ∈ ∈
3
}
2 1
x
y
x
−
=
+
ˈ ˆ กF
1
{( , ) | ,r x y x R y R−
= ∈ ∈
3
}
1 2
x
y
x
−
=
−
ˈ ˆ กF
. F ( ) 5f x x= +
2
25
( )
5
x
g x
x
−
=
−
F f g=
. {( , ) | 0 ,r x y x y Rπ= < < ∈ sin }x
y e x= ˈ ˆ กF F
.
2 2
( ) 4, 2; ( ) 2 3f x x x g x x x= − ≥ = + − F
2
2
4
( )( ) , 2
2 3
f x
x x
g x x
−
= ≥
+ −
. F ( ) 3f x x= −
3; 3
( )
3 ; 3
x x
g x
x x
− ≥
= 
− <
F f g=

47
47. ก
2 1
( ) 6, ( )
3
f x x g x
x
= + =
−
F F ก
ก. 2
6
( )( )
( 3)
f g x
x
=
−
o
. 2
1
( )( )
3
g f x
x
=
−
o
. fR R=
. {3} { | , 3}gR R x x R x= − = ∈ ≠
. {0} { | , 0}gR R x x R x= − = ∈ ≠

48
48. ก ˆ กF f g
3
( ) ;
2
x
f x x R
+
= ∈ ( ) ;g x x x R= ∈
3x = F
1 1
[( )( ) ( )(2)]/ ( 2)f g x f g x− −
− −o o F ก
ก. 2
. 6
. 1
.
1
2

49
49. F ˆ กF f g ˈ R , , 0c R c∈ <
ก ( )f x x c= − ( )g x c= − ก x F F
ก. ( ) ( )f x g x x+ ≠ ก x R∈
. f g+ ˈ ˆ กF 1 1−
. f gD R+ =
. F f g+ F ˈ ˆ กF

50
50. ก F {1,2}A = F F ก F
ก. F ก A A F ก 4
. ˆ กF ก A A F ก 4
. ˆ กF ก A A F ก 1
. F F ˆ กF ก A A F ˈ ˆ กF

51
51. F F ˈ F
ก.
2
{( , ) | 1} {( , ) | 0}x y R R x y x y R R y x∈ × − > ∩ ∈ × + <
. {( , ) | 2} {( , ) | 2 3 }x y I I y x x y R R y x∈ × = + ∩ ∈ × = −
. {( , ) | } {( , ) | }x y R R y x x y R R y x∈ × > ∩ ∈ × <
. {( , ) | 1 4} {( , ) | 2}x y R R x y x y R R y∈ × − ≤ − < ∩ ∈ × = −

52
52. ก F
2
{( , ) | 3}A x y R R y x= ∈ × < −
{( , ) | 2 3( 1) 4 }B x y R R y x x= ∈ × + + >
ก F F F ก F
ก. ( 1, 2)− − ˈ A B′∩
.
3 3
( , )
2 4
− ˈ A B′∩
.
3 3
( , )
2 4
− ˈ A B′−
. ( 1, 2)− − ˈ A B′−

53
53. F F F F ˈ ˆ กF
ก. {( , ) | }, {1,2,3}x y A A y x A∈ × > =
.
2
{( , ) | 1}x y R R x y∈ × =
. {( , ) | 2}x y R R y x∈ × = −
. {( , ) | 2}, { 2, 1,0,1,2}x y B B y x B∈ × = − = − −

54
54. F
2
( ) 25, ( ) 2f x x g x x= − =
2
( ) ( ) ( ) ( 25)(2 )h x f x g x x x= = − F ( )( )g h xo
ก. { | 5}x x ≥
. { | 5 0x x− ≤ ≤ 5}x ≥
. { | 5x x ≤ − 5}x ≥
. { | 0}x x ≥

55
55. F
1
( )( ) , ( ) 3
3
f g x x g x x= = −o ( ( )) 2 1g h x x= − F
ก.
1
( ) ( )f x g x−
= ( ) 6 4h x x= +
.
1
( ) ( )f x g x−
= ( ) 6 6h x x= +
. ( ) 3 9f x x= + ( ) 6 4h x x= +
. ( ) 3 9f x x= − ( ) 6 6h x x= +

56
56. ก F
2
{( , ) | 6 10}r x y R R x y y= ∈ × = − + F F ˈ
ก. 1
r
D R− = 1 { | 0}r
R y y− = ≥
. 1 { | 0}r
D x x− = ≥ 1
r
R R− =
. 1
r
D R− = 1 { | 1}r
R x x− = ≥
. 1 { | 1}r
D y y− = ≥ 1
r
R R− =

57
57. ก 3 F F F ก Y F
F F F ก X F F F 3x y− = F F
F ก ˈ F ก F
ก. {( , ) | 0,0 3x y R R x y∈ × ≥ ≤ ≤ 3}y x≥ −
. {( , ) | 0,0 3x y R R y x∈ × ≥ ≤ ≤ 3}x y≥ −
. {( , ) | 3 0x y R R y∈ × − ≤ ≤ 3}y x≤ −
. {( , ) | 3 0x y R R x∈ × − ≤ ≤ 3}x y≤ −

58
58. F F ˈ
ก. F A ˈ ก :f A B→ ˈ ˆ กF 1 1− F B ˈ ก
. F f ˈ ˆ กF 1 1− F F ˈ
1 1
f f f f− −
=o o
.
2
( )g x x= 0x ≥ F ˈ ˆ กF 1 1−
. ( )
x
f x e= ˈ ˆ กF 1 1−

59
59. ก F
2
( )
1
f x
x
=
− F F ˈ
ก. { | 1}fD x x= ≠ { | 2 0}fR x x= − ≤ <
. { | 1fD x x= ≠ 1}x ≠ − { | 2 0}fR x x= − ≤ ≤
. { | 1}fD x x= ≠ { | 2fR x x= ≤ − 0}x >
. { | 1fD x x= ≠ 1}x ≠ − { | 2fR x x= ≤ − 0}x >

60
60. ˆ กF F
( ) 1f x x= + , ( )g x x= ,
1
( )h x
x
=
F F ˈ ( ก กF f go Fก F g fR D⊂ )
ก. f ho F
. h go F
. g fo F
. h fo F

61
61. ก F {( , ) | }r x y R R y x x= ∈ × = F r
ก.
1 ; 0
{( , ) | }
; 0
x x
r x y R R y
x x
−
 ≥
= ∈ × = 
− <
.
1 ; 0
{( , ) | }
; 0
x x
r x y R R y
x x
−
 ≥
= ∈ × = 
− − <
.
1 ; 0
{( , ) | }
; 0
x x
r x y R R y
x x
−
− ≥
= ∈ × = 
− <
.
1 ; 0
{( , ) | }
; 0
x x
r x y R R y
x x
−
− ≥
= ∈ × = 
− − <

62
62. ก F
2 2
1 {( , ) | 1}r x y R R x y= ∈ × + =
2 2
1
{( , ) | 1}
1
r x y R R y
x
= ∈ × = −
+
F A ˈ 1r B ˈ F 2r F A B− F F
ก. [0,1] {1}∪
. (0,1] { 1}∪ −
. (0,1]
. { 1}−

63
63. F F
ก. F f ˈ ˆ กF ก A B g ˈ ˆ กF ก B C
F g fo ˈ ˆ กF ก A C
. F ( )f x x= 2
( )g x x= F g f f gD D≠o o
. F
2
( ) 4 3f x x x= − + ( )g x x= F f g g fR R≠o o
. F
2 1
( )
3
x
f x
+
= 3 2
( ) 3 3g x x x x= − + F
1 1 1 1
(1) (1)f g g f− − − −
=o o

64
64. F { | 0}R x R x+
= ∈ ≥ {0,1,2,3,...}N =
:f R R+ +
→ ( ) 2f x x=
(0) 1, ( 1) ( ( )),g g n f g n n N= + = ∈
F F
ก. g ˈ ˆ กF F ก N R+
. f go ˈ ˆ กF F ก N R+
. g f gR R= o
. ( ) 2,g n n N< ∀ ∈

65
65. F ( )
1
x
f x
x
=
+ F
1
( )f x−
F
ก.
1
x
x−
. 1
x
x−
. 1
x
x+
.
1
x
x+

66
66. F
2 2
1 {( , ) | 4 4}r x y R R x y= ∈ × + =
2 {( , ) | log }r x y R R y x= ∈ × =
F F
ก. 1 1r rD R⊂
. 2 2r rD R⊂
. 1 2r rD D⊂
. 1 2r rR R⊂

67
67. F {( , ) | 3 2}f x y R R y x= ∈ × = −
{( , ) | 2 7}g x y R R y x= ∈ × = +
F F
1 1
( )(2)g f− −
o F F
ก.
17
6
−
.
7
2
−
.
1
6
−
.
7
2

68
68. ˆ กF F F ˈ ˆ กF F
ก. 4logy x=
.
2
, 1x
y a a= >
. sin 7y x= −
.
3
5 2y x= − +

69
69.
2 2
( ) ( ) 4y y x x+ − + = ก ˈ F
ก.
.
.
.
Y
X
2
1
1−2− 1 2
1−
Y
X
1
1−2− 1 2
1−
Y
X
1
1−2− 1 2
1−
Y
X
1
1−2− 1 2
1−
2
2
2
2

70
70. ˆ กF F F ˈ F F F
“ก F A ≠ ∅ ˈ F :f A A→ F f ˈ ˆ กF F ” F
ˈ
ก. ( ) ,f n n n N= ∀ ∈ , N =
. ( ) 2 ,f n n n N= ∀ ∈ , N =
. ( )
1
n
f n
n

= 
+
.
1
2( )
2
n
f n
n
+

= 


F n ˈ ก
F n ˈ F ก
F n ˈ ก
F n ˈ F ก

71
71. ก F
2
( ) 10 , ( ) 1x
f x g x x= = −
{( , ) | ( )( )}r x y R R y f g x= ∈ × = o F F ก
ก. [ 1,1], [0,1]r rD R= − =
. [0,1], [1,10]r rD R= =
. [ 1,1], [1,10]r rD R= − =
. F F f go F

72
72. F
2
( )f x x= ,A R⊆ R=
1
( ) { | ( ) }f A x f x A−
= ∈
F F
ก.
1
([ 25,0]) {0}f −
− =
.
1
([ 1,1]) [ 1,1]f −
− = −
.
1
([0,1]) [ 1,1]f −
= −
.
1
([4,9]) [2,3]f −
=

73
73. ก F {1,2,3,4,5}A = ˆ กF :f A A→ F
, ( )x A f x x∈ > ( ) 3f x = F ก F F
ก. 24
. 29
. 72
. 120

74
74. F
2 ; [ 2,3]
( )
5 ; (3,8)
x x
f x
x x
 ∈ −
= 
− ∈
2 ; ( 2,0]
( )
4 ; (0,4]
x x
g x
x x
− ∈ −
= 
− ∈
F F A = F f B =
1
g−
F A B′∩ F F
ก. ( 2,0) [2,6]− ∪
. ( 2,0) (2,6)− ∪
. [2,6]
. ( 2,0)−

75
75. ก F {2,5,6,7,8}D = F F D ˈ F
F F ˈ ˆ กF
ก. {( , ) | sin ( 5)}
6
x y y x
π
= −
. {( , ) | 2}x y y x= −
.
2
{( , ) | 4 }x y y x x= −
. {( , ) |x y y = กก x F 4}

76
76. ก F
2
{( , ) | 4 }f x y R R y x= ∈ × = −
{( , ) | 2}g x y R R y x= ∈ × = −
{( , ) | 2 0h x y R R y x= ∈ × + + = 0}x ≤
F F F ˈ ˆ กF F F F
ก. ( )f g h∩ ∩
. ( )f g h∩ ∪
. ( )f h g∩ ∪
. ( )f g h∪ ∪

77
77. ก F f g ˈ ˆ กF ก R R
2
( ) 2
1 ; 1
( )
20 ; 1
x
f x
x
g x
x x
=
≤
= 
− >
F n ˈ ก F F F ( )( ) 0g f n >o F n F ก F
F
ก. 1
. 2
. 3
. 4

78
78. F :f R R+
→ R+
ˈ ก :g R R→
ก F
2
( )( ) 3[ ( )] 2 ( ) 1g f x f x f x= − +o
2
( ) 2g x x x= − + F F F
ก. ( )(1) 2g f =o
. ( )(1) 2gf =
. ( )(1) 2
g
f
=
. ( )(1) 2g f− =

79
79. ก F
( ) ; 3
( ) ( ( 1)) ; 3 0
1 ; 0
f x x
f x f f x x
x x
 < −

= + − ≤ <
 + ≥
F 5h > F
(3 ) ( )
( 2)
f h f h
f
+ − −
−
F F ก F

80
80. ก F
21
( ) 3 1
2
f x x= +
( ) 3g x x= −
2
( ) 5 6h x x x= − + +
F
g
U
h
= F f UR D∩ ˈ F F
ก. ( 4,1)−
. ( 1,5)−
. (2,7)
. (4,8)

81
81. ก ˆ กF f g ก R R
( ) 1
1
( )
( )
f x x
g x
f x
= +
=
( )( )g f xo F F ก F F
ก. 1 x+
. 2 x+
.
1
1 x+
.
1
2 x+

82
82. F f g ˈ ˆ กF ก
2
{( , ) | 2 5}
{( , ) | 2 3}
f x y R R x y
g x y R R x y
= ∈ × + =
= ∈ × − =
F g fo F F
ก.
2
{( , ) | 2}x y R R x y∈ × + =
.
2
{( , ) | 4 11}x y R R x y∈ × + =
.
2
{( , ) | 4 2 5}x y R R x x y∈ × + − =
.
2
{( , ) | 4 12 2 4 0}x y R R x x y∈ × − + + =

83
83. ก F R ˈ
F
2 2
{( , ) | 9 4 18 16 11 0}r x y R R x y x y= ∈ × + − + − = F r rD R∩
F ก F
ก. [ 1,3]−
. [ 5,1]−
. [ 1,1]−
. [ 5,3]−

84
84. F
1
( )
2
x
f x
x
−
=
−
( )( 2) 3 6f g x x+ = +o F (2)g F ก F
ก.
5
6
.
3
2
.
12
5
.
24
11

85
85. ก F {1,2}A = {1,2,3,...,10}B =
F { | : ,N f f A B f= → ˈ 1 1− x A∈ F F ( ) }f x x=
F N กก

86
86. F { 2, 1,0,1,2}A = − − F ˆ กF :f A A→
F ( ) 0f x > 0x < ( ) 0f x < 0x > F ก F
F
ก. 160
. 80
. 64
. 16

87
87. F R ˈ :f R R→ ก
1 ; 0
(1 ) 0 ; 0
1 ; 0
x x
f x x
x x
− − <

− = =
 − >
F
2
( )x y f y x∗ = − x y F F ( 2) (3)f− ∗ F
F F
ก. ( 4, 2]− −
. ( 2,2]−
. (2,4]
. (4,6)

88
88. F R ˈ :f R R→ :g R R→ ก
2 1
( ) x
f x a +
= ( ) 5g x bx= +
F
1
( )( 2) 27f g−
− =o ( )(0) 15fg = F 3 ( 1) 4 (2)f g− −
F F ก F
ก. -35
. -33
. 37
. 39

89
89. F I ˈ F : , :f I I g I I→ → ก
( ) 2 ;f x x= ก x I∈
0
( )
2
g x x


= 

F :F I I→ ก F g f f= −o F F ˈ F
ก. F F F
. F F F
. F F F
. F
F x ˈ
F x ˈ

90
90. F R ˈ ก F
2 3
{ | 4}
2
x
A x R
x
−
= ∈ <
+
F F
(1) F a b ˈ ก A F
2
a b+
ˈ ก A
(2) F :f A R→ ก
2
( )f x x= F F f [0, )∞
F F ก F
ก. ก
. ก ก
. ก ก ก
. ก ก

91
91. ก F R ˈ I ˈ
F
2
{ | 2 8}A x I x= ∈ − <
1
{ |1 0}B x R
x
= ∈ + >
F F F F ˈ ˆ กF ก A B∩ B
ก. {( 3,1),( 2,2),( 1,3),(1,4),(2,5)}− − −
. {( 3,0),( 2,1),(1, 1),(2, 2),(3, 3)}− − − − −
. {( 3,1),(0,2),(1,1),(2,3),(3,4)}−
. {( 3,1),( 2,4),(1,5),(2,2),(3,1)}− −

92
92. F
2
1 {( , ) | 2 0}r x y x y= + − ≤
2
2 {( , ) | ln 0}r x y y x= − ≥
F 1 2( )r r∩ F F
ก. [1,2]
. ( ,0]−∞
.
1
( ,1] [ ,1]
2
−∞ ∪
.
1
( , ] [1,2]
2
−∞ ∪

93
93. F ( ) 1f x x= − 1 2
( )( ) 4 1g f x x−
= −o F ก
( ) 0g x = ˈ F F
ก. [ 4, 1]− −
. [ 1,0]−
. [0,4]
. [4,6]

94
94. F
2
{( , ) |r x y y x= ≤ 2 }y x≥ F F 1
r−
F F
ก. [0,2]
. [0,4]
. ( ,0] [2, )−∞ ∪ ∞
. ( ,0] [4, )−∞ ∪ ∞

95
95. F ( ) (3 )(2 )f x x x= + −
1
( )
3
g x
x
=
+
F f g⋅
F F
ก. ∅
. ( ,2]−∞
. ( 3,2)−
. ( 3,2]−

96
96. F f g ˈ ˆ กF ก R R
F
3
( ) 1f x x= + 3 2
( )( ) 3 3 2f g x x x x= + + +o F F
1
( ) ( 7)g f −
−o F ก F F
ก. -1
. -2
. 1
. 3

97
97. F 1r 2r ˈ Fก
1
2
2
{( , ) | 3}
{( , ) | 9 0
r x y R R y x
r x y R R x y
= ∈ × ≤ −
= ∈ × + − ≤
F F ก
ก. 1 2r r⊂
. 2 1r r⊂
.
1
1 2r r −
⊂
.
1
2 1r r−
⊂
3}y ≥

98
98. ก F {1,2,3}A = { , }B a b=
F { | }S r r A B= ⊂ ×
{ |F r S r= ∈ ˈ ˆ กF ก 2}=
F ( )n F F ก F

99
99. F A ก 8 B ก 6 A ก B ก
F ก 3 F ˆ กF F ก ( )B A− ( )A B−
F ก F F
ก. 30
. 60
. 10
. 20

100
100. F {1,2,3,4,5}A = S ˈ ˆ กF f :f A A→
ˈ ˆ กF 1 1− F (1) 3f > F ก S F ก F F
ก. 40
. 48
. 56
. 72

101
101. ก F 2
( ) 2 1f x x x= + + 3 2
( ) 3 3 9g x x x x= + + + F
1
( )(7)f g−
o F F ก F
ก. 2−
. 1−
. 1
. 2

102
102. F I +
ˈ ก
ก F {( , ) | 2 12f x y x y= + = , }x y I +
∈ F f fo F ก F
F
ก. {(8,5),(4,4)}
. {(5,8),(4,4)}
. {(2,2),(4,4)}
. {(6,3),(4,4)}

103
103. F {0,1,2,3}A = ( )P A F A
F r ˈ F ก A ( )P A ก
{( , ) | 2,r a B a a B= ≥ ∉ 1 }a B+ ∉ F r กก

104
104. F F 2
4
{( , ) | 2 }
( 1) 4
r x y R R y
x
= ∈ × = −
− −
F F
F r
ก. ( ,2) [3, )−∞ ∪ ∞
. ( ,2) (3, )−∞ ∪ ∞
. ( ,2] [3, )−∞ ∪ ∞
. ( ,2] (3, )−∞ ∪ ∞

105
105. F ( ) 10 ,x
f x x= ˈ ก ,a b ˈ ก F f F
1
1
( )
( )
f ab
f b
−
− F F
ก. 10log a
. 101 log a+
. 1 logb a+
. 1 loga b+

106
106. F 2
{( , ) | 2 1}f x y R R y x x= ∈ × = + +
2
1
{( , ) | }
1
g x y R R y
x
= ∈ × =
−
( )h g f fg= +o F h F F
ก. { | 1}x x ≠
. { | ( 2) 0}x x x − ≠
. 2
{ | ( 1)( 2) 0}x x x− − ≠
. 2
{ | ( 1)( 2) 0}x x x x− + ≠

107
107. F
1
( )
1
f x
x
=
+
1x ≠ − F I ˈ ˆ กF ก ก F
( )( )g f f f I= +o F ( )g x F ก F F
ก. 1
.
2
( 1)
( 2)
x
x
+
+
.
2
( 1)
( 2)
x x
x
+ +
+
.
2
( 1)
( 2)
x x
x
+ −
+

108
108. ก F
2
2 , 1
( ) ( 1) , 1 2
( 1) , 2
x
f x x x
x x
≤ −

= − − < <
 + ≥
ก ( ) 4 0f x − = ˈ ˈ F F F
ก. ( 3,5)−
. ( 6, 1)− −
. ( 5,4)−
. (1,6)

109
109. ก F { |S x x I= ∈ 5}x ≤
3 2 2
4
4
( ) ; ,
4
x x x a
f x a S b S
x bx
− − +
= ∈ ∈
+ +
F ( , )a b S S∈ × F (1) 0f = F ก F
[Entrance 1 , 2541]
ก. 15
. 18
. 20
. 22

110
110. ก F 2
{( , ) | log( 1) log( 2) log(4 )}f x y y x x x= = + + + − −
1
{( , ) | 2x
g x y y −
= = 0}x ≥
F f gD R∩ ˈ F
[Entrance 1 , 2541]
ก. [0,1.5)
. [0.5,2.5)
. [1,3)
. [1.5,4)

111
111. F {1,2,3}A = { , , , }B a b c d= F ก
{ : |f A B f→ F ˈ ˆ กF 1 1}− F ก F
[Entrance 1 , 2541]
ก. 40
. 34
. 30
. 24

112
112. ก F r ˈ F
2
2
1
{( , ) | }
1
x
r x y y
x
−
= =
+
F F ก F
[Entrance 1 , 2542]
ก. 1[ 1,1], [ 1,1]r r
D D −= − = −
. 1[ 1,1], [0,1]r r
D D −= − =
. 1[0,1], [ 1,1]r r
D D −= = −
. 1[0,1], [0,1]r r
D D −= =

113
113. ก
0, 0
( )
1, 0
x
f x
x
<
= 
≥
F {( , ) | (1 )x
g x y y f e= = − 0}y > F F F ก
[Entrance 1 , 2542]
ก. g gD R ′⊂
. g gD R′ ⊂
. [1, )g gD R⊂ ∪ ∞
. [1, )g gD R⊂ ∩ ∞

114
114. F ( ) 1f x x= − F
30
2
10
( )( )
n
f f n
=
∑ o F F
[Entrance 1 , 2542]
ก. 9028
. 9030
. 9128
. 9170

115
115. F ( ) 4f x x=
2
( )
1
g x
x
=
−
F F x F
( )( ) ( )( )f g x g f x=o o F ก F
[Entrance 1 , 2542]

116
116. ก F ( )
1
x
f x
x
=
−
2
( ) 1g x x= − F g fA D= o
gB D= F A B′∪ F F
[Entrance 1 , 2542]
ก. { 1,1}R − −
. ( 1, )− ∞
.
1
( ,1) (1, )
2
∪ ∞
. ( 1,1) (1, )− ∪ ∞

117
117. F
1
( ) sin , ( ) cosf x x g x x−
= = ( ) ( )( )h x f g x= o
F F
(1) h ( ( )) ( )
2
g h x g x
π
− =
(2) h ˈ ˆ กF F
F F ˈ
[Entrance 1 , 2542]
ก. (1) ก (2) ก
. (1) ก (2)
. (1) (2) ก
. (1) (2)

118
118. F {1,2,3}A = {3,4}B =
F { | : |S f f A B A B f= ∪ → × ˈ ˆ กF F }
F ก S F ก F F
[Entrance 1 , 2542]
ก. 120
. 240
. 360
. 480

119
119. ก F
7
( ) ( ), 3 3
24
x
f x xπ
+
= − < ≤ ( 6) ( )f x f x+ =
ก x R∈ F 1
( ) sin , [0, ]g x A x A π−
= + ∈
2
cos
5
A = F
F 1
( )(5)g f−
o F ก F F
[Entrance 1 , 2542]
ก.
1
10
.
1
5
.
1
5
−
.
1
10
−

120
120. ก F 2
{( , ) | 9 }r x y y x= = − 2
1
{( , ) | }
9
s x y y
x
= =
−
F F
1. 1r s
D R −∩ = ∅
2. 1 (0, )r s
R D −∩ = ∞
F F ก
[Entrance 1 , 2543]
ก. (1) (2) ก
. (1) ก (2)
. (1) (2) ก
. (1) (2)

121
121. F , :f g R R→ ก ( )
1
x
f x
x
=
+
( )g x = F กก F F ก x
( F (1.01) 2, ( 6) 6, ( 7.99) 7g g g= − = − − = − ˈ F )
F ( ) ( )( )F x f g x= o ( ) ( )( )G x g f x= o F F F ˈ
[Entrance 1 , 2543]
ก. ( , )FD = −∞ ∞
. (0,1)FR =
. ( ) 1; 0G x x= >
. ( ) 0; 0G x x= <

122
122. F {1,2,3,4,5}A = { , }B a b= F
{ | :S f f A B= → ˈ ˆ กF } ก S F ก F F
[Entrance 1 , 2543]
ก. 22
. 25
. 27
. 30

123
123. F 2
( ) ( 1)f x x= + ( ) 1g x x= + F f g g fD R′∩o o
F F
[Entrance 1 , 2543]
ก. [0,1)
. [0,2)
. [1, )∞
. [2, )∞

124
124. F ( )( ) 3 14f g x x= −o
1
( 2) 2
3
f x x+ = − F
1
( )( )g f x−
o F ก F F
[Entrance 1 , 2543]
ก. 3 4x −
. 3 6x −
. 3 8x −
. 3 10x −

125
125. F ,A B F ˈ ก
{1,2,3,4,5,6}A =
{{1},{1,2},{1,2,3},{1,2,3,4}}B =
{ : | ( )F f B A f x x= → ∉ ก }x B∈
ก F F ก F F
[Entrance 1 , 2544]
ก. 24
. 60
. 100
. 120

126
126. ก F 2
1
{( , ) | }
1
r x y y
x
= =
−
F F
(1) ( , 1) (1, )rD = −∞ − ∪ ∞
(2)
1 1
{( , ) | }
x
r x y y
x
− +
= = ±
F F ก
[Entrance 1 , 2544]
ก. (1) (2)
. (1) (2)
. (1) (2)
. (1) (2)

127
127. ก F ( ) , 1
1
x
f x x
x
= ≠ −
+
( ) , 1
1
x
g x x
x
= ≠
−
F F
[Entrance 1 , 2544]
ก. 1
( ) ( ) , 1f g x x x−
= ≠o
. 1 1
( )( ) , 1f g x x x− −
= ≠ −o
.
1
( )( ) , 1
1 2
x
f g x x
x
−
= ≠
+
o
.
1
( )( ) , 1
1 2
x
g f x x
x
−
= ≠ −
+
o

128
128. ก F ( ) 2sin
2
x
f x = 2
( ) 1g x x= −
( )f g g fR D R∩ − o F F
[Entrance 1 , 2544]
ก. ( 1,1)−
. ( 2,2)−
. [2, 3] [1,2]− ∪
. [ 2, 1] ( 3,2]− − ∪

129
129. ก F {1,2,3,4}A =
{ : | ( ) 1S f A A f x x= → ≤ + ก }x A∈
ˆ กF ˈ ก S F ก F
[Entrance 1 , 2544]

130
130. F 3 2 2 2
{( , ) | 2 3 0}r x y R R x xy x y= ∈ × + − + =
F F 1
r−
F ก F
[Entrance 1 , 2544]
ก.
1 1
( , ]
3 2
−
.
1 1
[ , )
2 3
−
.
1 1
( , ) ( , )
3 3
−∞ − ∪ − ∞
. ( , )−∞ ∞

131
131. ก F 2
( ) 4f x x= − 2
1
( )
9
g x
x
=
−
F F ˈ ก g fR o
[Entrance 1 , 2544]
ก.
1
2
.
1
4
.
1
8
.
1
14

132
132. ก F ( 1) 3 2 ( )f x x f x+ = + + (3 1) 2 8g x x− = +
F (0) 1f = F 1
( (2))g f−
[Entrance 1 , 2544]
ก. 1−
. 0
. 1
. 2

133
133. ก F 1 {( , ) | 1}x y
r x y e +
= ≤
2 {( , ) | ln( 3 5) 0}r x y x y= − + ≥
ˈ ก 1 2r r∩ F ก x F ก F F
[Entrance 1 , 2545]
ก. 1.5 F
. 2 F
. 2.5 F
. 3 F

134
134. ก F I ˈ
F ,f g ˈ ˆ กF ก I I ก ( ) 2f x x=
( ) 2
x
g x
x


= 

g f f−o ˈ ˆ กF ก I I F F
[Entrance 1 , 2545]
ก. F
. F F F
. F F F
. F F F
x ˈ F
x ˈ

135
135. ก F ( ) 5 ( )f x g x= − ( ) 5 2g x x= +
F [ , ]f gD a b=o F 4( )a b+ F ก F F
[Entrance 1 , 2545]
ก. 15
. 20
. 25
. 30

136
136. ก F ,f g ˈ ˆ กF F 1
( ( )) 2f g x x−
= + ก x R∈
F F
(1) (2 ) (2( 1))f x g x= − ก x R∈
(2) 1
( ( ))g f x−
ˈ ˆ กF R
F F ก
[Entrance 1 , 2545]
ก. (1) ก (2) ก
. (1) ก (2)
. (1) (2) ก
. (1) (2)

137
137. ก F
21
( ) 36 4
3
f x x= − F { | [ 3,3]A x x= ∈ −
( ) {0,1, 2,3}}f x ∈ F ก A F ก F
[Entrance 1 , 2545]

138
138. ก F k ˈ F
{( , ) | }r x y R R x k x y k y+ +
= ∈ × + = + F F
(1) F 1k = F r ˈ ˆ กF
(2) F 1k = − F r ˈ ˆ กF
F F ก
[Entrance 1 , 2545]
ก. (1) ก (2) ก
. (1) ก (2)
. (1) (2) ก
. (1) (2)

139
139. ก F
2
2 , 1
( ) ( 1) , 1 2
1 , 2
x
f x x x
x x
≤ −

= − − < <
 + ≥
F k ˈ F F ( ) 5g x > F ( )( )g f ko F F ก F F
[Entrance 1 , 2545]
ก. 5
. 6
. 7
. 8

140
140. ก F ( ) , 0f x x x= ≥
,0 1
( )
1, 1
x x
g x
x x
≤ <
= 
+ ≥
F F
(1) 1
g f −
o ˈ ˆ กF fR
(2) 1
f g−
o ˈ ˆ กF gR
F F ก
[Entrance 1 , 2545]
ก. (1) ก (2) ก
. (1) ก (2)
. (1) (2) ก
. (1) (2)

141
141. ก F {1,2}, {1,2,3,...,10}A B= =
1:1
{ | :f f A B→ x A∈ ( ) }f x x=
ก F ก F F
[Entrance 1 , 2546]
ก. 16
. 17
. 18
. 19

142
142. ก F 2
( ) ( 1)f x x= − − ก 1x ≤
( ) 1g x x= − ก 1x ≤
F F
(1)
1
( ) 1f x x−
= − ก 0x ≤
(2)
1 1 1 3
( )( )
4 4
g f− −
− =o
F F ก
[Entrance 1 , 2546]
ก. (1) (2) ก
. (1) ก (2)
. (1) (2) ก
. (1) (2)

143
143. ก F f g ˈ ˆ กF ( ) 0f x < ก x
F 2
( )( ) 2[ ( )] 2 ( ) 4g f x f x f x= + −o
1 1
( )
3
x
g x− +
= F
F F
(1) g fo ˈ ˆ กF
(2) (100) (100) 300f g+ =
F F ก
[Entrance 1 , 2546]
ก. (1) (2) ก
. (1) ก (2)
. (1) (2) ก
. (1) (2)

144
144. ก F {( , ) | 0 ,0 5r x y x y= ≤ ≤ ≤ 2 2
2 6 8}x y x y− − + ≤
F F
(1) [0,3]rD =
(2) F 0 c< (3, )c r∈ F 5c =
F F ก
[Entrance 1 , 2546]
ก. (1) (2) ก
. (1) ก (2)
. (1) (2) ก
. (1) (2)

145
145. ก F 0a >
2
( ) , 0f x ax x= ≥
3
( )g x x=
F 1
( )(4) 2f g−
=o F
1
1
(64)
(64)
f
g
−
− F F ก F
[Entrance 1 , 2546]

146
146. ก F ,f g ˈ ˆ กF [0, )fD = ∞
1 2
( ) , 0f x x x−
= ≥
1 2
( ) ( ( )) 1 , 0g x f x x−
= + ≥
F 0a > ( ) ( ) 19f a g a+ = F 1 1
( ) ( )f a g a−
+ F ก F F
[Entrance 1 , 2546]
ก. 273
. 274
. 513
. 514

147
147. ก F 0a > 3
(10)
( )
1
x
a
g x
x
−
= 
−
F ( 2.5, )gR = − ∞ F F F
(1) 1
( 1) log 2g a−
− =
(2)
1
3
log(4 )
( )
1
x
g x
x
−

= 
−
F F ก
[Entrance 1 , 2546]
ก. (1) ก (2) ก
. (1) ก (2)
. (1) (2) ก
. (1) (2)
1x <
1x ≥
0x <
0x ≥

148
148. F
2
4
{( , ) | }
2
x
r x y y
x
−
= =
−
F F
(1) 4 rR∈
(2) 1 [0,4) (4, )r
R − = ∪ ∞
F F ก
[Entrance 1 , 2546]
ก. (1) ก (2) ก
. (1) ก (2)
. (1) (2) ก
. (1) (2)

149
149. ก F ( ) 10x
f x = 2
( ) 100 3g x x= −
F ก ˈ ก g fR o F F
[Entrance 1 , 2547]

150
150. ก F {( , ) |r x y x y= ≥ 2 2
2 3}y x x= + −
F F
(1) [1, )rD = ∞
(2) ( , )rR = −∞ ∞
F F ก
[Entrance 1 , 2547]
ก. (1) ก (2) ก
. (1) ก (2)
. (1) (2) ก
. (1) (2)

151
151. ก F 2
( )f x ax b= + ( 1) 6g x x c− = + , ,a b c ˈ F
F ( ) ( )f x g x= 1,2x = ( )(1) 8f g+ = F 1
( )(16)f g−
o F
F ก F F
[Entrance 1 , 2547]
ก.
31
9
.
61
9
. 10
. 20

152
152. ก F
1
( )
1 1
x
f x
x
−
= 
+ −
F F
(1) 1
( ) ( )f x f x−
≠ ก (1, )x ∈ ∞
(2) 0a ≥ 2 F 1
( )f a a−
=
F F ก
[Entrance 1 , 2547]
ก. (1) ก (2) ก
. (1) ก (2)
. (1) (2) ก
. (1) (2)
[0,1]x ∈
(1, )x ∈ ∞

153
153. ก F 2
( )
1
x
f x
x
=
−
( 1,1)x ∈ − F F
(1)
2
1
1 1 4
( ) 2
0
x
f x x
−
− − +

= 


(2) f ˈ ˆ กF F ( 1,1)−
F F ˈ
[Entrance 1 , 2547]
ก. (1) ก (2) ก
. (1) ก (2)
. (1) (2) ก
. (1) (2)
0x ≠
0x =

154
154. F {1,2,3,4}A = {1,2,3,4,5}B =
F f ˈ ˆ กF ก A B
(1) 2f =
(2)f m= m ˈ
F ˆ กF f ก F F ก F
[Entrance 1 , 2548]
1. 75
2. 150
3. 425
4. 500

155
155. ก F
5
( ) 1h x x= − 5
( )g x x= F f ˈ ˆ กF
( ( )) ( )f g x h x= F (5)f F F
[A-net ก F ʾ 2549]

156
156. ก F {1,2,{1,2},(1,2)}A = (1,2) F
( )B A A A= × − ก B F ก F
[A-net ก F ʾ 2549]

157
157. ก F
2
1 1 4
( ) 2
0
x
f x x
− + +

= 


F
1 2
( )
3
f a−
= F a F F ก F
[A-net ก F ʾ 2549]
0x ≠
0x =

158
158. ก F {1,2,3,4,5}A =
{ , }B a b=
ˆ กF ก A B ก ˆ กF
[A-net ก F ʾ 2549]

159
159. ก F 2 2
{( , ) | 16}r x y R R x y= ∈ × + =
2 2
{( , ) | 3 2 0}s x y R R xy x y= ∈ × + + + =
F F ˈ r sD D−
[A-net ʾ 2550]
1. [ 4, 1]− −
2. [ 3,0]−
3. [ 2,1]−
4. [ 1,2]−

160
160. ก F ,f g ˈ ˆ กF 3
( ) ( 1) 3f x x= − +
1 2
( ) 1, 0g x x x−
= − ≥ F 1
( ) 0g f a−
=o F 2
a F F
[A-net ʾ 2550]
1. [10,40]
2. [40,70]
3. [70,100]
4. [100,130]

161
161. ก F ( ) 3 5f x x= + 2
( ) 3 3 1h x x x= + − F g ˈ ˆ กF
F f g h=o F (5)g F F
[A-net ʾ 2550]

162
162. ก F f g ˈ ˆ กF 2
( ) 1f x x= + ( )g x ax=
(0,1)a ∈ F k ˈ F ( )( ) ( )( )f g k g f k=o o F
1
2
1
( )( )f g
k
−
o F F ก F F
[A-net ʾ 2551]
1. 1
2. 2
3. 3
4. 4

163
163. ก F f g ˈ ˆ กF 3
1 ; 0
( )
1 ; 0
x x
f x
x x
− <
= 
− ≥
2
( ) 4 13g x x x= + + F a ˈ ก ( ) 25g a =
1 1
( 2 ) (13 )f a f a− −
− + F F ก F F
[A-net ʾ 2551]
1. 0
2. 2
3. 4
4. 6

164
164. ก F {( , ) | ( 2)( 1) 1}r x y x y= − − =
2 2
{( , ) | ( 1) }s x y xy y= = + F F F ˈ r sR R∩
[A-net ʾ 2551]
1. ( , 1)−∞ −
2.
1
( 2, )
2
− −
3.
1
( , 2)
2
4. (1, )∞

165
165. ก F 2 2
{( , ) | 1}A x y x y= + >
2 2
{( , ) | 4 9 1}B x y x y= + <
2 2
{( , ) | 1}C x y y x= − >
F F
[A-net ʾ 2551]
1. A B A− =
2. B C B− =
3. ( )B A C∩ ∪ = ∅
4. ( )A B C∩ ∪ = ∅

166
166. ก F ( ) 3 1f x x= −
2
1
2
, 0
( )
, 0
x x
g x
x x
−
 ≥
= 
− <
F 1
( (2) ( 8))f g g−
+ − F ก F F
[PAT1 ʾ 2552]
1.
1 2
3
−
2.
1 2
3
+
3.
1 2
3
−
−
4.
1 2
3
+
−

167
167. ก F [ 2, 1] [1,2]A = − − ∪ {( , ) | 1}r x y A A x y= ∈ × − = −
F , 0a b > ,r ra D b R∈ ∈ F a b+ F ก F F
[PAT1 ʾ 2552]
1. 2.5
2. 3
3. 3.5
4. 4

168
168. ก F 2
( ) 1f x x= − ( , 1] [0,1]x ∈ −∞ − ∪
( ) 2x
g x = ( ,0]x ∈ −∞
F F ก
[PAT1 ʾ 2552]
1. g fR D⊂
2. f gR D⊂
3. f ˈ ˆ กF 1 1−
4. g F ˈ ˆ กF 1 1−

169
169. ก F {1,2,3,4}A = { , , }B a b c=
{ | :S f f A B= → ˈ ˆ กF } ก F ก F F
[PAT1 ʾ 2552]
1. 12
2. 24
3. 36
4. 39

170
170. ก F ( ) 5f x x= − 2
( )g x x= F a ˈ
( ) ( )g f a f g a=o o F ( )( )fg a F F ก F F
[PAT1 ก ก ʾ 2552]
1. 25−
2. 18−
3. 18
4. 25

171
171. ก F 2
( ) 1f x x x= + + ,a b ˈ F 0b ≠
F ( ) ( )f a b f a b+ = − F 2
a F F F
[PAT1 ก ก ʾ 2552]
1. (0,0.5)
2. (0.5,1)
3. (1,1.5)
4. (1.5,2)

172
172. ก F {( , ) | [ 1,1]r x y x= ∈ − 2
}y x=
F F
ก. 1
{( , ) | [0,1]r x y x−
= ∈ }y x= ±
. ก r ก 1
r−
ก 2
F F ก
[PAT1 ก ก ʾ 2552]
1. ก. ก . ก
2. ก. ก .
3. ก. . ก
4. ก. .

173
173. ก F n ˈ
F :{1,2,..., } {1,2,..., }f n n→ ˈ ˆ กF 1 1− F ก
(1) (2) ... ( ) (1) (2)... ( )f f f n f f f n+ + + =
F F ก ˈ F (1) ( )f f n− F ก F F
[PAT1 ก ก ʾ 2552]
1. 2
2. 5
3. 8
4. 11

174
174. ก F [ 2,2]S = − 2 2
{( , ) | 2 2}r x y S S x y= ∈ × + =
F F F F ˈ r rD R−
[PAT1 ʾ 2552]
1. ( 1.4, 1.3)− −
2. ( 1.3, 1.2)− −
3. (1.2,1.4)
4. (1.4,1.5)

175
175. F
1
( )f x
x
= ( ) 2 ( )g x f x= F 1
(3) (3)g f f g−
+o o F F
[PAT1 ʾ 2552]

176
176. F 3
( )f x x= ( )
1
x
g x
x
=
+
F 1 1
( )(2)f g− −
+ F F
[PAT1 ʾ 2552]

177
177. ก F 1
1
( )
1
x
y f x
x
+
= =
−
x ˈ F F ก 1
2 1 3 2( ), ( ),...y f y y f y= =
1( )n ny f y −= 2,3,4,...n =
2553 2010y y+ ก F F
[PAT1 ʾ 2553]
1.
1
1
x
x
−
+
2.
2
1
1
x
x
+
−
3.
2
1
2
x
x
+
4.
2
1 2
1
x x
x
+ −
−

178
178. F f g ˈ ˆ กF ก
2
1
( )
4
x
f x
x
−
=
−
( ) ( ) 1g x f x x= − −
F F
ก. (2, )gD = ∞
. F 0x > F ( ) 0g x = 1 F F
F F ก F
[PAT1 ʾ 2553]
1. ก. ก . ก
2. ก. ก F .
3. ก. F . ก
4. ก. .

179
179. F A ˈ ก F F ก F F ก 10
B ˈ ก F F ก F F ก 10
C ˈ ˆ กF :f A B→ ˈ ˆ กF F
. . . a ( )f a F F ก 1 ก F a A∈
ก C F ก F
[PAT1 ʾ 2553]

180
180. ก R ˈ
F 2
( ) 1f x x= − ( ) 2 1g x x= + ก x
F ( )(1)f g⊗ F ก F
[PAT1 ʾ 2553]
F :f R R→ :g R R→ ˈ ˆ กF
ก ก ก ⊗ f g
( )( ) ( ( )) ( ( ))f g x f g x g f x⊗ = −
ก x

181
181. F f g ˈ ˆ กF F ˈ
3
( )
6
x
f x
x
+
=
+
1 6
( )( )
1
x
f g x
x
− −
=
−
o
F ( ) 2g a = F a F F F
[PAT1 ก ก ʾ 2553]
1. [ 1,1)−
2. [1,3)
3. [3,5)
4. [5,7)

182
182. F R
F 1 2 3 4, , , ,f f f f g h ˈ ˆ กF ก R R
1 ( ) 1f x x= + 2 ( ) 1f x x= −
2
3 ( ) 4f x x= + 2
4 ( ) 4f x x= −
1 2( )( ) ( )( ) 2f g x f h x+ =o o
3 4( )( ) ( )( ) 4f g x f h x x+ =o o
F ( )(1)g ho F ก F
[PAT1 ก ก ʾ 2553]

183
183. F R F F F ˈ ˆ กF
[PAT1 ʾ 2553]
1. F 2
1 {( , ) | 4r x y R R x y= ∈ × = − 0}xy ≥
2. F 2 2
2 {( , ) | 4r x y R R x y= ∈ × + = 0}xy >
3. F 3 {( , ) | 1}r x y R R x y= ∈ × − =
4. F 4 {( , ) | 1}r x y R R x y= ∈ × − =

184
184. F I F :f I I→ ˈ ˆ กF
( 1) ( ) 3 2f n f n n+ = + + n I∈
F ( 100) 15,000f − = F (0)f F ก F
[PAT1 ʾ 2553]

185
185. F R
F {( , ) | 3 5}f x y R R y x= ∈ × = −
{( , ) | 2 1}g x y R R y x= ∈ × = +
F a R∈ 1 1
( )( ) 4g f a−
=o
F ( )(2 )f g ao F ก F
[PAT1 ʾ 2553]

186
186. F R F :f R R→ ˈ ˆ กF
F ก
1
1
x
f x
x
− 
= 
+ 
ก 1x ≠ − F F ก F
[PAT1 ʾ 2554]
1. ( )( )f f x x= − ก x
2.
1
( )
1
x
f x f
x
+ 
− =  
− 
ก 1x ≠
3.
1
( )f f x
x
 
= 
 
ก 0x ≠
4. ( )2 2 ( )f x f x− − = − − ก x

187
187. ก F I
F
4 3 2
5 2
2 75
( )
270
x x a x
f x
x b x
− + −
=
+ −
,a b I∈
F {( , ) | (3) 0}A x y I I f= ∈ × =
2 2
{( , ) | 2 3}B x y I I a ab b= ∈ × − + <
F ก A B∩ F ก F
[PAT1 ʾ 2554]

188
188. ก F R
F :f R R→ ˈ ˆ กF 2
( ) (1 ) 2xf x f x x x+ − = − x R∈
F F
54
25
( ( ))
x
x f x
=
+∑ F ก F
[PAT1 ʾ 2554]

189
189. ก F I
F :f I I→ ˈ ˆ กF
(1) (1) 1f =
(2) (2 ) 4 ( ) 6f x f x= +
(3) ( 2) ( ) 12 12f x f x x+ = + +
F F (7) (16)f f+ F ก F
[PAT1 ʾ 2554]

190
190. ก F
1
{( , ) | }
5 3
r x y R R y
x
= ∈ × =
− −
R r
[PAT1 ʾ 2554]
1. { | 2 8}x R x∈ − < <
2. { | 6 3}x R x∈ − < <
3. { | 0 3}x R x∈ < <
4. { | 8}x R x∈ <

191
191. F R F :f R R→ ˈ ˆ กF F
ก
0 , 1
( ) 1
, 1
1
x
f x x
x
x
= −

= −
≠ − +
F { | ( )( ) cot 75 }A x R f f x= ∈ = o
o
F F F ˈ F
[PAT1 ʾ 2554]
1. ( 3, 2)A ∩ − −
2. ( 4, 3)A ∩ − −
3. (2,3)A ∩
4. (3, 4)A ∩

192
192. ก F ( ) 1 3f x x= − S ˈ x F
ก ก ( )( )f f x x=o ก ก S
[PAT1 ʾ 2554]

193
193. ก F :f N N→ F ก ก
( ) ( ) ( ) 4f x y f x f y xy+ = + + (1) 4f = F (20)f
[PAT1 ʾ 2554]

194
194. ก R F
{( , ) | 1 0}r x y R R x y y x= ∈ × + − − = F F
ก. r ˈ F { | 1}rD x R x= ∈ ≠ −
. F 1
r−
ˈ ˆ กF
F F ก F
[PAT1 ʾ 2555]
1. ก. ก . ก
2. ก. ก F .
3. ก. F . ก
4. ก. .

195
195. ก F R ก 2
( ) 3g x x x= + + ก
x F :f R R→ ˈ ˆ กF F ก
2
2
( )( ) 2( )(1 ) 6 10 17
2( )( ) ( )(1 ) 6 2 13
f g x f g x x x
f g x f g x x x
+ − = − +
+ − = − +
o o
o o
F (383)f F ก F
[PAT1 ʾ 2555]

196
196. ก F R F I
F f g ˈ ˆ กF ก R R
3 2
( 5) 2f x x x x+ = − + ก x
1
(2 1) 4g x x−
− = + ก x
F F
(ก) ( )(0) 169f g− < −
( ) { | ( )( ) 5 0}x I g f x∈ + =o ˈ F
F F ก F
[PAT1 ʾ 2555]
1. (ก) ก ( ) ก
2. (ก) ก F ( )
3. (ก) F ( ) ก
4. (ก) ( )

197
197. ก F
2
2
2 8
{( , ) | }
1
x
r x y I I y
x
−
= ∈ × =
+
I
ก r rD R− F ก F F
[PAT1 ʾ 2555]
1. 2
2. 4
3. 5
4. 7

198
198. ก F {1, 2,3,..., }A k= k ˈ ก
F {( , ) | 0 7}B a b A A b a= ∈ × < − ≤
F k F ก F F ก B F ก 714
[PAT1 ʾ 2555]

199
199. F R ก F
{( , ) | 12 1 3}r x y R R x y= ∈ × − + + = F F
(ก) ( 1,8)r rD R∩ ⊂ −
( ) { | 8 12}r rD R x R x− = ∈ < ≤
F F ก
[PAT1 ʾ 2556]
1. (ก) ก ( ) ก
2. (ก) ก F ( )
3. (ก) F ( ) ก
4. (ก) ( )

200
200. F A B ˈ ก A B F ก 4 5
ก A B∪ F ก 7 F F
(ก) F A B∩ 4 F
( ) F ก A B− B A− 64 F
F F ก F
[PAT1 ʾ 2556]
1. (ก) ก ( ) ก
2. (ก) ก F ( )
3. (ก) F ( ) ก
4. (ก) ( )

201
201. F R F F
(ก) F 2 2
{( , ) | 4, 0}x y R R x y xy∈ × + = > ˈ ˆ กF
( ) F 2
2, 0
( )
, 0
x x
f x
x x
− ≤
= 
>
2
(3 1) 2 3g x x x− = + x R∈
F F 1
( )(25) 14g f −
=o
F F ก F
[PAT1 ʾ 2556]
1. (ก) ก ( ) ก
2. (ก) ก F ( )
3. (ก) F ( ) ก
4. (ก) ( )

202
202. ก F
1 1
,
2
( )
1 1 1
,
2 2
x
x
f x
x
x

<
= 
 + ≥

F
1
( ( ( )))
3
f f f −
ก F F
[PAT1 ʾ 2556]
1. 6−
2. 6
3. 3−
4. 3

203
203. , {0,1, 2,3,...}x y ∈ ก F ( , )F x y ˈ ก
(1, 1) , 0, 0
( , ) 1 , 0
( ( 1, ), 1), 0, 0
F y x y
F x y x y
F F x y y x y
− = ≠

+ =
 − − ≠ ≠
F (1, 2) (3,1)F F+ F ก F
[PAT1 ʾ 2556]

204
204. ก F R F :f R R→ ˈ ˆ กF F ก
( )( ) 4 (4 ( ))f f x x f x= + −o ก x F F (4)f F ก
F
[PAT1 ʾ 2556]

205
205. F R F f ˈ ˆ กF F ˈ
2
2 4 4
( )
1
x x
f x
x
+ +
=
+
1x ≠ F F ˆ กF f ˈ
F F
[PAT1 ʾ 2557]
1. 2
{ | 6 7 0}x R x x∈ + − ≥
2. 2
{ | 3 10 0}x R x x∈ + − ≥
3. 2
{ | 12 0}x R x x∈ + − ≥
4. 2
{ | 6 16 0}x R x x∈ − − ≥

206
206. F I F {( , ) | 21 4 }A x y I I xy y x= ∈ × − = −
F ก A F ก F F
[PAT1 ʾ 2557]
1. 5
2. 4
3. 3
4. 2

207
207. ก F 3 2
( ) 3f x x ax bx= + + + 2
( ) 3g x bx x a= + + a
b ˈ F (3) 0f = 2x − ( )f x F ก 5 F F
( )(1)g fo F ก F
[PAT1 ʾ 2557]

208
208. F R F :f R R→ :g R R→ ˈ ˆ กF
F ( )( ) 4 5f g x x= −o 1
( ) 2 1g x x−
= + ก x
F F
(ก) 1
4( )(2 1) ( ) 1f g x g x−
+ = +o
( ) 1 1 1
( ( ))( ) ( ) 1g f g x f x− − −
= +o o
F F ก F
[PAT1 ʾ 2557]
1. (ก) ก ( ) ก
2. (ก) ก F ( )
3. (ก) F ( ) ก
4. (ก) ( )

209
209. ก F R F :f R R→ :g R R→ ˈ
ˆ กF F ก ( ( )) 2 15f x g y x y+ = + + ก x y
F F
(ก) ( )( ) 2 15g f x x= +o ก x y
( ) (25 (57)) 75g f+ =
F F ก F
[PAT1 ʾ 2557]
1. (ก) ก ( ) ก
2. (ก) ก F ( )
3. (ก) F ( ) ก
4. (ก) ( )

210
210. F R a ˈ 0a ≠ F :f R R→
:g R R→ ˈ ˆ กF ( ) 2f x ax= +
3
( ) 3 ( 1)g x x x x= − − ก x F 1 1
( )(1) 1f g− −
=o F
( )( )g f ao F ก F
[PAT1 ʾ 2557]

211
211. F R F :f R R→ ˈ ˆ กF F
:g R R→ ˈ ˆ กF ( ) 2 ( ) 5g x f x= + ก x
F a ˈ 1 1
( )(1 ) ( )(1 )f g a g f a− −
+ = +o o F F 2
a F ก
F
[PAT1 ก ʾ 2557]

212
212. F R F S′ F S
F
2 2
{( , ) | 1 4}f x y R R y x y= ∈ × + − =
4
{( , ) | 1 }g x y R R y x= ∈ × = − F A ˈ F f B ˈ
g F F
(ก) A B′⊂
( ) ( ) ( )A B B A− ∩ − = ∅
F F ก F
[PAT1 ʾ 2558]
1. (ก) ก ( ) ก
2. (ก) ก F ( )
3. (ก) F ( ) ก
4. (ก) ( )

213
213. ก F R F ,f g h ˈ ˆ กF ก
R R 1
( ) 2 5,( )( ) 4f x x f g x x−
= − =o ( )( )g h xo F
1x − F F ก 21− F c ˈ ก F F ก
3 2
( ) 3 2h x c x x− = − − F F
(ก) ( )( ) 23f h c =o
( ) ( )( ) 35h g c+ =
F F ก F
[PAT1 ʾ 2558]
1. (ก) ก ( ) ก
2. (ก) ก F ( )
3. (ก) F ( ) ก
4. (ก) ( )

214
214. F R F F F F ˈ ˆ กF
[PAT1 ʾ 2558]
1. F 1 {( , ) | 1 0}r x y R R xy= ∈ × + =
2. F 2 {( , ) | tan }r x y R R y x= ∈ × =
3. F 2 2
3 {( , ) | 1}r x y R R x y= ∈ × = +
4. F 4 {( , ) | 2 }r x y R R y x= ∈ × = −
5. F
2
5 {( , ) | }
1
y
r x y R R x
y
= ∈ × =
+

215
215. F f g ˈ ˆ กF
9 , 0
( )
7 , 4
x x
f x
x x
 − ≤
= 
− >
2 , 1
( )
4 , 1
x x
g x
x x
+ <
= 
− ≥
F F
(ก) F 0x ≤ F ( )( ) 9 4g f x x= − −o
( ) F 4 6x< ≤ F ( )( ) 3g f x x= −o
( ) F 6x > F ( )( ) 9g f x x= −o
F F ก F
[PAT1 ʾ 2558]
1. F (ก) F ( ) ก F F ( )
2. F (ก) F ( ) ก F F ( )
3. F ( ) F ( ) ก F F (ก)
4. F (ก) F ( ) F ( ) ก F
5. F (ก) F ( ) F ( ) F

216
216. ก F I R
F
2
2
{( , ) | }
4 2 1
x
r x y R R y
x x
+
= ∈ × =
− − +
2
{ | }rA x x I D= ∈ ∩ F ก ก A F ก F F
[PAT1 ʾ 2558]
1. 6
2. 10
3. 19
4. 29
5. 30

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