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Function problem p

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ข้อสอบเข้ามหาวิทยาลัยระดับชั้นมัธยมปลาย เรื่องความสัมพันธ์และฟังก์ชัน
Onet,คณิต กข.,คณิต1,Anet,Pat1

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Function problem p

  1. 1. 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. 2 2. ˆ กF ( )y f x= F ก F [O-net ʾก ก 2548] 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. 3 3. F {(1,0),(2,1),(3,5),(4,3),(5,2)}f = F (2) (3)f f+ F F ก F [O-net ʾก ก 2548]
  4. 4. 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 [O-net ʾก ก 2548]
  5. 5. 5 5. F {1,2,3,4}A = {( , ) | }r m n A A m n= ∈ × ≤ F ก F r F ก F F [O-net ʾก ก 2549] 1. 8 2. 10 3. 12 4. 16
  6. 6. 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 [O-net ʾก ก 2549] 1. {3,4,10} 2. {2,3,15} 3. {0,3,10} 4. {4,5,9}
  7. 7. 7 7. ก ˆ กF F F ก X กก F 1 [O-net ʾก ก 2549] 1. 2 1y x= + 2. 2y x= − 3. 1y x= − 4. 1 2 x y   =    
  8. 8. 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 [O-net ʾก ก 2550] 1. 1 2. 2 3. 3 4. 4
  9. 9. 9 9. ก x F F ก ก 2 4 5 6y x x= − − + F ก X [O-net ʾก ก 2550] 1. 2 1 ( , ) 3 3 − − 2. 5 3 ( , ) 2 2 − − 3. 1 6 ( , ) 4 7 4. 1 3 ( , ) 2 2
  10. 10. 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 [O-net ʾก ก 2550] 1. -4 2. 0 3. 6 4. 14
  11. 11. 11 11. ก F 2 ( ) 2 15f x x x= − − F F [O-net ʾก ก 2550] 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. 12 12. ก F {1,2}A = { , }B a b= F F F ˈ) ก F A B× [O-net ʾก ก 2551] 1. (2, )b 2. ( , )b a 3. ( ,1)a 4. (1,2)
  13. 13. 13 13. F {1,99}A = F A F F ˈ ˆ กF [O-net ʾก ก 2551] 1. F ก 2. F F ก 3. 4. F
  14. 14. 14 14. ก F r F ก F F ก F [O-net ʾก ก 2551] 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. 15 15. F F ˈ ก ˆ กF 2 2 2 1 3 2 1 x x y x x x − = + + + − [O-net ʾก ก 2551] 1. 2− 2. 1− 3. 0 4. 1
  16. 16. 16 16. F a Fก ˆ กF (2 )x y a= F (3,16) F F [O-net ʾก ก 2551] 1. 2 2. 3 3. 4 4. 5
  17. 17. 17 17. F 2 ( ) 2f x x x= − + + F F ก F [O-net ʾก ก 2552] 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. 18 18. F F ˈ ˆ กF [O-net ʾก ก 2552] 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. 19 19. F ( ) 3f x x= − ( ) 2 4g x x= − + − F f gD R∪ F [O-net ʾก ก 2552] 1. ( ],3−∞ 2. [ )2,− ∞ 3. [ ]2,3− 4. ( ),−∞ ∞
  20. 20. 20 20. ก Fก ˆ กF f ˈ F 11 ( 11) 3 ( 3) (3)f f f− − − F [O-net ʾก ก 2552] 0-5-10 5 -5 X Y
  21. 21. 21 21. F F ˈ ˆ กF [O-net ʾก ก 2553] 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. 22 22. F F ˈ F ก ˈ [O-net ʾก ก 2553] 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. 23 23. F 2 ( ) 3 4f x x= − − F F F ก F [O-net ʾก ก 2553] 1. [ ]2,2fD = − [ ]0,3fR = 2. [ ]2,2fD = − [ ]1,3fR = 3. [ ]0,2fD = [ ]0,3fR = 4. [ ]0,2fD = [ ]1,3fR =
  24. 24. 24 24. F ( 2) 2 1f x x− = − F 2 ( )f x F F ก F F [O-net ʾก ก 2553] 1. 2 2 1x − 2. 2 2 1x + 3. 2 2 3x + 4. 2 2 9x +
  25. 25. 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 [O-net ʾก ก 2553] 1. ก. ก . ก 2. ก. ก . 3. ก. . ก 4. ก. .
  26. 26. 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. 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 [Entrance ก . ʾ 2520] 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. 28 28. F {1,2,3}, {2,3,4}A B= = ˆ กF 1 1− ก A B [Entrance ก . ʾ 2520] 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. 29 29. ก F {(1, 2),(0,0)}r = − F ( )P A F F r F F ( )P A [Entrance ก . ʾ 2521] ก. { ,{ 2},{ 2,0},{0, 2}}∅ − − − . { ,{1},{1,0},{0,1}}∅ . {{ 2},{ },{ 2,0}, }− ∅ − ∅ . { ,{1, 2},{0,0},{(1, 2),(0,0)}}∅ − −
  30. 30. 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 [Entrance ก . ʾ 2521] ก. . . . o o o4− 2− 3 4− 2− 3 o o • o3 2− 3 o o
  31. 31. 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 [Entrance ก . ʾ 2521] ก. 3 . 5 . 6 . 10 . F F ก
  32. 32. 32 32. ก F 1 1 ( 1) 1 2 2 f x x+ = − F F 1 (2)f − F F ก [Entrance ก . ʾ 2521] ก. 6 . 4 . 2 . 0 . F F ก
  33. 33. 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 [Entrance ก . ʾ 2521] 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. 34 34. F ,x y ˈ ก ก x y a+ = 0a > ก F F [Entrance ก . ʾ 2522] 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. 35 35. F 2 ( ) 7f x x= + x ˈ ( ) sing x x= 0 2x π≤ ≤ F F ก [Entrance ก . ʾ 2522] 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. 36 36. f ˈ ˆ กF 1 1− ก A B F F ก [Entrance ก . ʾ 2522] 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. 37 37. F A B= = {( , ) | 2}f x y A B y x= ∈ × = + F ก F [Entrance ก . ʾ 2523] 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. 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 [Entrance ก . ʾ 2523] 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. 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 [Entrance ก . ʾ 2523] 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. 40 40. ก F 2 2 {( , ) | 2 1x y x by x− + = , ,x y b ˈ } F F F ก F [Entrance ก . ʾ 2523] 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. 41 41. ก F 2 2 {( , ) | 0} 4 9 x y r x y= − = F F F ก F [Entrance ก . ʾ 2524] 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. 42 42. ก 2 log 100 ( 1) {( , ) | 5 }x f x y y − − = = F F ก F [Entrance ก . ʾ 2524] 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. 43 43. ก 2 2 1 {( , ) | 1}, {( , ) | } 1 f x y y x g x y y x = = − = = − F F ก F [Entrance ก . ʾ 2524] 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. 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 [Entrance ก . ʾ 2524] 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. 45 45. ก F {( , ) | log 0}x y R R xy∈ × < ก ( ) ก F F F (ก F F ˈ F ) [Entrance ก . ʾ 2525] ก. . . . Y X Y X Y X Y X
  46. 46. 46 46. F F ก [Entrance ก . ʾ 2525] 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. 47 47. ก 2 1 ( ) 6, ( ) 3 f x x g x x = + = − F F ก [Entrance ก . ʾ 2525] 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. 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 [Entrance ก . ʾ 2526] ก. 2 . 6 . 1 . 1 2
  49. 49. 49 49. F ˆ กF f g ˈ R , , 0c R c∈ < ก ( )f x x c= − ( )g x c= − ก x F F [Entrance ก . ʾ 2526] 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. 50 50. ก F {1,2}A = F F ก F [Entrance ก . ʾ 2526] 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. 51 51. F F ˈ F [Entrance ก . ʾ 2527] 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. 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 [Entrance ก . ʾ 2527] 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. 53 53. F F F F ˈ ˆ กF [Entrance ก . ʾ 2527] 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. 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 [Entrance ก . ʾ 2527] ก. { | 5}x x ≥ . { | 5 0x x− ≤ ≤ 5}x ≥ . { | 5x x ≤ − 5}x ≥ . { | 0}x x ≥
  55. 55. 55 55. F 1 ( )( ) , ( ) 3 3 f g x x g x x= = −o ( ( )) 2 1g h x x= − F [Entrance ก . ʾ 2527] 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. 56 56. ก F 2 {( , ) | 6 10}r x y R R x y y= ∈ × = − + F F ˈ [Entrance ก . ʾ 2528] 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. 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 [Entrance ก . ʾ 2528] 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. 58 58. F F ˈ [Entrance ก . ʾ 2528] 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. 59 59. ก F 2 ( ) 1 f x x = − F F ˈ [Entrance ก . ʾ 2528] 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. 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⊂ ) [Entrance ก . ʾ 2528] ก. f ho F . h go F . g fo F . h fo F
  61. 61. 61 61. ก F {( , ) | }r x y R R y x x= ∈ × = F r [Entrance ก . ʾ 2529] 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. 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 [Entrance ก . ʾ 2529] ก. [0,1] {1}∪ . (0,1] { 1}∪ − . (0,1] . { 1}−
  63. 63. 63 63. F F [Entrance ก . ʾ 2529] 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. 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 [Entrance ก . ʾ 2529] 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. 65 65. F ( ) 1 x f x x = + F 1 ( )f x− F [Entrance ก . ʾ 2529] ก. 1 x x− . 1 x x− . 1 x x+ . 1 x x+
  66. 66. 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 [Entrance ก . ʾ 2530] ก. 1 1r rD R⊂ . 2 2r rD R⊂ . 1 2r rD D⊂ . 1 2r rR R⊂
  67. 67. 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 [Entrance ก . ʾ 2530] ก. 17 6 − . 7 2 − . 1 6 − . 7 2
  68. 68. 68 68. ˆ กF F F ˈ ˆ กF F [Entrance ก . ʾ 2530] ก. 4logy x= . 2 , 1x y a a= > . sin 7y x= − . 3 5 2y x= − +
  69. 69. 69 69. 2 2 ( ) ( ) 4y y x x+ − + = ก ˈ F [Entrance ก . ʾ 2531] 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. 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 [Entrance ก . ʾ 2531] 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. 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 [Entrance ก . ʾ 2531] 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. 72 72. F 2 ( )f x x= ,A R⊆ R= 1 ( ) { | ( ) }f A x f x A− = ∈ F F [Entrance ก . ʾ 2531] 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. 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 [Entrance ก . ʾ 2531] ก. 24 . 29 . 72 . 120
  74. 74. 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 [Entrance ก . ʾ 2532] ก. ( 2,0) [2,6]− ∪ . ( 2,0) (2,6)− ∪ . [2,6] . ( 2,0)−
  75. 75. 75 75. ก F {2,5,6,7,8}D = F F D ˈ F F F ˈ ˆ กF [Entrance ก . ʾ 2532] 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. 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 [Entrance ก . ʾ 2533] ก. ( )f g h∩ ∩ . ( )f g h∩ ∪ . ( )f h g∩ ∪ . ( )f g h∪ ∪
  77. 77. 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 [Entrance ก . ʾ 2533] ก. 1 . 2 . 3 . 4
  78. 78. 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 [Entrance ก . ʾ 2533] ก. ( )(1) 2g f =o . ( )(1) 2gf = . ( )(1) 2 g f = . ( )(1) 2g f− =
  79. 79. 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 [Entrance ก . ʾ 2533]
  80. 80. 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 [Entrance ก . ʾ 2534] ก. ( 4,1)− . ( 1,5)− . (2,7) . (4,8)
  81. 81. 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 [Entrance ก . ʾ 2534] ก. 1 x+ . 2 x+ . 1 1 x+ . 1 2 x+
  82. 82. 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 [Entrance ก . ʾ 2535] 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. 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 [Entrance ก . ʾ 2535] ก. [ 1,3]− . [ 5,1]− . [ 1,1]− . [ 5,3]−
  84. 84. 84 84. F 1 ( ) 2 x f x x − = − ( )( 2) 3 6f g x x+ = +o F (2)g F ก F [Entrance ก . ʾ 2535] ก. 5 6 . 3 2 . 12 5 . 24 11
  85. 85. 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 [Entrance ก . ʾ 2535]
  86. 86. 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 [Entrance ก . ʾ 2536] ก. 160 . 80 . 64 . 16
  87. 87. 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 [Entrance ก . ʾ 2536] ก. ( 4, 2]− − . ( 2,2]− . (2,4] . (4,6)
  88. 88. 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 [Entrance ก . ʾ 2536] ก. -35 . -33 . 37 . 39
  89. 89. 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 [Entrance ก . ʾ 2536] ก. F F F . F F F . F F F . F F x ˈ F x ˈ
  90. 90. 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 [Entrance ก . ʾ 2536] ก. ก . ก ก . ก ก ก . ก ก
  91. 91. 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 [Entrance ก . ʾ 2537] 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. 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 [Entrance ก . ʾ 2537] ก. [1,2] . ( ,0]−∞ . 1 ( ,1] [ ,1] 2 −∞ ∪ . 1 ( , ] [1,2] 2 −∞ ∪
  93. 93. 93 93. F ( ) 1f x x= − 1 2 ( )( ) 4 1g f x x− = −o F ก ( ) 0g x = ˈ F F [Entrance ก . ʾ 2537] ก. [ 4, 1]− − . [ 1,0]− . [0,4] . [4,6]
  94. 94. 94 94. F 2 {( , ) |r x y y x= ≤ 2 }y x≥ F F 1 r− F F [Entrance ก . ʾ 2538] ก. [0,2] . [0,4] . ( ,0] [2, )−∞ ∪ ∞ . ( ,0] [4, )−∞ ∪ ∞
  95. 95. 95 95. F ( ) (3 )(2 )f x x x= + − 1 ( ) 3 g x x = + F f g⋅ F F [Entrance ก . ʾ 2538] ก. ∅ . ( ,2]−∞ . ( 3,2)− . ( 3,2]−
  96. 96. 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 [Entrance ก . ʾ 2538] ก. -1 . -2 . 1 . 3
  97. 97. 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 [Entrance ก . ʾ 2538] ก. 1 2r r⊂ . 2 1r r⊂ . 1 1 2r r − ⊂ . 1 2 1r r− ⊂ 3}y ≥
  98. 98. 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 [Entrance ก . ʾ 2538]
  99. 99. 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 [Entrance ก . ʾ 2540] ก. 30 . 60 . 10 . 20
  100. 100. 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 [Entrance ก . ʾ 2540] ก. 40 . 48 . 56 . 72
  101. 101. 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 [Entrance ก . ʾ 2540] ก. 2− . 1− . 1 . 2
  102. 102. 102 102. F I + ˈ ก ก F {( , ) | 2 12f x y x y= + = , }x y I + ∈ F f fo F ก F F [Entrance ก . ʾ 2540] ก. {(8,5),(4,4)} . {(5,8),(4,4)} . {(2,2),(4,4)} . {(6,3),(4,4)}
  103. 103. 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 [Entrance ก . ʾ 2540]
  104. 104. 104 104. F F 2 4 {( , ) | 2 } ( 1) 4 r x y R R y x = ∈ × = − − − F F F r [Entrance ก . ʾ 2541] ก. ( ,2) [3, )−∞ ∪ ∞ . ( ,2) (3, )−∞ ∪ ∞ . ( ,2] [3, )−∞ ∪ ∞ . ( ,2] (3, )−∞ ∪ ∞
  105. 105. 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 [Entrance ก . ʾ 2541] ก. 10log a . 101 log a+ . 1 logb a+ . 1 loga b+
  106. 106. 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 [Entrance ก . ʾ 2541] 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. 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 [Entrance ก . ʾ 2541] ก. 1 . 2 ( 1) ( 2) x x + + . 2 ( 1) ( 2) x x x + + + . 2 ( 1) ( 2) x x x + − +
  108. 108. 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 [Entrance ก . ʾ 2541] ก. ( 3,5)− . ( 6, 1)− − . ( 5,4)− . (1,6)
  109. 109. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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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