แแแแแแแแแแแแแแแแแแแแแแแแแแ แแแแ แแแแ แ 31202 แแแแแแ แแแแแแแแ แแแแแแแแแแแ 2 แแแแแแแแแแ ……….. แแแแแแแแแแแแแแแแแแแแแแแแแแแ แแแแแแแแแ แแแแ – แแแแ........................ ............................ ............................ ....... แแแ แแแ......................แ. 4/……………………..
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
แบบฝึ�กทั�กษะคณิ�ตศาสตร์�เพิ่��มเต�ม ร์หั�ส ค 31202
เร์��อง ฟั�งก�ชั�นภาคเร์!ยนทั!� 2 ปี$การ์ศ%กษา ………..
โร์งเร์!ยนกาญจนาภ�เษกวิ�ทัยาลั�ย เพิ่ชัร์บ+ร์ณิ�
ชั��อ – สก,
ลั.......................................................................................
เลัขทั!�......................ม. 4/……………………..
โร์งเร์!ยนกาญจนาภ�เษกวิ�ทัยาลั�ย เพิ่ชัร์บ+ร์ณิ�
ส.าน�กงานเขตพิ่�/นทั!�การ์ศ%กษาม�ธยมศ%กษา เขต 40
ส.าน�กงานคณิะกร์ร์มการ์การ์ศ%กษาข�/นพิ่�/นฐาน
กร์ะทัร์วิงศ%กษาธ�การ์
จงหาผลคู�ณคูาร์�ที เซี ยนต่�อไปน �
ข2อทั!�
เซต A เซต B ผลัค+ณิคาร์�ทั!เซ!ยนของเซต A
แลัะเซต B1 {1, 2} {3, 4,
5} {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}
2 {a} {1, 2} {(a, 1), (a, 2)}3 {3, 4} {a, b,
c}4 {2, 4} {5, 6}5 {a, b} {3, 4, 5,
6}6 {1, 2,
3} {5, 7, 9}
7 {a, b, {2, 4}
แบบฝึ�กทั�กษะทั!� 1
c}8 {x, y} {1, 2,
3}9 {1, 2} {1, 2}10 {m, n} {x, y}
ให�น�กเร์ ยนเต่�มคู�าต่อบลงในช่�องว่�างแต่�ละข้�อให�ถู�กต่�องสมบ�ร์ณ�
ข2อทั!�
ค.าถาม ค.าตอบ
1. คู�� อ� นดั�บสามแปดั เข้ ยนแทีนดั�ว่ยส�ญล�กษณ�ใดั
2. (10, 6) อ�านว่�าอย�างไร์ ม สมาช่�กต่�ว่หน�าคู+อจ�านว่นใดั
3. คู��อ�นดั�บ (2, 3) และ (√4 , 3) เที�าก�นหร์+อไม�
4. จงหาคู�าข้องต่�ว่แปร์ในแต่�ละข้�อต่�อไปน � 4.1 (x, y) = (6, 9) 4.2 (x – 2, 4) = (8, y + 2) 4.3 (-3, a) = (b – 4, 6) 4.4 (x + y, x – y) = (6, 4) 4.5 (2x, y) = (16, 2)
4.1 ………………………….4.2 ………………………….4.3 ………………………….4.4 ………………………….4.5 ………………………….
ให�น�กเร์ ยนเต่�มคู�าต่อบลงในช่�องว่�างแต่�ละข้�อให�ถู�กต่�องสมบ�ร์ณ�
1. ก�าหนดั A = {3} และ B = {5, 6} แล�ว่1.1 A B =
………………………………………………………………….1.2 B A =
………………………………………………………………….
1.3 A A = ………………………………………………………………….
1.4 B B = ………………………………………………………………….
2. ก�าหนดั A = {1, 3} และ B = {5, 7, 9} แล�ว่2.1 A B =
………………………………………………………………….2.2 B A =
………………………………………………………………….2.3 A A =
………………………………………………………………….2.4 B B =
………………………………………………………………….
3. ถู�า A = {2, 4} และ B = {6, 8, 10, 12} แล�ว่ จงหา
3.1 จ�านว่นสมาช่�กข้อง A B =……………………………………………
3.2 จ�านว่นสมาช่�กข้อง B B =……………………………………………
ให�น�กเร์ ยนเข้ ยนกร์าฟข้องคูว่ามส�มพั�นธ์�แต่�ละข้�อต่�อไปน �ให�ถู�กต่�องสมบ�ร์ณ�
1. จงเข้ ยนกร์าฟข้อง r = {(2, 4), (3, 7), (1, 5)}
แบบฝึ�กทั�กษะทั!� 2
2. จงเข้ ยนกร์าฟข้อง r = {(0, 0), (-1, -1), (-2, -2), (-3, -3), (-4, -4)}
3. จงเข้ ยนกร์าฟข้อง r = {(x, y) I I | x2 + y2 = 4}
xy
4. จงเข้ ยนกร์าฟข้อง r = {(x, y) I I | y = x + 3}
xy
4. จงเข้ ยนกร์าฟข้อง r = {(x, y) | 2 x 5 }
1. ก�าหนดั r = {(1, 2), (2, 4), (3, 5), (4, 6)}1.1 Dr =
…………………………………………………………………1.2 Rr =
………………………………………………………………… 2. ก�าหนดั r = {(-2, 5), (3, 7), (6, 6), (6, 2), (4, 8)}
2.1 Dr =
…………………………………………………………………2.2 Rr =
………………………………………………………………… 3. ก�าหนดั r = {(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)}
5. จงเข้ ยนกร์าฟข้อง r = {(x, y) | y = x2 - 2 }
xy
ให�น�กเร์ ยนหาโดัเมนและเร์นจ�ข้องคูว่ามส�มพั�นธ์�ในแต่�ละข้�อต่�อไปน �
แบบฝึ�กทั�กษะทั!� 3
1. ก�าหนดั r = {(1, 2), (2, 4), (3, 5), (4, 6)}1.1 Dr =
…………………………………………………………………1.2 Rr =
………………………………………………………………… 2. ก�าหนดั r = {(-2, 5), (3, 7), (6, 6), (6, 2), (4, 8)}
2.1 Dr =
…………………………………………………………………2.2 Rr =
………………………………………………………………… 3. ก�าหนดั r = {(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)}
ให�น�กเร์ ยนเต่�มคู�าต่อบลงในช่�องว่�างแต่�ละข้�อต่�อไปน �ให�ถู�กต่�องสมบ�ร์ณ�
ข2อทั!�
ควิามส�มพิ่�นธ�ทั!�ก.าหันดใหั2 โดเมน เร์นจ�
1 {(1, a), (2, b), (3, c), (4, d)}
2 {(2, 10), (3, 20), (4, 30), (5, 40)}
3 {(1, 3), (2, 4), (3, 5), (4, 6)}
4 {(0, 0), (1, 1), (2, 2), (3, 3)}
5 {(1, 4), (2, 5), (3, 6), (4, 7)}
6 {(1, 1), (2, 4), (3, 9), (4, 16)}
7 {(0, 0), (1, 2), (2, 4), (3, 6), (4, 8)}
8 {(x, y) | y = x - 4}9 {(x, y) I I | y =
x2 + 1}10 {(x, y) | y = √x - 8}
จงหาต่�ว่ผกผ�นข้องคูว่ามส�มพั�นธ์� r พัร์�อมที��งโดัเมนและเร์นจ�
ข2อทั!�
ควิามส�มพิ่�นธ�ทั!�ก.าหันดใหั2
ต�วิผกผ�นของควิามส�มพิ่�นธ�
โดเมน r-1 เร์นจ� r-1
1 {(1, a), (2, b), (3, c), (4, d)}
2 {(2, 10), (3, 20), (4, 30), (5, 40)}
3 {(1, 3), (2, 4), (3, 5), (4, 6)}
4 {(0, 0), (1, 1), (2, 2), (3, 3)}
5 {(1, 4), (2, 5), (3, 6), (4, 7)}
6 {(1, 1), (2, 4), (3, 9), (4, 16)}
7 {(0, 0), (1, 2), (2, 4), (3, 6), (4, 8)}
8 {(x, y) | y = x - 4}
9 {(x, y) I I | y = x2 + 1}
10
{(x, y) | y = √x - 8
}11
{(x, y) | y = x−12 }
12
{(x, y) | y = 3x+ 2}
แบบฝึ�กทั�กษะทั!� 4
ให�น�กเร์ ยนกาเคูร์+0องหมาย ลงในต่าร์างต่�อไปน �ให�ถู�กต่�อง
ข2อทั!� ควิามส�มพิ่�นธ� ฟั�งก�ชั�นเปี8น ไม:เปี8น
1 r = {(1, 2), (3, 7), (4, 9), (8, 6), (10, 8)}
2 r = {(3, 2), (4, 2), (6, 2)}3 r = {(1, 3), (2, 4), (3, 6),
(3, 8)}4 r = {(x, y) | y = x2 - 1}5 r = {(x, y) | xy = 2}6 r = {(x, y) | y2 = x +
3}7 r = {(x, y) | y = x3}8 r = {(x, y) | y = x2 +
3}9 r = {(2, 3), (2, 4), (3, 6),
(4, 9)}10 r = {(x, y) | x = y2 +
2y + 1}
ให�น�กเร์ ยนต่ร์ว่จสอบคูว่ามส�มพั�นธ์�แต่�ละข้�อต่�อไปน �เป2นฟ3งก�ช่�นหร์+อไม�แล�ว่เข้ ยนคู�าต่อบลงในต่าร์างให�ถู�กต่�อง
ข2อทั!�
ควิามส�มพิ่�นธ� ค.าตอบ
1 r1 = {(3, 2), (-1, 4), (0, 2), (5, -3)}
2 r2 = {(1, -1), (-1, -1), (0, 1), (3, 1)}
3 r3 = {(4, 2), (4, 3), (5, 4), (6, 6)}
4 r4 = {(6, 1), (6, 1), (7, 2),
(8, 3), (9, 4)}5 r5 = {(2, 1), (2, 3), (3, 4),
(4, 5), (5, 6)}6 r6 = {(x, y) | y = 3x – 1}7 r7 = {(x, y) | y = 3x2– x
+ 3}8 r8 = {(x, y) | y = 2x2 +
1}9 r9 = {(x, y) | y = 4x2}10 r10 = {(x, y) | x2 + y2 =
4 , x 0}ให�น�กเร์ ยนเล+อกคู�าต่อบที 0น�กเร์ ยนเห4นว่�าถู�กที 0ส5ดัเพั ยงคู�าต่อบเดั ยว่
ก�าหนดั f = {(1, 2), (2, 4), (3, 6), (4, 8)}
ใชั2ตอบค.าถามข2อ 1 - 3 1. เข้ ยน f ให�อย��ในร์�ปสมการ์ไดั�ต่ร์งก�บข้�อใดั
ก. y = 2x
ข้. y = 3x
คู. y = 2x + 1
ง. y = x + 1
2. เข้ ยนฟ3งก�ช่�น f ในร์�ปข้องกร์าฟไดั�ต่ร์งก�บ ข้�อใดั
ก.
ข้.
คู.
ง. Y 1 0 X
3. เ ข้ ย น ฟ3 ง ก� ช่� น f ใ น ร์� ปต่าร์างไดั�ต่ร์งก�บข้�อใดั
ก.
ข้.
คู.
ง.
x 2 4 6 8f(x)
1 2 3 4
x 0 2 3 4f(x)
2 4 6 8
x 2 4 6 4f(x)
1 2 3 8
x 1 2 3 4f(x)
2 4 6 8
ให�น�กเร์ ยนเต่�มคู�าต่อบแต่�ละข้�อลงในต่าร์างให�ถู�กต่�องสมบ�ร์ณ�ข2อทั!�
ฟั�งก�ชั�น โดเมนของฟั�งก�ชั�น
เร์นจ�ของฟั�งก�ชั�น
1 f = {(2, 5), (3, 7), (4, 9), (5, 11)}
{2, 3, 4, 5}
{5, 7, 9, 11}
2 f = {(a, 1), (b, 2), (c, 3), (d, 4)}
3 f = {(m, 2), (n, 4), (p, 6)}
4 f = {(4, 7), (6, 9), (8, 11)}
5 f = {(2, 1), (3, 2), (4, 3), (5, 4)}
6 f = {(a, 10), (b, 10), (c, 10)}
7 f = {(3, 6), (5, 10), (7, 14)}
8 f = {(1, 4), (2, 5), (3, 6), (4, 7)}
ให�น�กเร์ ยนเต่�มคู�าต่อบลงในต่าร์างต่�อไปน �ให�ถู�กต่�องสมบ�ร์ณ�1. ให� f(x) = 2x2 – x + 5 และ g(x) = x + 1 จงหา 1.1 (f + g )(x) 1.2 (f – g)(-2) 1.3 (fg)(3)
1.4 ( fg)(0)
………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
แบบฝึ�กทั�กษะทั!� 5
………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………2. ให� f(x) = 3x – 5 และ g(x) = 1
x+3 จงหา 2.1 (f + g) (-3) 2.2 (fg)(6)………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………3. ให� f(x) = 2x + 10 และ g(x) = √ x−2 จงหา 3.1 (f + g)(x) 3.2 (fg)(2) 3.3 (f-g)(3)
3.4 ( fg)(x)
………………………………………………………………………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
1. ให� f(x) = 3x – 5 และ g(x) = 1 – x2 จงหาคู�าข้อง1.1 f(g(0)) =…………………………........ 1.2 g(f(0))
=…………………………………. =……………………………….
=…………………………………. =……………………………….
=…………………………………. =……………………………….
=………………………………….1.3 f(f(4)) =…………………………........ 1.4 g(g(1))
=………………………………… =……………………………….
=…………………………………. =……………………………….
=…………………………………. =……………………………….
=………………………………….1.5 (f ๐ g)(-2) =…………………………........ 1.6
(g ๐ f)(-2) =……………………………… =……………………………….
=…………………………………. =……………………………….
=…………………………………. =……………………………….
=………………………………….1.7 (f ๐ f)(-1) =…………………………........ 1.8
(g ๐ g)(2) =……………………………… =……………………………….
=………………………………….
แบบฝึ�กทั�กษะทั!� 6
=………………………………. =………………………………….
=………………………………. =………………………………….
1.9 (f ๐ g)(x) =…………………………........ 1.10
(g ๐ f)(x) =…………………………… =……………………………….
=…………………………………. =……………………………….
=…………………………………. =……………………………….
=………………………………….1.11 (f ๐ f)(x) =…………………………........ 1.12
(g ๐ g)(x) =……………………………… =……………………………….
=…………………………………. =……………………………….
=…………………………………. =……………………………….
=………………………………….
2. จงหา f ๐ g , g ๐ f , f ๐ f และ g ๐ g พัร์�อมที��งโดัเมนข้องแต่�ละฟ3งก�ช่�น
2.1 f(x) = 3x + 2 , g(x) = 4x -1 2.2 f(x) =
5x – 6 , g(x) = x3f ๐ g =………………………………………. f ๐ g
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
g ๐ f =………………………………………. g ๐ f =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
f ๐ f =………………………………………. f ๐ f =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
g ๐ g =………………………………………. g ๐ g =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
2.3 f(x) = x2 , g(x) = x + 5 2.4 f(x) = x3 + 1 , g(x) = x + 2
f ๐ g =………………………………………. f ๐ g =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
g ๐ f =………………………………………. g ๐ f =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
f ๐ f =………………………………………. f ๐ f =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
g ๐ g =………………………………………. g ๐ g =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
2.5 f(x) = 1x, g(x) = 2x + 3 2.6 f(x) =
x2 , g(x) = √ x+5
f ๐ g =………………………………………. f ๐ g =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
g ๐ f =………………………………………. g ๐ f =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
f ๐ f =………………………………………. f ๐ f =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
g ๐ g =………………………………………. g ๐ g =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
2.7 f(x) = x , g(x) = 2x + 5 2.8 f(x) = x + 4 , g(x) = x – 4
f ๐ g =………………………………………. f ๐ g =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
g ๐ f =………………………………………. g ๐ f =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
f ๐ f =………………………………………. f ๐ f =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
g ๐ g =………………………………………. g ๐ g =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
1. สมมต่�ว่�า f เป2นฟ3งก�ช่�นหน70งต่�อหน70ง1.1 ถู�า f(3) = 10 จงหา f-1 (10) =
………………………………………1.2 ถู�า f-1(3) = -4 จงหา f- (-4) =
………………………………………1.3 ถู�า f(10) = 18 จงหา f-1 (18) =
………………………………………1.4 ถู�า f-1(4) = 3 จงหา f- (3) =
…………………………………………1.5 ถู�า f(x) = 5 + 2x จงหา f-1 (10) =
………………………………………
แบบฝึ�กทั�กษะทั!� 7
1.6 ถู�า f(x) = x2 + 4x เม+0อ x ≥ -2 จงหา f-1 (112) = …………………………………….2. จงหาฟ3งก�ช่�นผกผ�นข้อง f 2.1 f(x) = 3x + 1 2.2 f(x) = 5 - x
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….ดั�งน��น f-1(x) =……………………………………. ดั�งน��น f-1(x) =……………………………
2.3 f(x) = 5x + 7 2.4 f(x) = 5 - 3x
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….ดั�งน��น f-1(x) =……………………………………. ดั�งน��น f-1(x) =……………………………
2.5 f(x) = x5 2.6 f(x) = 1
x2
, x > 0 =……………………………………….
=……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….ดั�งน��น f-1(x) =……………………………………. ดั�งน��น f-1(x) =……………………………
2.7 f(x) = 5−2 x1+3 x 2.8 f(x) = 3 –
4x5
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….ดั�งน��น f-1(x) =……………………………………. ดั�งน��น f-1(x) =……………………………
2.9 f(x) = √2−5 x 2.10 f(x) =
x2 + x , x ≥ - 12
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….ดั�งน��น f-1(x) =……………………………………. ดั�งน��น f-1(x) =……………………………
2.11 f(x) = 5 – x2 , x ≥ 0 2.12 f(x) = (4 – x3)5
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….ดั�งน��น f-1(x) =……………………………………. ดั�งน��น f-1(x) =……………………………
2.13 f(x) = x6 2.14 f(x) = 3 – x3
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….
=………………………………………. =……………………………………….ดั�งน��น f-1(x) =……………………………………. ดั�งน��น f-1(x) =……………………………
Related Documents
![3) 2 1. - tanchoom.go.th¹ผนสี่ป_-2561-2564... · [1] แผนพัฒนาท องถิ่นสี่ป พ.ศ. 2561 - 2564 เพิ่มเติม (ฉบับที่](https://static.cupdf.com/doc/110x72/605680dcdb6e69796a67cc8e/3-2-1-aoeaaaaa-2561-2564-1-aaoeaaaaaaa-aaaaaaaaaa.jpg)
