บทเรียนความสัมพันธ์ฟังก์ชัน ม.4

⌦ 43
⌦

ĝ„‹· „¦¦¤š¸É 6

1. ‹Š¡·‹µ¦–µªµn ‡ªµ¤­´¤¡œ´ ›r˜°n ޜʸÁžœ} ¢Š{ „r œ´ ®¦º°Å¤n

1.^(1,2),(2,3),(3,4),(5,6)` ˜°………………….

2. ^(1,2),(3,4),(3,7),(4,3)` ˜°………………….

3. ^(2,4),(3,4),(3,4),(5,6)` ˜°………………….

4. ^(2,4),(4,6),(6,8),(7,8)` ˜°………………….

5. ^(2,1),(3,1),(5,1),( 1,1)` ˜°………………….

2. Ä®œo „´ Á¦¸¥œ¡‹· µ¦–µªnµ‡ªµ¤­¤´ ¡œ´ ›r˜n°ÅžœÊÁ¸ ž}œ¢Š{ „r ´œ®¦º°Å¤n

1. ®¯­(x,y)/ y 1 ¿¾½ ˜°………………….
x
2. ^(x,y)/ x y2 1` ˜°………………….
3. ^(x,y)/ y x2 5` ˜°………………….
4. ^(x,y)/ x y2` ˜°………………….

5. ®¯­(x,y)/ y x 1 1¿¾½ ˜°………………….

6. ^(x,y)/ y 2 x` ˜°………………….
7. ^(x,y)/ x y2 2x` ˜°………………….

3. Ä®œo „´ Á¦¥¸ œ­¦ž» ‡ªµ¤®¤µ¥…°Š‡ªµ¤­¤´ ¡œ´ ›ršÉÁ¸ ž}œ¢Š{ „r œ´ ¨³‡ªµ¤­´¤¡œ´ ›rš¸ÅÉ ¤Án ž}œ¢Š{ „rœ´

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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
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8. „·‹„¦¦¤Á­œ°Âœ³
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⌦ 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))
ϫσ ༡ 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
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
……………………………………………………………………………………………………………….
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⌦ 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´ f 1 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 f 1 = {(2,1),(3,2),(4,3)}

58 ⌫ ⌫  ⌦
 ⌫         

g = {(1,2),(2,3),(3,2)}
‹³Å—o g 1 = {(2,1),(3,2),(2,3)}

h = {(1,2),(3,2),(4,1)}
‹³Å—o h 1 = {(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 f 1 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 f 1 = {(2,3),(3,4),(1,5)}
g 1 = {(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 Df 1 = {1,2,3}
Rf = {1,2,3} ¨³ Rf 1 = {3,4,5}

11. ‹µ„…°o 10 ‡¦¼Â¨³œ´„Á¦¥¸ œnª¥„œ´ ­¦ž»
‹³Å—o = RDf f 1 ¨³ = DRf f 1

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 f 1
‹³Å—o f 1 = {(2,1),(3,2),(5,3)}

⌦ 59
⌦

Ä®œo „´ Á¦¸¥œÁ…¥¸ œ„¦µ¢…°Š f ¨³ f 1 ¨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 ,D 1 ¨³ R 1
ff

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 Df g = {2,3,4} , Rf g = {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 g 1 (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 1 x
5. „ε®œ— f(x) 2. 3. 4.
x 1 x 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

x 3

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

1 x

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

⌦ 75
⌦

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