ความสัมพันธ์และฟังก์ชัน

44 ⌫ ⌫  ⌦  ⌫         

⌦ 45 ⌦ 4. „¦³ªœ„µ¦‹´—„µ¦Á¦¸¥œ¦o¼ 1. ‡¦Â¼ ¨³œ„´ Á¦¸¥œššªœ‡ªµ¤¦¼oÁ¦º°É ŠÃ—Á¤œÂ¨³Á¦œ‹r…°Š‡ªµ¤­¤´ ¡œ´ ›r 2.‡¦­¼ ¦ž» ªµn ‡ªµ¤­´¤¡´œ›šr Áɸ žœ} ¢Š{ „rœ´ ×Á¤œ…°Š‡ªµ¤­¤´ ¡´œ›r‹³Á¦¥¸ „ªnµÃ—Á¤œ…°Š¢{Š„r œ´ ¨³Á¦œ‹r…°Š‡ªµ¤­´¤¡´œ›r‹³Á¦¥¸ „ªnµÁ¦œ‹…r °Š¢{Š„r´œ 3. ‡¦„¼ 宜—Ëš¥r˜°n ŞœÄ¸Ê ®œo „´ Á¦¥¸ œ®µÃ—Á¤œÂ¨³Á¦œ‹r…°Š¢Š{ „rœ´ A = {1 , 2 , 3} ‡µÎ ˜° ×Á¤œ‡º° A Á¦œ‹‡r °º {a,b} B = {a , b , c , d} A = {1 , 2 , 3 , 4 , 5} ‡Îµ˜° ×Á¤œ‡°º A Á¦œ‹r‡°º B B = {a , b , c , d} A = {1 , 2 , 3 , 4} ‡Îµ˜° ×Á¤œ‡º° A Á¦œ‹‡r º° {a,b,c,d} B = {a , b , c , d , e} A = {1 , 2 , 3 , 4} ‡Îµ˜° ×Á¤œ‡º° A Á¦œ‹‡r º° B B = {a , b , c , d } 4. ‹µ„Ëš¥˜r ª´ °¥nµŠÄ®oœ„´ Á¦¥¸ œ¡‹· µ¦–µ„µ¦‹´ ‡¼¦n ³®ªµn Š­¤µ·„…°ŠÁŽ˜ A ¨³­¤µ„· …°ŠÁŽ˜ B ¨³°„×Á¤œÂ¨³Á¦œ‹…r °Š¢Š{ „rœ´ 6. ‡¦¼Â¨³œ´„Á¦¸¥œnª¥„œ´ ­¦ž» Ä®oŗªo nµÃ—Á¤œ…°Š¢{Š„rœ´ ‡°º ÁŽ˜ A ¨³Á¦œ‹r…°Š¢Š{ „rœ´ Ážœ} ­´ÁŽ˜…°ŠÁŽ˜ B ŽŠÉ¹ Á¦¸¥„ªµn ¢Š{ „r œ´ ‹µ„ A Ş B ¨³Ä­o ´¨´„¬–ršœ—ªo ¥ f : AoB 7. Ä®œo „´ Á¦¸¥œ«„¹ ¬µÁ¡·É¤Á˜¤· ‹µ„ĝ‡ªµ¤¦¼šÉ¸ 7 8. Ä®oœ„´ Á¦¥¸ œ f„š´„¬³Ã—¥šµÎ  „f ®—´ ‹µ„ĝ„‹· „¦¦¤šÉ¸ 7 5. ®¨nŠ„µ¦Á¦¥¸ œ¦¼o 1. ĝ‡ªµ¤¦¼ošÉ¸ 7 2. ĝ„·‹„¦¦¤š¸É 7 3. ®o°Š­¤—» æŠÁ¦¥¸ œ 4. Internet

46 ⌫ ⌫  ⌦  ⌫          6. „¦³ªœ„µ¦ª´—¨³ž¦³Á¤œ· Ÿ¨ 1. ž¦³Á¤œ· Ÿ¨‹µ„„µ¦šÎµÂ „f ®´— 2. ž¦³Á¤·œŸ¨‹µ„„µ¦šµÎ š—­° 7. ´œš„¹ ®¨´Š„µ¦­°œ ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. 8. „·‹„¦¦¤Á­œ°Âœ³ ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ……………………………………………………………………………………………………………….

⌦ 47 ⌦

48 ⌫ ⌫  ⌦  ⌫         

⌦ 49 ⌦ ¢Š{ „rœ´ ®œŠ¹É ˜°n ®œÉŠ¹ „µÎ ®œ—Ä®o f {(m,7), (n,8), (t,9)} ¨³ f {(m,7), (n,8), (t,7)} ‹³Á®Èœªnµ f ¨³ g ˜µn Š„ÁÈ žœ} ¢{Š„r œ´ ¥ŠÉ· Ş„ªµn œ´Êœ¢Š{ „rœ´ f ¤¸­¤µ·„˜ª´ ®¨´Š®œÉŠ¹ ˜ª´ ‹´ ‡n¼„´­¤µ·„˜´ª®œoµÁ¡¥¸ Š®œ¹ŠÉ ˜ª´ Ášnµœœ´Ê —´Š¦ž¼ Df f Rf m 7 n 8 t 9 ­ªn œ¢Š{ „r œ´ g ¤­¸ ¤µ·„˜´ª®¨´Š®œ¹ÉŠ˜ª´ ‡°º 7 ‹´ ‡¼„n ´­¤µ·„˜´ª®œµo ­°Š˜´ª‡º° m ¨³ t —Š´ ¦ž¼ Dg g Rg m n 7 t 8 9 ‹³Á®œÈ ªµn f ¤¨¸ „´ ¬–³…°Š¢Š{ „r´œ®œŠÉ¹ ˜n°®œŠÉ¹ Ĝ…–³š¸É g Ťnč¨n ´„¬–³…°Š¢{Š„r œ´ ®œŠÉ¹ ˜n°®œŠÉ¹ šœ¥· µ¤ Ä®o f Áž}œ¢Š{ „r ´œ f Ážœ} ¢{Š„r ´œ®œÉ¹Š˜n°®œŠ¹É ‹µ„ A Ş B „˜È n°Á¤Éº° f Ážœ} ¢{Š„r œ´ ‹µ„ A Ş B ­Îµ®¦´­¤µ·„ x1 ¨³ x2 Ä—Ç Äœ A ™µo f(x1 ) f(x2 ) ¨ªo x1 x2 Á…¸¥œÂšœ—ªo ¥ f ; A 1-1 B

50 ⌫ ⌫  ⌦  ⌫          ˜ª´ °¥nµŠš¸É 3 „ε®œ—Ä®o f {(1,7), (2,8), (3,9), (4,8)} ‹ŠÂ­—Šªµn f ŤnÁžœ} ¢{Š„r œ´ 1 – 1 ª›· ¸šµÎ ¡·‹µ¦–µÂŸœ£µ¡…°Š f {(1,7), (2,8), (3,9), (4,8)} f 17 28 39 4 f ŤnÁž}œ¢Š{ „r œ´ 1 – 1 Áœ°Éº Š‹µ„¤¸ (2,8)  f ¨³ (4,8)  f ŽÉŠ¹ ­¤µ„· ˜ª´ ®¨Š´ ‡º° 8 Á®¤°º œ„œ´ ˜­n ¤µ„· ˜´ª®œµo ˜µn Š„œ´ ‡º° 2 z 4 ˜ª´ °¥nµŠšÉ¸ 4 „µÎ ®œ—Ä®o f(x) x  1 ‹ŠÂ­—Šªµn f ŤnÁžœ} ¢{Š„r œ´ 1 – 1 ª›· ¸šµÎ „µ¦š‹É¸ ³Â­—Šªµn f ŤnÁž}œ¢Š{ „rœ´ 1 – 1 Á¦µ˜o°Š®µ x1 ¨³ x2 š¸É x1 ŤnÁšµn „´ x2 ˜nšÎµÄ®o f(x1 ) ¨³ f(x2 ) ¤¸‡µn Ášµn „´œ Á¨º°„ x1 ¨³ x2 šÉ¤¸ ¸‡nµ­¤¼¦–rÁšnµ„´œ Ánœ x1 2 ¨³ x2 2 ™oµ x1 2 ¨oª f(x1 ) f(2) 2  1 2  1 3 ™µo x2 2 ¨ªo f(x2 ) f(2)  2  1 2  1 3 ‹³Á®œÈ ªnµ¤‡¸ n°¼ ´œ—´ (2,3) ¨³ (2,3) °¥n¼Äœ f ˜n 2 z 2 —´Šœ´Êœ f ŤÄn ¢n {Š„rœ´ 1 – 1 ˜´ª°¥µn ŠšÉ¸ 5 ‹Š¡­· ‹¼ œªr nµ f Ážœ} ¢{Š„r œ´ 1 – 1 Á¤Éº° f(x) 3x  4 ª›· ¸šÎµ Ä®o f(x1) f(x2 ) ‹³Å—ªo µn 3x1  4 3x2  4 œµÎ - 4 ª„šŠ´Ê ­°Š…µo Š ‹³Å—o 3x1  4  (4) 3x2  4  (4) 3x1 3x2 x2 œµÎ 1 ‡–¼ šŠÊ´ ­°Š…µo Š 3 ‹³Å—o x1 x2 ‹³Á®œÈ ªµn ™oµ f(x1 ) f(x2 ) ¨oª x1 —Š´ œ´Êœ f Ážœ} ¢Š{ „rœ´ 1 – 1

⌦ 51 ⌦ ĝ„·‹„¦¦¤šÉ¸ 7 1. ‹Š¡·‹µ¦–µªnµ…o°Ä—Ážœ} ¢Š{ „rœ´ ‹µ„ R ޚɴª™Š¹ R 1. f(x) 9x  4 2. f(x) 3 x  2 3. f (x) 4x2 1 4. f(x) 7x  1 5. f(x) x3 6. f(x) x2  2x  5 7. f(x) 3 8. f(x) x2, x  0 9. f(x) 1 x 10. f(x) x 2. ‹ŠÁ…¸¥œÁ‡¦ºÉ°Š®¤µ¥ 9®œoµ…o°š¸É™¼„ ¨³ 8 ®œoµ…o°š¸ÉŸ—· „ε®œ—Ä®o A {1,2,3} , B {4,5} ¨³ C {4,5,6} ………….1) {(1,4),(2,5),(3,5)} Áž}œ¢Š{ „r œ´ ‹µ„ A ޚɪ´ ™¹Š B ………….2) {(1,4),(2,5),(3,5)} Áž}œ¢{Š„r ´œ‹µ„ A Şšª´É ™Š¹ C ………….3) {(1,4),(2,4),(3,5)} Ážœ} ¢{Š„r œ´ ‹µ„ A Şš´Éª™¹Š B ………….4) {(1,4),(2,5),(3,6)} Ážœ} ¢Š{ „r ´œ‹µ„ A ޚɪ´ ™¹Š C 3. …°o ė˜°n ŞœÁ¸Ê ž}œ¢Š{ „rœ´ 1 – 1 1) f(x) x 2) f(x) x2  1 3) f(x) x  5 4) f(x) 3x  2 5) f(x) x  1 6) f(x) 1 x 7) f(x) x 4 8) f(x) x2 9) f(x) x3 10) f(x)  x

52 ⌫ ⌫  ⌦  ⌫          Ÿœ„µ¦‹—´ „µ¦Á¦¸¥œ¦¼oš¸É 8 Á¦ºÉ°Š ¢Š{ „r œ´ ž¦³„° Êœ´ ¤´›¥¤«¹„¬µžše ɸ 4 ª·µ ‡–˜· «µ­˜¦r Áª¨µ 4 Éª´ äŠ Ÿ¨„µ¦Á¦¥¸ œ¦šo¼ ‡¸É µ—®ªŠ´ ­µ¤µ¦™®µ¢{Š„rœ´ ž¦³„°…°Š¢Š{ „r œ´ ­°Š¢{Š„rœ´ š¸„É µÎ ®œ—Ä®oŗo 1. ‹—» ž¦³­Š‡„r µ¦Á¦¥¸ œ¦o¼ 1. °„‡ªµ¤®¤µ¥…°Š¢{Š„r ´œž¦³„°Å—o 2. °„Å—ªo µn ‹³®µ¢{Š„r ´œž¦³„°…°Š¢Š{ „r œ´ ­°Š¢Š{ „r´œš„¸É µÎ ®œ—Ä®Åo —o®¦º°Å¤n 3. ®µ¢Š{ „r œ´ ž¦³„°…°Š¢{Š„r ´œ­°Š¢Š{ „r´œš¸„É 宜—Ä®Åo —o 4. °„×Á¤œÂ¨³Á¦œ‹r…°Š¢Š{ „r œ´ ž¦³„°š„¸É 宜—Ä®Åo —o 2. œª‡ªµ¤‡—· ®¨´„ gof(x) Ážœ} ¢Š{ „r œ´ šÉ¸­¦oµŠ…ʹœÄ®¤n Ážœ} ¢Š{ „r œ´ ‹µ„ÁŽ˜ A ŞÁŽ˜ C ×¥ ×Á¤œ¤µ‹µ„ A ¨³Á¦œ‹r¤µ‹µ„ C Af BgC 3. ÁœºÊ°®µ­µ¦³ œ¥· µ¤ Ä®o f ¨³ g Ážœ} ¢Š{ „r œ´ ¨³ Rf ˆ Dg z I ¢{Š„r œ´ ž¦³„°…°Š f ¨³ g Á…¸¥œÂšœ —oª¥ gof „ε®œ—×¥ (gof)(x) = g(f(x)) ­µÎ ®¦´š„» x ŽÉ¹Š f(x)  Dg 4. „¦³ªœ„µ¦‹´—„µ¦Á¦¥¸ œ¦¼o 1. ‡¦Â¼ ¨³œ„´ Á¦¸¥œššªœ„µ¦®µ‡nµ…°Š¢{Š„rœ´ f(x) 2. ‡¦„¼ µÎ ®œ—Ÿœ£µ¡ ¢Š{ „rœ´ f ¨³ g ץčÂo ŸnœÃž¦Šn Ä­ —´Š¦¼ž AB C 1f a g p 2b q 83 c r 3. ‹µ„Ÿœ£µ¡‹³Å—o 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

⌦ 53 ⌦ ‹µ„ f ¨³ g š„ɸ 宜—Ä®‹o ³Å—o g(f(1)) = g(a) = p g(f(2)) = g(c) = q g(f(3)) = g(b) = p 4. °µ‹­¦µo Š¢{Š„r´œ…œÊ¹ Ä®¤Án ¦¥¸ „ªnµ¢{Š„rœ´ ž¦³„° gof ( ‹Ã¸ °Á°¢ ) Áž}œ¢Š{ „rœ´ ‹µ„ A Ş C (gof)(1) = g(f(1)) (gof)(2) = g(f(2)) (gof)(3) = g(f(3)) œ´œÉ ‡º° gof = {(1,p),(2,q),(3,p)} 5. ‡¦¼„µÎ ®œ—Ÿœ£µ¡¢Š{ „r œ´ f ¨³ g Ä®œo „´ Á¦¥¸ œ®µ gof ˚¥˜r ª´ °¥µn Š Af B gC 4 79 5 8 10 6 ‹³Å—o gof = {(4,9),(5,10),(6,9)} 6. ‡¦Â¼ ¨³œ´„Á¦¸¥œªn ¥„œ´ ­¦ž» ‡ªµ¤®¤µ¥…°Š‡µÎ ªnµ¢Š{ „r ´œž¦³„° 7. ‡¦Â¼ ¨³œ´„Á¦¥¸ œššªœÃ—Á¤œÂ¨³Á¦œ‹…r °Š¢Š{ „rœ´ 8. ‹µ„Ëš¥˜r ª´ °¥nµŠÄœ…°o 2 ¨³ 5 Ä®œo ´„Á¦¥¸ œ®µÃ—Á¤œÂ¨³Á¦œ‹r…°Š¢Š{ „rœ´ f ¨³ g ¨oª ªn ¥„œ´ ¡‹· µ¦–µªµn ×Á¤œ…°Š gof Áž}œ°¥µn ŠÅ¦ 9. ‡¦¼Â¨³œ´„Á¦¸¥œnª¥„´œ­¦»žªnµ‹³®µ¢{Š„r´œž¦³„°…°Š¢{Š„r´œ­°Š¢{Š„r´œš¸É„µÎ ®œ—Ä®oŗo °¥nµŠÅ¦ 10. Ä®œo „´ Á¦¥¸ œ«¹„¬µÁ¡¤É· Á˜¤· ‹µ„ĝ‡ªµ¤¦¼šo ¸É 8 11. Ä®oœ„´ Á¦¥¸ œ „f š´„¬³Ã—¥šÎµÂ f„®—´ Ĝĝ„·‹„¦¦¤šÉ¸ 8

54 ⌫ ⌫  ⌦  ⌫          5. ®¨Šn „µ¦Á¦¥¸ œ¦¼o 1. ĝ‡ªµ¤¦šo¼ ɸ 8 2. ĝ„‹· „¦¦¤š¸É 8 3. ®°o Š­¤»—æŠÁ¦¸¥œ 4. ­º ‡oœšµŠ Internet 6. „¦³ªœ„µ¦ª´—¨³ž¦³Á¤œ· Ÿ¨ 1. ž¦³Á¤œ· Ÿ¨‹µ„„µ¦šµÎ  „f ®´— 2. ž¦³Á¤œ· Ÿ¨‹µ„„µ¦šµÎ š—­° 7. œ´ š„¹ ®¨´Š„µ¦­°œ ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. 8. „‹· „¦¦¤Á­œ°Âœ³ ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ……………………………………………………………………………………………………………….

⌦ 55 ⌦ ĝ‡ªµ¤¦oš¼ ɸ 8 ¢{Š„rœ´ ž¦³„° ™oµ¤¸¢{Š„rœ´ °¥nµŠœo°¥®œ¹ÉŠ¢{Š„r œ´ Á¦µ­µ¤µ¦™­¦oµŠ¢Š{ „rœ´ Ä®¤Ån —o°¸„œ°„Á®œ°º ‹µ„„µ¦ª„ ¨ ‡¼– ®¦º°®µ¦¢Š{ „rœ´ ‡º°„µ¦œÎµ¢Š{ „rœ´ ¤µž¦³„°„œ´ —oª¥ÁŠ°ºÉ œÅ…šÉ„¸ µÎ ®œ—Ä®o ¢Š{ „rœ´ Ä®¤nšÅ¸É —‡o º° ¢Š{ „r ´œ ž¦³„° ŽÉ¹Š¢{Š„r´œž¦³„°œ¸‹Ê ³¤¸šµš­µÎ ‡´ÄœÁ¦É°º Š„µ¦®µ°œ¡» œ´ ›Ãr —¥Ä„o ‘¨¼„ÃŽn šœ·¥µ¤ „µÎ ®œ—Ä®o f ¨³ g Ážœ} ¢{Š„rœ´ ŽÉŠ¹ ¢Š{ „r œ´ ž¦³„°…°Š f ¨³ g Áž}œ¢Š{ „r´œ‹µ„ {x  Df / f(x)  Dg } Ş¥´ŠÁ¦œ‹r…°Š g ¨³ (x, z)  gof „˜È °n Á¤º°É ¤¸ y Ž¹ÉŠ (x, y)  f ¨³ (y, z)  g ˜ª´ °¥nµŠš¸É 1 „ε®œ—Ä®o f {(a,1), (b,2), (c,5)} g {(1,7), (2,8), (3,9)} Áœ°Éº Š‹µ„ (a,1)  f ¨³ (1,7)  g —´ŠœœÊ´ (a,7)  gof (b,2)  f ¨³ (2,8)  g —Š´ œœÊ´ (b,8)  gof ­nªœ (c,5)  f ˜nŤn¤‡¸ n¼°œ´ —´ š¸É¤¸ 5 Ážœ} ¡·„—´ ¦„…°Š g —´Šœ´Êœ‹¹ŠÅ¤¡n ·‹µ¦–µ ‹³Å—ªo µn gof {(a,7), (b,8)} ˜´ª°¥µn ŠšÉ¸ 2 „µÎ ®œ—Ä®o f(x) 2x  1 ¨³ g(x) x2  2 ª·›š¸ µÎ ‹Š®µ Dgof , Dfog , (gof )( x) ¨³ fog(x) ¡‹· µ¦–µ Dgof ¨³ Dfog ‹µ„šœ¥· µ¤ Dgof {x  Df / f(x)  Dg } Ĝš¸œÉ ¸Ê Df  R ¨³ Dg  R —Š´ œÊœ´ Dgof {x  R / 2x  1  R} R ¨³ Dfog {x  Dg / g(x)  Dg } {x  R / x 2  2  R} R Áœ°Éº Š‹µ„¤¸ Dgof ¨³ Dfog —Š´ œ´Êœ¤¸ gof ¨³ fog ¡‹· µ¦–µ (gof )( x) g(f(x)) g(2x  1) Á¡¦µ³ f(x) 2x  1 (2x  1) 2  2 Á¡¦µ³ g(x) x2  2 ®¦º° g(A) A2  2 Á¤É°º A 2x  1 (4 x 2  4 x  1)  2 4x2  4x 1 ¡‹· µ¦–µ (fog)( x) f ( g( x)) f(x2  2) Á¡¦µ³ g(x) x2  2 2(x2  2)  1 Á¡¦µ³ f(x) 2x  1 ®¦°º f(A) 2A  1 Á¤°Éº A x2  2 (2x 2  4  1) 2x2  3

56 ⌫ ⌫  ⌦  ⌫         

⌦ 57 ⌦ Ÿœ„µ¦‹´—„µ¦Á¦¥¸ œ¦šo¼ ¸É 9 Á¦º°É Š ¢Š{ „rœ´ °·œÁª°¦­r œÊ´ ¤›´ ¥¤«„¹ ¬µžše ɸ 4 ª·µ ‡–˜· «µ­˜¦r Áª¨µ 2 Éª´ äŠ Ÿ¨„µ¦Á¦¸¥œ¦šo¼ ¸É‡µ—®ª´Š ®µ¢Š{ „rœ´ °·œÁª°¦­r ¨³Á…¸¥œ„¦µ¢…°Š¢Š{ „r œ´ °·œÁª°¦r­Å—o 1. ‹»—ž¦³­Š‡„r µ¦Á¦¥¸ œ¦o¼ 1. ®µ°œ· Áª°¦r­…°Š¢Š{ „rœ´ š„¸É 宜—Ä®Åo —o 2. °„‡ªµ¤®¤µ¥…°Š¢{Š„r ´œ°œ· Áª°¦­r ŗo 3. °„Å—ªo nµ¢Š{ „r ´œš¸„É 宜—Ä®¤o ¸¢{Š„r œ´ °·œÁª°¦r­®¦°º Ťn 4. ®µÃ—Á¤œÂ¨³Á¦œ‹…r °Š¢Š{ „r œ´ °·œÁª°¦­r ŗo 5. Á…¸¥œ„¦µ¢…°Š¢{Š„r œ´ °œ· Áª°¦­r ŗo 2. œª‡ªµ¤‡·—®¨´„ ™oµ„µÎ ®œ—¢{Š„rœ´ Ä®o­µ¤µ¦™®µ°œ· Áª°¦r­…°Š¢Š{ „rœ´ ŗo ˜°n ·œÁª°¦­r …°Š¢{Š„rœ´ Ťn‹µÎ Ážœ} ˜°o Š Ážœ} ¢Š{ „rœ´ Á­¤°Åž ‹³Á¦¥¸ „°œ· Áª°¦r­…°Š¢{Š„r ´œš¸ÉÁžœ} ¢{Š„r œ´ ªnµ ¢Š{ „rœ´ °œ· Áª°¦­r ( Inverse Function ) 3. Áœ°Êº ®µ­µ¦³ ™µo „µÎ ®œ—¢Š{ „r´œÄ®o­µ¤µ¦™®µ°œ· Áª°¦r­…°Š¢{Š„r´œÅ—o ˜°n ·œÁª°¦r­…°Š¢{Š„r œ´ Ť‹n εÁžœ} ˜o°Š Áž}œ¢{Š„r œ´ Á­¤°Åž ‹³Á¦¥¸ „°œ· Áª°¦­r …°Š¢{Š„r´œš¸ÉÁžœ} ¢{Š„r ´œªµn ¢Š{ „rœ´ °·œÁª°¦­r ( Inverse Function ) š§¬‘¸ š Ä®o f Áž}œ¢Š{ „rœ´ f1 Ážœ} ¢{Š„r œ´ °œ· Áª°¦­r „Ș°n Á¤ºÉ° f Áž}œ¢{Š„r ´œ 1-1 4. „¦³ªœ„µ¦‹—´ „µ¦Á¦¸¥œ¦o¼ 1. ‡¦¼Â¨³œ„´ Á¦¥¸ œššªœ°œ· Áª°¦r­…°Š‡ªµ¤­¤´ ¡´œ›r 2. ‡¦¼„µÎ ®œ—Ëš¥r˜´ª°¥µn ŠÄ®œo „´ Á¦¥¸ œ®µ°·œÁª°¦­r f = {(1,2),(2,3),(3,4)} ‹³Å—o f1 = {(2,1),(3,2),(4,3)}

58 ⌫ ⌫  ⌦  ⌫          g = {(1,2),(2,3),(3,2)} ‹³Å—o g1 = {(2,1),(3,2),(2,3)} h = {(1,2),(3,2),(4,1)} ‹³Å—o h1 = {(2,1),(2,3),(1,4)} 3. ‹µ„Ëš¥˜r ª´ °¥nµŠÄ®oœ„´ Á¦¥¸ œ¡‹· µ¦–µªnµ°·œÁª°¦­r …°Š f , g ¨³ h Ážœ} ¢Š{ „rœ´ ®¦°º Ťn 4. ‡¦¼°„ªµn f1 Á¦¸¥„¢{Š„r œ´ °œ· Áª°¦­r 5. ‡¦Â¼ ¨³œ´„Á¦¸¥œnª¥„œ´ ­¦»ž‡ªµ¤®¤µ¥…°Š¢{Š„rœ´ °œ· Áª°¦r­ 6. ‡¦¼„ε®œ—Ëš¥r˜´ª°¥nµŠÄ®oœ„´ Á¦¸¥œ¡·‹µ¦–µ Ánœ f = {(3,2),(4,3),(5,1)} g = {(4,1),(5,3),(6,2)} h = {(2,3),(3,5),(4,1)} ‹µ„Ëš¥˜r ´ª°¥nµŠÄ®œo ´„Á¦¥¸ œ˜°‡µÎ ™µ¤˜n°Åžœ¸Ê 1. ¢Š{ „r œ´ Ĝ…o°Ä—Ážœ} ¢Š{ „r œ´ 1-1 2. ¢Š{ „r´œÄ—š¤É¸ ¢¸ Š{ „r ´œ°œ· Áª°¦r­ 7. ‡¦Â¼ ¨³œ„´ Á¦¥¸ œnª¥„œ´ ­¦»žªnµ¢{Š„r œ´ š¤É¸ ¨¸ „´ ¬–³°¥µn ŠÅ¦š¤¸É ¸¢{Š„r œ´ °œ· Áª°¦­r 8. ‡¦Â¼ ¨³œ„´ Á¦¥¸ œššªœÃ—Á¤œÂ¨³Á¦œ‹r…°Š¢{Š„rœ´ 9. ‡¦¼„µÎ ®œ—Ëš¥r˜ª´ °¥nµŠ f = {(3,2),(4,3),(5,1)} g = {(4,1),(5,3),(6,2)} ‹µ„Ëš¥˜r ª´ °¥nµŠÄ®œo „´ Á¦¥¸ œ®µ°œ· Áª°¦­r ‹³Å—o f1 = {(2,3),(3,4),(1,5)} g1 = {(4,1),(5,3),(6,2)} 10. Ä®œo „´ Á¦¥¸ œªn ¥„´œ®µÃ—Á¤œÂ¨³Á¦œ‹r ‹³Å—o Df = {3,4,5} ¨³ Df1 = {1,2,3} Rf = {1,2,3} ¨³ Rf1 = {3,4,5} 11. ‹µ„…°o 10 ‡¦¼Â¨³œ´„Á¦¥¸ œnª¥„œ´ ­¦ž» ‹³Å—o = RDf f1 ¨³ = DRf f1 12 ‡¦Â¼ ¨³œ„´ Á¦¥¸ œššªœ„¦µ¢…°Š‡ªµ¤­¤´ ¡œ´ ›r ‡¦¼„µÎ ®œ—Ëš¥˜r ª´ °¥nµŠ Áœn f = {(1,2),(2,3),(3,5)} Ä®oœ„´ Á¦¸¥œ®µ f1 ‹³Å—o f1 = {(2,1),(3,2),(5,3)}

⌦ 59 ⌦ Ä®œo „´ Á¦¸¥œÁ…¥¸ œ„¦µ¢…°Š f ¨³ f1 ¨Šœ¦³œµ¡„· —´ Œµ„‹³Å——o Š´ ¦ž¼ y y=x 5 f 4 f 1 3 2 1 0 12 345 x 13. Ä®oœ„´ Á¦¸¥œ«¹„¬µÁ¡É·¤Á˜¤· ‹µ„ĝ‡ªµ¤¦š¼o ¸É 9 14.  f„š„´ ¬³Ã—¥Ä®œo ´„Á¦¸¥œšÎµÂ f„®—´ ‹µ„ĝ„·‹„¦¦¤šÉ¸ 9 5. ®¨Šn „µ¦Á¦¸¥œ¦¼o 1. ĝ‡ªµ¤¦o¼š¸É 9 2. ĝ„·‹„¦¦¤š¸É 9 3. ®°o Š­¤»—æŠÁ¦¥¸ œ 4. ­º‡œo ‹µ„ Internet 6. „¦³ªœ„µ¦ª´—¨³ž¦³Á¤œ· Ÿ¨ 1. ž¦³Á¤œ· Ÿ¨‹µ„„µ¦šÎµÂ f„®´— 2. ž¦³Á¤œ· Ÿ¨‹µ„„µ¦šµÎ š—­° 7. œ´ š¹„®¨Š´ „µ¦­°œ ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. 8. „·‹„¦¦¤Á­œ°Âœ³ ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ……………………………………………………………………………………………………………….

60 ⌫ ⌫  ⌦  ⌫          ĝ‡ªµ¤¦oš¼ ɸ 9

⌦ 61 ⌦

62 ⌫ ⌫  ⌦  ⌫         

⌦ 63 ⌦

64 ⌫ ⌫  ⌦  ⌫          ĝ„‹· „¦¦¤š¸É 9 1. „µÎ ®œ—¢Š{ „r œ´ f ‹Š®µ°œ· Áª°¦r­…°Š¢Š{ „rœ´ ˜n°Åžœ¡Ê¸ ¦o°¤šŠÊ´ Á…¥¸ œ„¦µ¢ 1. f (x) 5x 1 2. f (x) 3 3. f (x) x2 1 4. f (x) (x  2)2 5. f (x) 4  3x 6. f (x)  x 1 x2 1 7. f (x) x 8. f (x) x2 ,0 d x d1 3 9. f (x) x 10. f (x)  16  x2 ,0 d x d 4 2. ‹µ„…°o 1 °·œÁª°¦­r …°Š¢Š{ „r œ´ Ĝ…o°Ä—µo ŠšÁ¸É ž}œ¢Š{ „r œ´ ˜° …………………………………………………………………………………..

⌦ 65 ⌦ 3. „ε®œ—¢{Š„r´œ f ‹Š¡·‹µ¦–µªnµ ¢{Š„r´œ f ˜n°Åžœ¸Ê¤¸¢{Š„r´œ°·œÁª°¦r­®¦º°Å¤n ™oµ¤¸ ‹Š®µ f 1 ,Df ,Rf ,Df 1 ¨³ Rf 1 1. f (x) 2x  3 2. f (x) 3  x 3. f (x) x2  9 4. f (x) (x 1)2 5. f (x) 3x2 6. f (x) x  2 x 1 7. f (x) x 1 8. f (x) x2 ,1 d x d 0 1 9. f (x) x 10. f (x) 9  x2 ,0 d x d 3

66 ⌫ ⌫  ⌦  ⌫          Ÿœ„µ¦‹´—„µ¦Á¦¥¸ œ¦¼oš¸É 10 Á¦º°É Š ¡¸‡–˜· …°Š¢Š{ „r´œ Ê´œ¤´›¥¤«„¹ ¬µžeš¸É 4 ª· µ ‡–˜· «µ­˜¦r Áª¨µ 4 ´ªÉ äŠ Ÿ¨„µ¦Á¦¸¥œ¦š¼o ɸ‡µ—®ªŠ´ ®µ¡¸ ‡–˜· …°Š¢{Š„rœ´ š¸É„µÎ ®œ—Ä®Åo —o 1. ‹—» ž¦³­Š‡r„µ¦Á¦¥¸ œ¦o¼ 1. °„‡ªµ¤®¤µ¥…°Š¡¸ ‡–·˜…°Š¢{Š„r ´œ˜Ê´ŠÂ˜n 2 ¢{Š„r ´œ…ʹœÅž 2. ®µ¢Š{ „rœ´ š¸ÁÉ „—· ‹µ„„µ¦ª„ ¨ …°Š¢Š{ „r œ´ ŗo 3. ®µ¢Š{ „r œ´ šÉ¸Á„—· ‹µ„„µ¦‡–¼ ®µ¦ …°Š¢Š{ „r ´œÅ—o 4. ®µÃ—Á¤œÂ¨³Á¦œ‹…r °Š¡¸ ‡–˜· …°Š¢{Š„r´œÅ—o 2. œª‡ªµ¤‡·—®¨´„ ™µo ¤¸¢{Š„r ´œ˜Š´Ê ˜n®œÉ¹Š¢Š{ „r ´œ…¹œÊ Ş Á¦µ°µ‹œÎµ¢Š{ „r œ´ Á®¨nµœ¤¸Ê µ­¦µo Š¢{Š„r œ´ Ä®¤Ån —o ×¥„µ¦œµÎ ‡nµ…°Š¢{Š„r ´œ¤µª„ ¨ ‡–¼ ®¦º°®µ¦„œ´ Ž¹ŠÉ ¤¸ÁŠ°Éº œÅ…˜µ¤šœ·¥µ¤ 3. Áœ°ºÊ ®µ­µ¦³ ™µo ¤¢¸ {Š„r ´œ˜Š´Ê ˜®n œŠ¹É ¢Š{ „r œ´ …ʹœÅž Á¦µ°µ‹œÎµ¢{Š„rœ´ Á®¨µn œ¸¤Ê µ­¦oµŠ¢{Š„r œ´ Ä®¤Ån —o ×¥„µ¦œµÎ ‡nµ…°Š¢Š{ „r œ´ ¤µª„ ¨ ‡–¼ ®¦°º ®µ¦„œ´ ŽŠÉ¹ ¤¸ÁŠº°É œÅ…˜µ¤šœ·¥µ¤˜°n ޜʸ šœ·¥µ¤ Ä®o f ¨³ g Ážœ} ¢{Š„r´œ œ¥· µ¤¢{Š„r œ´ f  g , f  g , f ˜ g ¨³ f —Š´ œÊ¸ g 1. (f  g)( x) f(x)  g(x) 2. (f  g)( x) f(x)  g(x) 3. (f ˜ g)( x) f(x) ˜ g(x) f f (x) Á¤°Éº g(x) z 0 4. (x) = g g(x) ×¥šÉ¸š»„Ç ­¤µ„· x Ĝ×Á¤œ…°Š¢Š{ „r œ´ Ĝ…o° 1 – …o° 3 œ¸°Ê ¥¼nšŠ´Ê Ĝ×Á¤œ…°Š¢Š{ „rœ´ f ¨³ g œœ´É ‡°º š»„Ç ­¤µ·„ x  Df ˆ Dg

⌦ 67 ⌦ 4. ( f )( x) f(x) Á¤É°º g(x) z 0 g g(x) ×¥šÉš¸ »„Ç ­¤µ„· x Ĝ×Á¤œ…°Š¢{Š„r ´œ f °¥nš¼ Ê´ŠÄœÃ—Á¤œ…°Š¢Š{ „r´œ f ¨³ g g š¸É g(x) z 0 œœ´É ‡º°š»„Ç ­¤µ·„ x  Df ˆ Dg  {x  / g(x) z 0} 4. „¦³ªœ„µ¦‹—´ „µ¦Á¦¸¥œ¦o¼ 1. ‡¦¼Â¨³œ„´ Á¦¸¥œššªœ‡nµ…°Š¢Š{ „rœ´ f šÉ¸ x 2. ‡¦¼„µÎ ®œ—Ëš¥˜r ´ª°¥µn Š f = {(1,2),(2,4),(3,6)} g = {(1,1),(2,2),(3,3)} ‹µ„Ëš¥r˜´ª°¥nµŠ ‹³Å—o f(1) = 2 f(2) = 4 f(3) = 6 g(1) = 1 g(2) = 2 g(3) = 3 3. ‹µ„…°o 2 ‡¦„¼ µÎ ®œ—Ä®o (f+g)(1) = f(1) + g(1) = 2+1 =3 Ä®oœ„´ Á¦¥¸ œ®µ (f+g)(2) ¨³ (f +g)(3) ‹³Å—o f+g = {(1,3),(2,6),(3,9) 4. ‡¦Â¼ ¨³œ„´ Á¦¸¥œ­¦ž» ‡ªµ¤®¤µ¥…°Š¡¸‡–˜· …°Š¢Š{ „r´œ 5. ‡¦Â¼ ¨³œ´„Á¦¥¸ œššªœ‡ªµ¤®¤µ¥…°Š¡¸‡–˜· …°Š¢{Š„rœ´ 6. ‹µ„Ëš¥r˜´ª°¥µn Š…°o 2 ‡¦¼Ä®oœ·¥µ¤ f – g ‹³Å—o f – g = {(1,1),(2,2),(3,3)} 7. ‡¦¼„µÎ ®œ—Ëš¥˜r ´ª°¥nµŠÁ¡·É¤Á˜·¤Ä®oœ„´ Á¦¥¸ œ®µ f+g ¨³ 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. ‡¦Â¼ ¨³œ„´ Á¦¥¸ œššªœ„µ¦ª„ ¨…°Š¢Š{ „r œ´ 9. ‹µ„Ëš¥˜r ´ª°¥nµŠ…°o 2 ‡¦Ä¼ ®oœ¥· µ¤ f ˜ g ¨³ f g ‹³Å—o f ˜ g = {(1,2),(2,8),(3,18)} f = {(1,2),(2,2),(3,2)} g 10.  f„š´„¬³Ã—¥Ä®oœ´„Á¦¸¥œšµÎ  „f ®—´ Ĝĝ„·‹„¦¦¤ 11. ‡¦¼Â¨³œ´„Á¦¸¥œššªœ¡¸‡–˜· …°Š¢Š{ „r´œ

68 ⌫ ⌫  ⌦  ⌫          ‡¦Ä¼ ®Ão ‹š¥r˜´ª°¥µn Š f = {(1,2),(2,4),(3,6),(4,7)} g = {(2,3),(3,1),(4,2),(5,3)} Ä®oœ„´ Á¦¸¥œ®µ f + g ‹³Å—o f + g = {(2,7),(3,7),(4,9)} Ä®œo „´ Á¦¥¸ œ®µÃ—Á¤œÂ¨³Á¦œ‹…r °Š f + g ‹³Å—o Dfg = {2,3,4} , Rfg = {7,9} 12. Ä®œo „´ Á¦¥¸ œ«„¹ ¬µÁ¡¤·É Á˜¤· ‹µ„ĝ‡ªµ¤¦o¼šÉ¸ 10 13.  „f š´„¬³Ä®œo „´ Á¦¥¸ œšÎµÂ „f ®´—‹µ„ĝ„‹· „¦¦¤šÉ¸ 10 5. ®¨Šn „µ¦Á¦¥¸ œ¦o¼ 1. ĝ‡ªµ¤¦¼oš¸É 10 2. ĝ„·‹„¦¦¤š¸É 10 3. ®o°Š­¤»—æŠÁ¦¸¥œ 4. ­º‡oœ‹µ„ Internet 6. „¦³ªœ„µ¦ª´—¨³ž¦³Á¤·œŸ¨ 1. ž¦³Á¤œ· Ÿ¨‹µ„„µ¦šµÎ  f„®´— 2. ž¦³Á¤·œŸ¨‹µ„„µ¦šµÎ š—­° 7. ´œš¹„®¨Š´ „µ¦­°œ ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………. 8. „‹· „¦¦¤Á­œ°Âœ³ ……………………………….……………………………………………………………………………… ………………………………………………………………………………………………………………. ……………………………………………………………………………………………………………….

⌦ 69 ⌦ ĝ‡ªµ¤¦¼ošÉ¸ 10 ¡¸‡–˜· …°Š¢Š{ „rœ´ ™oµ¤¸¢{Š„rœ´ ˜Š´Ê ˜n®œÉŠ¹ ¢Š{ „r ´œ…œ¹Ê Ş Á¦µ°µ‹œÎµ¢{Š„rœ´ Á®¨µn œÊ¤¸ µ­¦µo Š¢Š{ „r ´œÄ®¤Ån —o ×¥„µ¦œµÎ ‡µn …°Š¢{Š„r´œ¤µª„ ¨ ‡–¼ ®¦º°®µ¦„œ´ Ž¹ŠÉ ¤Á¸ ŠÉº°œÅ…˜µ¤šœ¥· µ¤˜n°ÅžœÊ¸ šœ·¥µ¤ Ä®o f ¨³ g Áž}œ¢{Š„r ´œ œ¥· µ¤¢Š{ „rœ´ f  g , f  g , f ˜ g ¨³ f —´ŠœÊ¸ g 1. (f  g)( x) f(x)  g(x) 2. (f  g)( x) f(x)  g(x) 3. (f ˜ g)( x) f(x) ˜ g(x) ×¥šÉ¸š»„Ç ­¤µ·„ x Ĝ×Á¤œ…°Š¢{Š„rœ´ Ĝ…o° 1 – …°o 3 œ°Ê¸ ¥n¼šŠÊ´ Ĝ×Á¤œ…°Š¢{Š„r œ´ f ¨³ g œœÉ´ ‡º°š„» Ç­¤µ·„ x  Df ˆ Dg 4. ( f )( x) f(x) Á¤Éº° g(x) z 0 g g(x) ×¥š¸Éš»„Ç ­¤µ„· x Ĝ×Á¤œ…°Š¢Š{ „rœ´ f °¥nš¼ ŠÊ´ Ĝ×Á¤œ…°Š¢{Š„r œ´ g f ¨³ g š¸É g(x) z 0 œÉ´œ‡º°š„» Ç­¤µ„· x  Df ˆ Dg  {x  / g(x) z 0} ‹µ„šœ¥· µ¤ ‹³Á®Èœªµn „°n œš‹¸É ³œµÎ ‡µn …°Š¢{Š„r´œ¤µª„ ¨ ‡¼– ®¦º°®µ¦„œ´ Á¦µ‹³˜°o Š®µ ×Á¤œ…°Š¢Š{ „rœ´ ­°Š¢Š{ „r ´œšÉ¸‹³œµÎ ¤µ—εÁœœ· „µ¦ª„ ¨ ‡¼– ®¦º°®µ¦„´œ„°n œ ×¥œµÎ ÁŒ¡µ³­¤µ·„šÉ¸ ŽÎµÊ „œ´ ¤µÁž}œ­¤µ·„…°ŠÃ—Á¤œ…°Š¢Š{ „r œ´ Ä®¤n ­µÎ ®¦´ „µ¦®µ¦¢Š{ „r œ´ œœÊ´ ¤Á¸ ŠºÉ°œÅ…Á¡É·¤Á˜¤· ªnµ ­¤µ„· Ĝ ×Á¤œÄ®¤nœœÊ´ ‹³˜°o ŠÅ¤nšÎµÄ®‡o nµ…°Š¢Š{ „r œ´ š¸ÁÉ ž}œ˜ª´ ®µ¦Ážœ} «¼œ¥r ˜ª´ °¥nµŠšÉ¸ 1 „µÎ ®œ—Ä®o f(x) x 3  2 ¨³ g(x) x  1 ª·›š¸ ε ‹Š®µ (f  g)( x) , (f  g)( x) , (f ˜ g)( x) ¨³ ( f )( x) g ¡‹· µ¦–µ Df R ¨³ Dg {x  R / x t 1} —´ŠœÊœ´ Df ˆ Dg R ˆ {x  R / x t 1} {x  R / x t 1} ‹³Å—ªo nµÃ—Á¤œ…°Š¢Š{ „rœ´ Ä®¤‡n º° {x  R / x t 1}

70 ⌫ ⌫  ⌦  ⌫          ¡‹· µ¦–µ (f  g)( x) f(x)  g(x) (x3  2)  x  1 ¡·‹µ¦–µ x3  2  x 1 (f  g)( x) f(x)  g(x) (x3  2)  x  1 ¡·‹µ¦–µ x3  2  x 1 (f ˜ g)( x) f(x) ˜ g(x) (x3  2) x  1 ¡‹· µ¦–µ ( f )( x) f(x) Á¤Éº° g(x) z 0 g g(x) x3  2 Á¤Éº° x  1 ! 0 ®¦º° x ! 1 x 1 ×Á¤œ…°Š f ‡º° {x  R / x ! 1} g ˜´ª°¥µn Šš¸É 2 „µÎ ®œ—Ä®o f(x) 1 ¨³ g(x) 2  x ‹Š®µ (f  g)( x) ª›· š¸ µÎ x  10 ¡·‹µ¦–µ Df {x  R / x ! 10} ¨³ Dg {x  R / x d 2} ‹³Å—ªo nµ Df ˆ Dg I ÁœÉº°Š‹µ„ Df ¨³ Dg Ťn¤­¸ ¤µ„· ¦ªn ¤„´œÁ¨¥ —Š´ œÊ´œ (f  g)( x) Ášµn „´ÁŽ˜ªnµŠ ˜´ª°¥µn ŠšÉ¸ 3 „µÎ ®œ—Ä®o f(x) 2x  3 ¨³ g(x) x2 ª·›¸šÎµ ‹Š®µ (f  g)( 1) , (f  g)(0) , (f ˜ g)(1) ¨³ ( f )(3) g Áœ°Éº Š‹µ„ Df R ¨³ Dg R —´ŠœœÊ´ Df ˆ Dg R ‹³Å—ªo nµ (f  g)( 1) f(1)  g(1) 5  1 4 (f  g)( 0) f(0)  g(0) 3  0 3

⌦ 71 ⌦ (f ˜ g)(1) f(1) ˜ g(1) 1 (1) ˜ 1 ×Á¤œ…°Š f R  {0} g ( f )(3) f(3) g g(3) 3 9 1 3 ˜ª´ °¥µn ŠšÉ¸ 4 „µÎ ®œ—Ä®o f {(1,2), (1,3), (2,4), (4,3)} ‹Š®µ 5f ª›· ¸šÎµ ¡·‹µ¦–µ 5f ¢Š{ „r ´œ 5f ¤µ‹µ„¢Š{ „rœ´ gf Á¤É°º g ‡º°¢{Š„r´œ‡Š˜ª´ g(x) ‹³Á®Èœªµn ×Á¤œ…°Š g ‡º°ÁŽ˜…°Š‹Îµœªœ‹¦Š· R 5 œÉ´œ‡°º Df ˆ Dg R ˆ {1,1,2,4} {1,1,2,4} Df —Š´ œÊ´œ 5f {(1,5 u 2), (1,5 u 3), (2,5 u 4), (4,5 u 3)} {(1,10), (1,15), (2,20), (4,15)}

72 ⌫ ⌫  ⌦  ⌫          ĝ„·‹„¦¦¤š¸É 10 1. „µÎ ®œ—Ä®o f {(0,1), (1,3), (2,0), (5,4)} g {(1,2), (0,1), (2,3), (5,0)} ‹Š®µ f  g , f  g , f ˜ g , f ¨³ (3)f g 2. „µÎ ®œ—Ä®o f(x) 5  3x ¨³ g(x) x 2  1 ‹Š®µ (f  g)( x) , (f ˜ g)( x) ¨³ ©¨§¨ f ¹¸¸·( x) ‹Š®µÃ—Á¤œÂ¨³Á¦œ‹…r °Š¢{Š„rœ´ Á®¨µn œ¸Ê g 3. „µÎ ®œ—Ä®o f(x) 2x 2  5 ¨³ g(x) 4  x 2 ‹Š®µ (f  g)(1) , (f ˜ g)( 2) ¨³ ©§¨¨ f ·¹¸¸( 2) g 4. „µÎ ®œ—Ä®o f(x) 5  3x Á¤ºÉ°  4  x d 3 ¨³ g(x) x  1 Á¤º°É  2 d x  5 ‹Š®µ (f  g)( x) , (f  g)( x) , (f ˜ g)( x) ¨³ §¨¨© f ·¸¸¹( x) g 5. „µÎ ®œ—Ä®o f(x) x 2  4 ¨³ g(x) x  2 ‹Š®µ f  g , f  g , f ˜ g ¨³ f g

⌦ 73 ⌦ ˚¥Ár ­¦¤· š„´ ¬³ 1. „µÎ ®œ— x t 1 ¨³ (fog)( x) 4x 2  8x ¨³ f(x) x2  4 ¨ªo g1 (4) ¤¸‡µn ˜¦Š„´…°o ė 1. 1 2. 2 3. 3 4. 4 2. „ε®œ— r {(x, y)  R u R / y x 2  4 x  5 Á¤º°É x 2  2x  3  0} ™oµÄ®o A = ×Á¤œ…°Š r ¨³ B = ×Á¤œ…°Š r 1 ¨ªo Aˆ Bc Áž}œÁŽ˜˜¦Š„´…°o ė 1. (1,1] 2. [3,10) 3. (1,3) 4. (1,10) 3. ™oµ f(x) 2x  3 Á¤Éº°  2 d x d 4 ¨ªo Á¦œ‹r…°Š f( x ) ¤‡¸ nµ˜¦Š„´…°o ė 1. > 2,4@ 2. > 1,11@ 3. >3,11@ 4. >3,4@ 4. „ε®œ— f(x) x  g(x) ¨³ g(x) x ˜ f(x) ¨oª ¨©§¨ f ¸¸¹·( x) Ášµn „´ …°o ė g 1. x 1 x 1x 5. „ε®œ— f(x) 2. 3. 4. x 1x x2 13 2x  1 ¨³ (f 1og)( x) 2x  3 ¨oª (g 1of)( 4) ¤‡¸ nµ˜¦Š„´ …o°Ä— 1. 2x  4 4  13 15  15 6. „µÎ ®œ— f(x) 2. 3. 4. 10 4 10 2x3  x  A ™oµ (3,2) Áž}œ‹»—°¥n¼ œ„¦µ¢ f 1 ¨ªo ‡µn A ¤‡¸ nµ˜¦Š„´…o°Ä— 1. – 15 2. 15 3. – 54 4. 54 7. „ε®œ— (fog)( x) 4x 2  4x  5 ¨³ g 1 (x) x  3 ¨oª f(x) Ášµn „´ …°o ė 2 1. x 2  4x  8 2. x 2  8x  10 3. x 2  8x  20 4. x 2  4 x  6 8. ™oµ f(x) (3  x)(2  x) ¨³ g(x) 1 ¨oªÃ—Á¤œ…°Š f ˜ g ‡º°ÁŽ˜Äœ…°o ė x3 1. I 2. (f,2] 3. (3,2) 4. (3,2] 9. Ä®o I  Áž}œÁŽ˜…°Š‹µÎ œªœÁ˜È¤ª„ „ε®œ—Ä®o f {(x, y) / x  2y 12 ¨³ x, y  I } ¨ªo fof ÁšnµÁŽ˜Äœ…o°Ä— 1. {(8,5), (4,4)} 2. {(5,8), (4,4)} 3. {(2,2), (4,4)} 4. {(6,3), (4,4)} 10. „ε®œ—Ä®o f(x) x ¨³ g(x) x 2  1 ™µo A Dgof ¨³ B Dg 1x ¨ªo A ‰ Bc ˜¦Š„´…o°Ä— 1. R  {1,1} 2. (1, f) 3. (1 ,1) ‰ (1, f) 4. (1,1) ‰ (1, f) 2

74 ⌫ ⌫  ⌦  ⌫         

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