수학의뿘-수학(하) 꾴죀 1.pdf, page 1-520 @ Normalize ( 수학의뿘 …

정답및해설

수학(하)

16수하(409-449)해ok 2016.9.30 9:48 AM 페이지409 Apogee

410

1(x-2)¤ +(y+1)¤ =25

(x+3)¤ +(y-4)¤ =3

2(0 0)

'8=2'2

(-1 5)

4

3x¤ -2x+y¤ -5=0

(x-1)¤ +y¤ =6

(1 0) '6

x¤ +y¤ +2x-8y-8=0

(x+1)¤ +(y-4)¤ =25

(-1 4) 5

(1 0) '6

(-1 4) 5

4O O'

(2 1) (-1 -3)

"√(2+1)¤ +(1+3)¤ =5

r=2 r'=2 r+r'=4<5 O O'

0

r=2 r'=3 r+r'=5 O O'

1

r=5 r'=3 r-r'=2<5<r+r'=8

O O'

2

r=6 r'=1 r-r'=5 O O'

1

r=7 r'=1 r-r'=6>5 O O'

0

0 1 2 1 0

5x¤ +y¤ -4x+2y-4=0

x¤ +y¤ -10x-4y+20=0

(x¤ +y¤ -4x+2y-4)-(x¤ +y¤ -10x-4y+20)=0

6x+6y-24=0

y=-x+4

y=-x+4

유제 1-1(x-3)¤ +y¤ =9 (3 0)

(3 0) (1 2)

"√(1-3)¤ +(2-0)¤ =2'2

(3 0)

2'2

(x-3)¤ +y¤ =8

유제 1-2(1 a) 5

(x-1)¤ +(y-a)¤ =5¤

(-3 0)

(-3-1)¤ +(0-a)¤ =25 a¤ =9

a=3 a>0

유제 1-3AB

{ } (1 3)

A

"√(4-1)¤ +(1-3)¤ ='1å3

(1 3)

'1å3

(x-1)¤ +(y-3)¤ =13

1+5

2

4+(-2)

2

원의 방정식

개념확인코너 본문 p.015

유제 본문 p.016

정답및해설


411

유제 1-4A (1 4) (5 2)

A{ } A(3 3)

A (1 4)

"√(1-3)¤ +(4-3)¤ ='5

(x-3)¤ +(y-3)¤ =5

x=2

y=1 y=5

B(2 1) C(2 5)

B(2 5) C(2 1)

ABC

ABC=;2!;_4_1=2

유제 1-5y (0 b)

r

x¤ +(y-b)¤ =r¤

(3 0) (2 -1)

3¤ +(0-b)¤ =r¤ b¤ +9=r¤ yy

2¤ +(-1-b)¤ =r¤ b¤ +2b+5=r¤ yy

b=2 r¤ =13

x¤ +(y-2)¤ =13

유제 1-6y=x+3

(a a+3) r

(x-a)¤ +(y-a-3)¤ =r¤

(-1 2) (1 4)

(-1-a)¤ +(2-a-3)¤ =r¤

2a¤ +4a+2=r¤ yy

(1-a)¤ +(4-a-3)¤ =r¤

2a¤ -4a+2=r¤ yy

a=0 r¤ =2

x¤ +(y-3)¤ =2

유제 1-72 (3 0) x

(3 2) (3 -2)

OB

A

C5

3

1

2 3 x

y

4+2

2

1+5

2

(3 2)

(3 -2)

2

(x-3)¤ +(y+2)¤ =4또는

(x-3)¤ +(y-2)¤ =4

유제 1-8(a 1) y

(x-a)¤ +(y-1)¤ =a¤

(2 3)

(2-a)¤ +(3-1)¤ =a¤

4-4a+a¤ +4=a¤ 4a=8

a=2

유제 1-94

r r>0 (r -r)

(x-r)¤ +(y+r)¤ =r¤

(r -r) x-y-2=0

r-(-r)-2=0 r=1

(x-1)¤ +(y+1)¤ =1

유제 1-10

2

r r>0

(-r r)

(x+r)¤ +(y-r)¤ =r¤

(-1 2)

(-1+r)¤ +(2-r)¤ =r¤

r¤ -6r+5=0 (r-1)(r-5)=0

r=1 r=5

1 5

p_1¤ +p_5¤ =26p

유제 1-11x¤ +y¤ +4x-2ky=4

x¤ +4x+4+y¤ -2ky+k¤ =4+4+k¤

(x+2)¤ +(y-k)¤ =k¤ +8

x

y

(-r, r)

(-r, r)

(-1, 2)

O

2

-2

(3, 2)

(3, -2)

3

y

xO


412

(-2 k)

"√k¤ +8

(-2 k) '5

"√(-2)¤ +k¤ ='5 4+k¤ =5

k¤ =1

"√k¤ +8='ƒ1+8=3

유제 1-12x¤ +y¤ +2ax-4ay+10a-8=0

(x+a)¤ +(y-2a)¤ =5a¤ -10a+8

5a¤ -10a+8=5(a-1)¤ +3

a=1

3 3p

유제 1-13x¤ +y¤ +Ax+By+C=0

(0 2) (0 6) (4 2)

- 4B=-32 B=-8

B=-8 C=12

B=-8 C=12 A=-4

x¤ +y¤ -4x-8y+12=0

(x-2)¤ +(y-4)¤ =8

(2, 4) 2'2

x¤ +y¤ +Ax+By+C=0

(0 0) (1 -1) (1 2)

A-B=-2 A+2B=-5

A=-3 B=-1

x¤ +y¤ -3x-y=0

{x-;2#;}¤ +{y-;2!;}¤ =;2%;

C=0 yy

A-B+C=-2 yy

A+2B+C=-5 yy

‡

2B+C=-4 yy

6B+C=-36 yy

4A+2B+C=-20 yy

‡

{;2#;, ;2!;}

유제 1-14x¤ +y¤ +4x-2y+k=0

(x+2)¤ +(y-1)¤ =5-k

5-k>0

k<5

k 1 2 3 4 4

유제 1-15x¤ +y¤ +6x+2ky+8k=0

(x+3)¤ +(y+k)¤ =k¤ -8k+9

18p

(k¤ -8k+9)p=18p

k¤ -8k+9=18

k¤ -8k-9=0

(k+1)(k-9)=0

k=9 k>0

유제 1-16x¤ +y¤ =r¤ (0 0) A(2 4)

"√2¤ +4¤ =2'5

A

r

2'5+r

2'5+r=2'5+1

r=1

유제 1-17"√(a+6)¤ +(b+1)¤ ="√{a-(-6)}¤ +√{b-(-1)}¤

(-6 -1) P(a b)

(2 5) (-6 -1)

"√(2+6)¤ +(5+1)¤ ='∂100=10

3

(-6 -1) P

10+3=13

(-6, -1) (2, 5)

10 3P

r

(0, 0) A(2, 4)

' 52

'1å0

2


413

유제 1-18P (x y)

PA”="√(x+3)¤ +y¤ PB”="√(x-2)¤ +y¤

PA” PB”=3 2 2PA”=3PB”

4PA” ¤ =9PB” ¤

4{(x+3)¤ +y¤ }=9{(x-2)¤ +y¤ }

4x¤ +24x+36+4y¤ =9x¤ -36x+36+9y¤

x¤ +y¤ -12x=0

(x-6)¤ +y¤ =36

(6 0) 6

P

2p_6=12p

유제 1-19P (x y)

AP”="√(x+1)¤ +y¤ BP”="√(x-1)¤ +y¤

AP” BP”=2 1 AP”=2BP”

AP” ¤ =4BP” ¤

(x+1)¤ +y¤ =4{(x-1)¤ +y¤ }

x¤ +2x+1+y¤ =4x¤ -8x+4+4y¤

x¤ +y¤ -:¡3º:x+1=0

{x-;3%;}¤ +y¤ =:¡9§:

PAB

P

PAB

△PAB…;2!;_2_;3$;=;3$;

유제 1-20x¤ +y¤ -6x+8y+21=0

(x-3)¤ +(y+4)¤ =4

(0 0) (3 -4)

5 a 2

a+2=5

a=3

|a-2|=5

a-2=—5

a a=7

A B

P

O x

y

1-13

5

3

4

|a-2|<5<a+2

⁄ |a-2|<5 -5<a-2<5

a 0<a<7

¤ 5<a+2 a>3

⁄ ¤ 3<a<7

유제 1-21

(x¤ +y¤ -4)-(x¤ +y¤ -4x+ky)=0

4x-ky-4=0

y=;k$;x-;k$;

y=x+3

;k$;_1=-1 k=-4

유제 1-22

(x¤ +y¤ +3x+ay-6)-(x¤ +y¤ -2x+2y-2)=0

5x+(a-2)y-4=0 yy

x¤ +y¤ -2x+2y-2=0

O' (x-1)¤ +(y+1)¤ =4

O O'

O' (1 -1)

5-(a-2)-4=0

a=3

유제 1-23x¤ +y¤ =4 x¤ +y¤ -4x+3y-9=0

O O' A B

OO' AB C

(x¤ +y¤ -4)-(x¤ +y¤ -4x+3y-9)=0

4x-3y+5=0

x¤ +y¤ =4 O(0 0)

A

BC

O

2

O'

OO'

㉠

16수하(409-449)해ok 2016.9.30 8:8 PM 페이지413 Apogee

414

OC”= =;5%;=1

OCA O’A”=2 OC”=1

AC”=øπOA” ¤ -OC” ¤ ='ƒ4-1='3

AB”=2AC”=2'3

유제 1-24x¤ +y¤ =9 x¤ +y¤ +3x-4y+k=0

O O' A B

OO' AB C

AB”=2'5 AC”=;2!;AB”='5

OA”=3 OCA

OC”=øπOA” ¤ -AC” ¤ ="√3¤ -('5)¤ =2

(x¤ +y¤ -9)-(x¤ +y¤ +3x-4y+k)=0

-3x+4y-9-k=0

x¤ +y¤ =9 O(0 0)

OC”=2

OC”= =2 |-9-k|=10

-9-k=—10

k=1 k>0

유제 1-25(x-1)¤ +(y-4)¤ =13

x¤ +y¤ -2x-8y+4=0

(x¤ +y¤ -6x+2)+k(x¤ +y¤ -2x-8y+4)=0

yy

(1 0)

-3+3k=0 k=1

k=1

(x¤ +y¤ -6x+2)+(x¤ +y¤ -2x-8y+4)=0

2x¤ +2y¤ -8x-8y+6=0

x¤ +y¤ -4x-4y+3=0

유제 1-26

|-9-k|

"ç(-3)¤ +4¤

A3

B

OC

O'

' 5

|5|

"√4¤ +(-3)¤

(x¤ +y¤ +2x-4y-2)+k(x¤ +y¤ -2x-1)=0

yy

-2-k=0 k=-2

k=-2

(x¤ +y¤ +2x-4y-2)-2(x¤ +y¤ -2x-1)=0

-x¤ -y¤ +6x-4y=0

x¤ +y¤ -6x+4y=0

(x-3)¤ +(y+2)¤ =13

'1å3

p_('1å3)¤ =13p

1-1(x-1)¤ +(y+3)¤ =4

(3 2) (2 2)

"≈(2-3≈)¤ +(2≈-2)¤ =1

(x-3)¤ +(y-2)¤ =1

(-2 0) (2 0)

{ } (0 0)

(2 0) (0 0)

2

x¤ +y¤ =4

1-2(x+1)¤ +(y-3)¤ =9

(-1, 3)

(-1 3) (2 1)

"√(2+1)¤ +(1-3)¤ ='1å3

(x+1)¤ +(y-3)¤ =13

1-3(2 3)

(-6 -3) 10

3 r

3+r=10

r=7

1-4y=mx (-2 3)

0+0

2

-2+2

2

연습문제 본문 p.034


415

3=m_(-2)

m=-;2#;

1-5y=x

(a, a) '2

(x-a)¤ +(y-a)¤ =('2 )¤

(1 3)

(1-a)¤ +(3-a)¤ =2, a¤ -4a+4=0

(a-2)¤ =0 a=2

(x-2)¤ +(y-2)¤ =2

1-6x¤ +y¤ -2x-4y+1=0

(x-1)¤ +(y-2)¤ =4

(1 2) 2

a+b+r=1+2+2=5

1-7x¤ +y¤ +2ax+2y-4=0

(x+a)¤ +(y+1)¤ =a¤ +5

(2 b)

a=-2 b=-1

r

r¤ =a¤ +5=(-2)¤ +5=9

a¤ +b¤ +r¤ =4+1+9=14

1-8x¤ +y¤ -2x+4y+k=0

(x-1)¤ +(y+2)¤ =5-k

y

'ƒ5-k=1 k=4

1-9x¤ +y¤ +Ax+By+C=0

(0 0) (1 3) (4 2)

C=0 yy

A+3B+C=-10 yy

4A+2B+C=-20 yy

‡

A+3B=-10 4A+2B=-20

A=-4 B=-2

x¤ +y¤ -4x-2y=0

1-10x¤ +y¤ -8x+2y-3k+2=0

(x-4)¤ +(y+1)¤ =3k+15

3k+15>0

k>-5

1-11x¤ +y¤ +2x-8y-10=0

(x+1)¤ +(y-4)¤ =27 A(-1 4)

x¤ +y¤ -6x-1=0

(x-3)¤ +y¤ =10 B(3 0)

A B

AB

{ } (1 2)

(1 2) (-1 4)

"√(-1-1)¤ +(4-2)¤ =2'2

(1 2) 2'2

(x-1)¤ +(y-2)¤ =8

x¤ +y¤ -2x-4y-3=0

a=-2 b=-4 c=-3

a+b+c=-9

1-12y=x+3

(a a+3) x

|a+3|

(x-a)¤ +(y-a-3)¤ =(a+3)¤

(1, 2)

(1-a)¤ +(-a-1)¤ =(a+3)¤

a¤ -6a-7=0

(a+1)(a-7)=0

a=-1 a=7

5

a=-1

(x+1)¤ +(y-2)¤ =4

4+0

2

-1+3

2


416

1-13O O' (-2 -1) (1 3)

5 1 2

PQ

5+1+2=8

PQ

5-1-2=2

PQ

8+2=10

1-14(x-7)¤ +(y-5)¤ =4 2

(7 5) A(-5 0) 13

d

13+2=15 13-2=11

11…d…15

d

⁄ d=12 13 14 P 2

¤ d=11 15 P 1

d P

3_2+2_1=8

1-152

r r>0

(-r, r)

y

xO

(-r, r)

(-r, r)

(-2, 1)

A(-5, 0)

132

(7, 5)PP'

O'P'

15 2

P QQ'

O

(x+r)¤ +(y-r)¤ =r¤

(-2, 1)

(-2+r)¤ +(1-r)¤ =r¤

r¤ -6r+5=0

(r-1)(r-5)=0

r=1 r=5

(-1, 1), (-5, 5)

"√(-5+1)¤ +(5-1)¤ =4'2

1-16x¤ +y¤ -8x-6y-2k+30=0

(x-4)¤ +(y-3)¤ =2k-5

2k-5>0 k>;2%; yy

1

3

2k-5<3¤ , 2k<14

k<7 yy

;2%;<k<7

k

3+4+5+6=18

1-17x¤ +y¤ -8kx+4ky+20k-9=0

(x-4k)¤ +(y+2k)¤ =20k¤ -20k+9

20k¤ -20k+9 k

20k¤ -20k+9=20(k¤ -k)+9

20k¤ -20k+9=20{k-;2!;} ¤ +4

k=;2!;

(x-2)¤ +(y+1)¤ =4

(2 -1) 2

a+b+r=2+(-1)+2=3

1-18AP”:BP”=3:1 AP”=3BP”

AP” ¤ =9BP” ¤

P (x y)

(x-2)¤ +y¤ =9 {(x+2)¤ +y¤ }


417

x¤ +y¤ +5x+4=0

{x+ }¤+y¤ =;4(;

APB

APB… _4_ =3

1-19(x-1)¤ +(y-1)¤ =1

x¤ +y¤ -2x-2y+1=0

(x¤ +y¤ -1)+k(x¤ +y¤ -2x-2y+1)=0 yy

(0 0)

-1+k=0 k=1

k=1

(x¤ +y¤ -1)+(x¤ +y¤ -2x-2y+1)=0

x¤ +y¤ -x-y=0

a=-1 b=-1 c=0

a+b+c=-2

1-20

A B P (1 0) x

3

(x-1)¤ +(y-3)¤ =9

x¤ +y¤ -2x-6y+1=0

AB x¤ +y¤ =9

x¤ +y¤ -2x-6y+1=0

(x¤ +y¤ -9)-(x¤ +y¤ -2x-6y+1)=0

OP

A

B-3 3

(1, 3)

-3

1 x

y

3

2

1

2

y

xO

A(2, 0)

B(-2, 0)

P 3;2;

{ , 0}5-;2;

5

2

x+3y-5=0

1-21x y y=x

y=-x

y=x¤ -6 y=x

y=-x

⁄ x¤ -6=x x¤ -x-6=0

(x+2)(x-3)=0 x=-2 x=3

¤ x¤ -6=-x x¤ +x-6=0

(x+3)(x-2)=0 x=-3 x=2

⁄ ¤ m=4

(-2 -2) (3 3) (-3 3) (2 -2)

2 3 3 2

2_p_3¤ +2_p_2¤ =26p

n=26

m+n=4+26=30

1-22AB M

M { } M(3 4)

B’M”="√(3-2)¤ +(4-5)¤ ='2

PAB

P’A”¤ +PB”¤ =2(P’M”¤ +B’M”¤ )=2(P’M”¤ +2)

PA”¤ +PB”¤ P’M”¤

P P' PM

M(3, 4)

x¤+y¤ =1O

P'

P

B(2, 5)

A(4, 3)

x

y

3+5

2

4+2

2

O x

y

y=x y=-x

y=x¤ -6


418

P’M”…P’'M”=P’'O”+O’M”=1+"√3¤ +4¤ =6

P’A”¤ +P’B”¤

PA”¤ +PB” ¤ =2(P’M” ¤ +2)…2(6¤ +2)=76

1-23

B

ABCD P (x y)

CP”¤ +DP”¤ =AP”¤

(x-1)¤ +y¤ +(x-1)¤ +(y-1)¤ =x¤ +(y-1)¤

x¤ +y¤ -4x+2=0

(x-2)¤ +y¤ =2

P (2 0)

'2

B(0 0) P

P P'(2+'2 0)

2+'2

1-24

P Q

a b

PR”=8-a-b

QR”=9-a-b

PRQ

(8-a-b)¤ +(9-a-b)¤ =(a+b)¤

a+b=t 0<t…8

(8-t)¤ +(9-t)¤ =t¤

A

P

R

a

a

B C

D9

8 Q b

b

2+'22-'2

(2, 0)B(O) P' x

y

(x-2)¤ +y¤ =2

A(0, 1)

B(0, 0) C(1, 0)

D(1, 1)

P(x, y)

O

y

x

t¤ -34t+145=0 (t-5)(t-29)=0

t=a+b=5 0<t…8

CD”=8 2a…8, 2b…8

a…4, b…4 yy

a+b=5 a…4, b…4

5-b…4, 5-a…4

aæ1, bæ1 yy

1…a…4, 1…b…4

S

S=pa¤ +pb¤

S=p{a¤ +(5-a)¤ }

a+b=5

S=p[2{a-;2%;}2+:™2∞:]

1…a…4 a=1 a=4

S

p[2{1-;2%;}2+:™2∞:]=17p

1-25O O'

A B OO' AB C

AB”=2'6 AC”=;2!;AB”='6

O’'A”=2'2 O'CA

O’'C”=øπO'A”¤ -AC”¤ ="√(2'2)¤ -('6)¤ ='2

(x¤ +y¤ +2x+2y-k)-(x¤ +y¤ -2x-2y-6)=0

4x+4y+6-k=0

O' (1 1)

O’'C”='2

O’'C”= ='2 |14-k|=8

14-k=—8 k=6 k=22

k 28

|4+4+6-k|

"√4¤ +4¤

a=1 a=4a=5

2


419

1y=2x+2 x¤ +y¤ =2

x¤ +(2x+2)¤ =2

5x¤ +8x+2=0

D

=4¤ -5¥2=6>0

y=x-2 x¤ +y¤ =2

x¤ +(x-2)¤ =2

x¤ -2x+1=0

D

=(-1)¤ -1¥1=0

y=- x+3 x¤ +y¤ =2

x¤ +{- x+3} 2=2

5x¤ -12x+28=0

D

=(-6)¤ -5¥28=-104<0

2ax-y+2=0 x¤ +y¤ =1

(0 0) ax-y+2=0

1

=1

"√a¤ +1=2

a¤ +1=4 a¤ =3

a='3 a=-'3

'3 -'3

|0+0+2|

"√a¤ +(-1)¤

D

4

1

2

1

2

D

4

D

4

3x¤ +y¤ =9 2

y=mx—r"√m¤ +1 m=2, r=3

y=2x—3"√2¤ +1=2x—3'5

y=2x—3'5

4x¤ +y¤ =1 -1

y=mx—r"√m¤ +1 m=-1 r=1

y=-x—"√(-1)¤ +1=-x—'2

y=-x—'2

5x¤ +y¤ =5 (1 2)

x¡x+y¡y=r¤ x¡=1 y¡=2 r¤ =5

x+2y=5

x+2y=5

6x¤ +y¤ =4 ('3 -1)

x¡x+y¡y=r¤

x¡='3, y¡=-1, r¤ =4

'3x-y=4

'3-y=4

유제 2-1y='3x+k x¤ +y¤ =4

4x¤ +2'3kx+k¤ -4=0

D

=('3k)¤ -4(k¤ -4)=-k¤ +16

D>0

=-k¤ +16>0

(k+4)(k-4)<0

-4<k<4

D=0

=-k¤ +16=0

(k+4)(k-4)=0

k=-4또는 k=4

D

4

D

4

D

4

원과 직선


유제 본문 p.044


420

D<0

=-k¤ +16<0

(k+4)(k-4)>0

k<-4또는 k>4

유제 2-2x¤ +y¤ -2x-4y+1=0

(x-1)¤ +(y-2)¤ =4

(1 2) x+2y+k=0

d

d=

d=

2

d>2

>2

|k+5|>2'5

k<-5-2'5또는 k>-5+2'5

유제 2-3(1 -1) ax+y+a=0

d

d= =

'2

d='2

='2

|2a-1|='2"√a¤ +1

2a¤ -4a-1=0

a -;2!;

유제 2-4(2 3) 2x-3y+k=0

d

d= =

13p '1å3

d='1å3

='1å3|k-5|

'1å3

|k-5|

'1å3

|4-9+k|

"√2¤ +(-3)¤

|2a-1|

"√a¤ +1

|2a-1|

"√a¤ +1

|a¥1+1¥(-1)+a|

"√a¤ +1¤

|k+5|

'5

|k+5|

'5

|1¥1+2¥2+k|

"√1¤ +2¤

D

4

|k-5|=13

k=-8 k=18

k

-8+18=10

유제 2-5(0 -4) y 3

r(r>0)

(-r -4)

(-r -4)

3x-4y+8=0

r

=r

|-3r+24|=5r

-3r+24=—5r

r=3 r>0

(x+3)¤ +(y+4)¤ =9

유제 2-6x¤ +y¤ -4x-2y+4=0

(x-2)¤ +(y-1)¤ =1

(2 1)

x+y-5=0

d

d= ='2

최댓값 '2+1 최솟값 '2-1

유제 2-7ABP P

AB

AB x+y=4 x+y-4=0

(0, 0) AB

=2'2

x¤ +y¤ =4 2

P AB

2'2-2

AB”="√(-4)¤ +4¤ =4'2

ABP

;2!;¥4'2¥(2'2-2)=8-4'2

|-4|

øπ1¤ +1¤

|2+1-5|

"√1¤ +1¤

O

1

2

x+y-5=0

x

y

' 2

|-3r+16+8|

"√3¤ +(-4)¤


421

유제 2-8

A B

O(0 0) 2x-y-5=0

H

O’H”= ='5

OHA

A’H”="√3¤ -('5)¤ =2

AB”=2AH”=4

유제 2-9

A B

C(1 2) x-y+2=0

H

CH”= =

HBC B’H”=;2!;AB”=

r=CB”=æ≠{ }¤ +{ } ¤ =2

유제 2-10

x-3y+6=0 y= x+2

m

¥m=-1 m=-3

x¤ +y¤ =4 2

y=-3x—2"√(-3)¤ +1=-3x—2'∂10

1

3

1

3

'2

2

'1 å4

2

'1å4

2

'2

2

|1-2+2|

"√1¤ +(-1)¤

A

H

BC

O

2

x-y+2=0

1-2 x

y

|-5|

"√2¤ +(-1)¤

O

3 A

H

B

다른 풀이

y=-3x+n

3x+y-n=0 (0, 0)

2

=2 n=—2'∂10

y=-3x—2'∂10

유제 2-11(2, -1)

2x-y=5 yy

(a, b)

ax+by=5 yy

2a-b=0 b=2a

= =2 ( a+0)

유제 2-12x¤ +y¤ =25 (-3 4)

-3x+4y=25

3x-4y+25=0

O O

O (5 5) 3x-4y+25=0

O

=4

p_4¤ =16p

유제 2-13x¤ +y¤ -2x-4y=0

(x-1)¤ +(y-2)¤ =5

C(1, 2) CP

=2

m

4-2

2-1

x

(x-1)¤ +(y-2)¤ =5

2

O

P(2, 4)

1

y

C(1, 2)

|15-20+25|

"√3¤ +(-4)¤

2a

a

b

a

|-n|

"√3¤ +1¤


422

2¥m=-1 m=-;2!;

y-4=-;2!;(x-2)

y=-;2!;x+5

y=-;2!;x+5 x 10, y 5

a=10, b=5

a+b=15

다른 풀이

(x-1)¤ +(y-2)¤ =5 (2, 4)

(2-1)(x-1)+(4-2)(y-2)=5

y=-;2!;x+5

y=-;2!;x+5 x 10, y 5

a=10, b=5

a+b=15

유제 2-14(x-4)¤ +y¤ =1

(4, 0)

x¤ +y¤ =4 (x¡, y¡)

x¡x+y¡y=4 yy

(4, 0)

4x¡=4 x¡=1

(x¡, y¡) x¤ +y¤ =4

x¡¤ +y¡¤ =4

x¡=1 y¡=—'3

(1, '3), (1, -'3)

y=- x+ y= x-

m¡m™={- }¥{ }=-;3!;1

'3

1

'3

4

'3

1

'3

4

'3

1

'3

24

2

-2

-2

O

y

(x¡, y¡)

x

다른 풀이

(x-4)¤ +y¤ =1

(4, 0)

x¤ +y¤ =4 m

y=m(x-4) mx-y-4m=0

(0, 0) mx-y-4m=0

2

=2

|-4m|=2"√m¤ +1, 16m¤ =4m¤ +4

m=—

m¡m™=-;3!;

유제 2-15P T

PT”='1å1

CT” PT”

CTP

CTP=90˘

r r>0

CT”=r

CP”="√{-4-(-2)}¤ +√(5-1)¤ ='2å0

r="√('2å0)¤ -('1å1)¤ =3

유제 2-16

C T

CT” PT”

CTP CTP=90˘

P(4 a)

C(1 2)

CP”="√(4-1)¤ +√(a-2)¤ ="√a¤ -4a+13

PT”=4 CT”='2

CP” ¤ =PT” ¤ +CT” ¤

a¤ -4a+13=16+2 a¤ -4a-5=0

(a+1)(a-5)=0

a=5 a>0

P(4, a)

4

C(1, 2)

T

P(-4, 5)

C(-2, 1)

T'∂ 11

1

'3

|-4m|

"√m¤ +(-1)¤


423

2-1(0 0) d

y=x+2 x-y+2=0

d= ='2<'5

2

d= ='5

1

d= =2'2>'5

0

2-2x¤ +y¤ =r¤ x-2y+2'5=0

(0 0)

r

r= =2

2-3y=2x+a x¤ +y¤ =5

x¤ +(2x+a)¤ =5 5x¤ +4ax+(a¤ -5)=0

D D>0

=(2a)¤ -5(a¤ -5)=25-a¤ >0

-5<a<5

다른 풀이

x¤ +y¤ =5 (0, 0) 2x-y+a=0

'5

<'5

|a|<5 -5<a<5

2-4x¤ +y¤ =r¤ 3x+4y-15=0

(0 0) 3x+4y-15=0

r

=3>r

r>0

0<r<3

|-15|

"√3¤ +4¤

|a|

"√2¤ +(-1)¤

D

4

|2'5|

"√1¤ +(-2)¤

|-4|

"√1¤ +(-1)¤

|-5|

"√2¤ +1¤

|2|

"√1¤ +(-1)¤

a=0 b=3

a¤ +b¤ =9

2-5r r>0

x¤ +y¤ +ax+by+c=0 3x+4y-1=0

(1, 2)

r

r= =2

(x-1)¤ +(y-2)¤ =2¤

x¤ +y¤ -2x-4y+1=0

a=-2, b=-4, c=1

a+b+c=-5

2-6(1 -1)

x+y+8=0

=4'2

P

x+y+8=0

4'2+3

4'2-3

(4'2+3)(4'2-3)=23

2-7x¤ +y¤ =4 2

y=2x—2"√2¤ +1=2x—2'5

x¤ +y¤ =13 (-2 3)

-2x+3y=13

2x-3y+13=0

2-8x¤ +y¤ =5 y=mx+n

y=2x-3 m=2

x¤ +y¤ =5 2

y=2x—'5"≈2¤ +1=2x—5

n=—5

m¤ +n¤ =4+25=29

|1-1+8|

"√1¤ +1¤

x+y+8=0

P

P3

(1, -1)3

' 24

|3+8-1|

"√3¤ +4¤



424

2-9x¤ +y¤ =80

(8 -4)

8x-4y=80

2x-y=20

x=0 y=-20

y=0 x=10

;2!;¥10¥20=100

2-10(-1 3) (2 4)

=;3!;

-3

y-4=-3(x-2)

3x+y-10=0

a=3 b=-10

a+b=-7

다른 풀이

(x+1)¤ +(y-3)¤ =10 (2 4)

(2+1)(x+1)+(4-3)(y-3)=10

3x+y-10=0

a=3 b=-10

a+b=-7

2-112

x¤ +y¤ =4

y=mx-4

(m¤ +1)x¤ -8mx+12=0

D

Dæ0

=(-4m)¤ -12(m¤ +1)=4(m¤ -3)æ0

mæ'3 m>0

다른 풀이

(0, 0)

mx-y-4=0 2

D

4

4-3

2-(-1)

O 10

-20

2x-y=20

(8, -4)

x

y

…2

"√m¤ +1æ2

m¤ +1æ4 mæ'3 m>0

2-12(2 0)

(2 b) |b|

(x-a)¤ +(y-b)¤ =b¤

4x-3y+16=0

(2 b) 4x-3y+16=0

|b|

= =|b|

|24-3b|=5|b|

b¤ +9b-36=0

(b+12)(b-3)=0

b=-12 b=3

12 3

p¥12¤ -p¥3¤ =135p

2-13

A B

y=x+k

H

AH”=;2!;AB”=;2!;¥2=1

OAH

OH”="√2¤ -1¤ ='3 yy

x¤ +y¤ =4 (0, 0) y=x+k

x-y+k=0 OH”

OH”= = yy

= '3= , |k|='6

k=—'6

2-14x¤ +y¤ =5 yy

y=x-3 yy

|k|

'2

|k|

'2|k|

"√1¤ +(-1)¤

O

2

2

y y=x+k

x-2

-2

HA

B

|24-3b|

5

|8-3b+16|

"√4¤ +(-3)¤

|-4|

"√m¤ +(-1)¤


425

x¤ -3x+2=0 (x-1)(x-2)=0

x=1 x=2

x=1 y=-2

x=2 y=-1

(1 -2) (2 -1)

⁄ (1 -2) x-2y=5

¤ (2 -1) 2x-y=5

a+b+c+d=1+(-2)+2+(-1)=0

2-153 (x-1)¤ +(y+1)¤ =25

y=3x+k (k )

(1 -1) 3x-y+k=0

5

= =5

|k+4|=5'1å0

k=-4-5'1å0 k=-4+5'1å0

y

(-4-5'1å0)+(-4+5'1å0)=-8

다른 풀이

3 (x-1)¤ +(y+1)¤ =25

y-(-1)=3(x-1)—5"√3¤ +1

y=3x-4—5'1å0

y

(-4-5'1å0)+(-4+5'1å0)=-8

2-16x¤ +y¤ =4 ('3 -1)

'3x-y=4

(x-a)¤ +y¤ =1

(a 0) '3x-y-4=0

1

= =1

|'3a-4|=2

a= a=2'3

a

¥2'3=42'3

3

2'3

3

|'3a-4|

2

|'3a-4|

"√('3)¤ +(-1)¤

|k+4|

'1å0

|3+1+k|

"√3¤ +(-1)¤

2-17A(0 -6) B(8 0)

;8{;+ =1

3x-4y-24=0

x¤ +y¤ -4x-6y+9=0

(x-2)¤ +(y-3)¤ =4

C(2 3) C AB

H

CH”= =6

PAB AB

AB”="√6¤ +8¤ =10

PH” PAB

PAB

;2!;¥10¥(6+2)=40

2-18x¤ +y¤ =9 (x¡ y¡)

x¡x+y¡y=9 yy

(6 0)

6x¡=9 x¡=;2#;

(x¡ y¡) x¤ +y¤ =9

x¡¤ +y¡¤ =9 yy

x¡=;2#;

;4(;+y¡¤ =9 y¡=—

;2#;x+ y=9

x+'3y-6=0

;2#;x- y=9

x-'3y-6=0

3'3

2

3'3

2

O 6 x

y

' 32

' 3-2

3'3

2

|6-12-24|

"√3¤ +(-4)¤

O

BC

A

H

P

3

-6

82 x

y

y

-6


426

y 2'3 -2'3

;2!;¥4'3¥6=12'3

다른 풀이

m

(6 0) y=m(x-6)

mx-y-6m=0

(0 0) mx-y-6m=0

x¤ +y¤ =9 3

=3

|-6m|=3"√m¤ +1

3m¤ =1 m=—

x+'3y-6=0

x-'3y-6=0

y 2'3 -2'3

;2!;¥4'3¥6=12'3

2-19C

B, D

AC”

="√(-2-2)¤ +(3-1)¤

=2'5 yy

ABCD

AC

AC”='2r yy

= '2r=2'5

r='ß10

2-20

C A C

B D

60˘

BAC= DAC

=30˘

O

1

x

y

A(2, -3)

D

BC

30˘

30˘

r

A(-2, 3)

C(2, 1)

D

B

'3

3

|-6m|

"√m¤ +(-1)¤

ABC

AC”="√(0-2)¤ +√{1-(-3)}¤ =2'5

AB”= AC”= ¥2'5='1å5

2-21y=m(x+1) m

(-1 0)

⁄

⁄ (x-1)¤ +y¤ =1 (1 0)

mx-y+m=0

1

⁄ = =1

⁄ |2m|="√m¤ +1

⁄ 3m¤ =1 m= m>0

¤ x m=0

⁄ ¤ m

0<m<

a=0 b=

a¤ +b¤ =;3!;

2-22x¤ +y¤ -2x+2y=0

(x-1)¤ +(y+1)¤ =2 yy

=k k

y+3=k(x+1) yy

k

(-1 -3)

k

O1

-1

(-1, -3)

kx-y+k-3=0

x

y

y+3

x+1

'3

3

'3

3

'3

3

|2m|

"√m¤ +1

|m+m|

"√m¤ +(-1)¤

O-1 1

y=m(x+1)

x

y

'3

2

'3

2


427

(1 -1)

'2

= ='2

|2k-2|='2"√k¤ +1

k¤ -4k+1=0 yy

k M m

Mm=1

2-23

OPQ

OP”=p, PQ”=q p>0 q>0

q¤ =p¤ -1

p¤ q¤

p x¤ +y¤ =1 O(0, 0)

x+y-2=0

='2

q¤ ('2)¤ -1=1

q 1 q>0

PQ 1

참고

(x-a)¤ +(y-b)¤ =r¤ P(x¡, y¡)

T

CP”¤ =PT”¤ +CT”¤

PT”="√CP”¤ -CT”¤

l="√(x¡-a)¤ +√(y¡-b)¤ -r¤

T

rl

P(x¡, y¡) C(a, b)

|-2|

"√1¤ +1¤

Q

P

O

1

x+y-2=0

p

q

|2k-2|

"√k¤ +1

|k+1+k-3|

"√k¤ +(-1)¤

2-24P(a 5)

A(x¡ y¡) B(x™ y™)

x¡x+y¡y=1 x™x+y™y=1

(a 5)

ax¡+5y¡=1 ax™+5y™=1

ax+5y=1 A B

AB

ax+5y=1

a

Q {0 ;5!;} Q

;5!;

O

Q

5

-1

1

1

-1 x

yP(a, 5)

A(x¡, y¡)

B(x™, y™)


428

1(0, 0) (0+2, 0-1)

(2, -1)

(1, 3) (1+2, 3-1)

(3, 2)

(-2 1) (-2+2 1-1)

(0, 0)

(2, -1) (3, 2) (0, 0)

2(4 -3) (4-2 -3+3)

(2, 0)

(4, -3) (4+1, -3+0)

(5, -3)

(4, -3) (4+0, -3-5)

(4, -8)

(2, 0) (5, -3) (4, -8)

3x x+2, y y-1

y=2x y-1=2(x+2)

y=2x+5

x-2y+3=0

(x+2)-2(y-1)+3=0

x-2y+7=0

x¤ +y¤ =4

(x+2)¤ +(y-1)¤ =4

(x+1)¤ +(y-2)¤ =1

{(x+2)+1}¤ +{(y-1)-2}¤ =1

(x+3)¤ +(y-3)¤ =1

y=2x+5

x-2y+7=0

(x+2)¤ +(y-1)¤ =4

(x+3)¤ +(y-3)¤ =1

y -51111⁄

x 11111⁄

x -2, y 311111555444⁄

x 2, y -1111113333⁄

x 2, y -111111331⁄

x 2, y -111111331⁄

4(-4, 3) (-4, -3) y -y

(-4, 3) (4, 3) x -x

(-4, 3) (4, -3)

x -x, y -y

(-4, 3) (3, -4) x y, y x

(-4, -3) (4, 3)

(4, -3) - (3, -4)

5y=2x-1 -y=2x-1 y -y

y=-2x+1

y=2x-1 y=2(-x)-1 x -x

y=-2x-1

y=2x-1 -y=2(-x)-1

x -x, y -y

y=2x+1

y=2x-1 x=2y-1

x y, y x

y=;2!;x+;2!;

y=-2x+1 y=-2x-1

y=2x+1 y=;2!;x+;2!;

6(x-2)¤ +(y+1)¤ =1

(x-2)¤ +(-y+1)¤ =1 y -y

(x-2)¤ +(y-1)¤ =1

(x-2)¤ +(y+1)¤ =1

(-x-2)¤ +(y+1)¤ =1 x -x

(x+2)¤ +(y+1)¤ =1

(x-2)¤ +(y+1)¤ =1

(-x-2)¤ +(-y+1)¤ =1

x -x, y -y

(x+2)¤ +(y-1)¤ =1

(x-2)¤ +(y+1)¤ =1

(y-2)¤ +(x+1)¤ =1

x y, y x

(x+1)¤ +(y-2)¤ =1

(x-2)¤ +(y-1)¤ =1

(x+2)¤ +(y+1)¤ =1

(x+2)¤ +(y-1)¤ =1

(x+1)¤ +(y-2)¤ =1

y=x112⁄

121⁄

y121⁄

x121⁄

y=x112⁄

121⁄

y121⁄

x121⁄

y=x112⁄

121⁄y121⁄x121⁄도형의이동



429

유제 3-1(2, 3) x 2 y

a (b, 1)

2+2=b, 3+a=1

a=-2, b=4

(1, 6) x 2 y

-2

(1+2, 6-2) (3, 4)

(p, q)

(p, q) x 2 y

-2 (-2, 1)

p+2=-2, q-2=1

p=-4, q=3

(-2, 1)

(-4, 3)

유제 3-22x+y-3=0 x -2 y

k x

x+2 y y-k

2(x+2)+(y-k)-3=0

2x+y+1-k=0

2x+y+5=0

1-k=5 k=-4

유제 3-3y=ax+b x -3 y

1 x

x+3, y y-1

y-1=a(x+3)+b

y=ax+3a+b+1

y=;3!;x-2 y

-1 y

a_;3!;=-1, 3a+b+1=-2

a=-3, b=6

a+b=3

유제 3-4(-3 2) (-2 4) x

1 y 2

y=x¤ +4x-1

=(x+2)¤ -5

x x-1 y y-2

y-2=(x-1+2)¤ -5

y=(x+1)¤ -3

(-1 -3)

a=-1 b=-3

a+b=-4

유제 3-5y=x¤ -2kx+k+2

=(x-k)¤ -k¤ +k+2

x 2 y

-1 x x-2 y

y+1

y+1=(x-2-k)¤ -k¤ +k+2

y=(x-2-k)¤ -k¤ +k+1

(k+2 -k¤ +k+1) y=2x-9

-k¤ +k+1=2(k+2)-9

k¤ +k-6=0

(k+3)(k-2)=0

k=-3 k=2

k 2

유제 3-6(x+1)¤ +(y-2)¤ =4 x m

y n x

x-m y y-n

(x-m+1)¤ +(y-n-2)¤ =4

x¤ +y¤ =4

-m+1=0 -n-2=0

m=1, n=-2

m+n=-1

y

x¤ +y¤ =4

(x+1)¤ +(y-2)¤ =4

xO

2

-1

유제 본문 p.071


430

유제 3-7(-2 1) (1 2) x

3 y 1

(x-a)¤ +(y-a+1)¤ =4 x x-3 y

y-1

(x-3-a)¤ +(y-1-a+1)¤ =4

(x-3-a)¤ +(y-a)¤ =4

y=2x-3

(3+a a)

a=2(3+a)-3 a=-3

유제 3-8(-2, 1) y

(2, 1)

(-2, -1)

유제 3-9A(2, 1) y=x

B(1, 2)

A(2, 1) y=-x

C(-1, -2)

BC

BC”="√(-1-1)¤ +√(-2-2)¤

BC='∂20=2'5

유제 3-10x¤ +(y-3)¤ =1 x

O

yy=xy=-x

A(2, 1)

C(-1, -2)

x

B(1, 2)

x

y

O

1

-1

-2 2

y축 대칭

원점 대칭

y -y

x¤ +(-y-3)¤ =1

x¤ +(y+3)¤ =1

y=x x y,

y x

y¤ +(x+3)¤ =1

(x+3)¤ +y¤ =1

다른 풀이

(0, 3) (0, -3) (-3, 0)

(x+3)¤ +y¤ =1

유제 3-11y=2x¤ -4x+1

x -x y -y

-y=2(-x)¤ -4(-x)+1

y=-2x¤ -4x-1

x y -y

-y=-2x¤ -4x-1, y=2x¤ +4x+1

y=2(x+1)¤ -1

a=-1, b=-1

a+b=-2

유제 3-12y=x¤ -4x+k y 3

y y-3

y-3=x¤ -4x+k y=x¤ -4x+k+3

x y -y

-y=x¤ -4x+k+3

y=-(x-2)¤ -k+1 yy

-k+1

-k+1=5 k=-4

유제 3-13x¤ +(y-2)¤ =4 x 2 y

-3 x x-2 y

y+3

(x-2)¤ +(y+3-2)¤ =4

(x-2)¤ +(y+1)¤ =4

y=x x y

y x

(y-2)¤ +(x+1)¤ =4

(x+1)¤ +(y-2)¤ =4

y=x1211⁄x121⁄


431

(1 k)

(1+1)¤ +(k-2)¤ =4 k=2

유제 3-14P(-1, 3) (2, a)

Q(b+1, 5) (2, a) PQ

=2 b=4

=a a=4

a+b=8

유제 3-15

y=x¤ -2x+5

=(x-1)¤ +4

(1, 4)

y=-x¤ +6x-7

=-(x-3)¤ +2

(3, 2)

(1, 4), (3, 2)

(a, b)

=a a=2

=b b=3

a+b=5

유제 3-16(3, 1) y=2x

(a, b)

{ , }

y=2x

=2_

2a-b+5=0 yy

(3 1) (a b) y=2x

_2=-1

a+2b-5=0 yy

a=-1, b=3

(3 1) y=2x

(-1, 3)

b-1

a-3

3+a

2

1+b

2

1+b

2

3+a

2

4+2

2

1+3

2

3+5

2

(-1)+(b+1)

2

유제 3-17x¤ +y¤ -6x+2y+6=0

(x-3)¤ +(y+1)¤ =4

(x-3)¤ +(y+1)¤ =4 (x+1)¤ +(y+5)¤ =4

y=ax+b

A(3 -1) B(-1 -5)

y=ax+b AB

AB

=1

a_1=-1

a=-1 yy

y=ax+b AB

{ , } (1, -3)

-3=a+b

b=-2 ( )

ab=(-1)_(-2)

=2

유제 3-18A(1, -1) y A'

A'(-1, -1)

AP”=A’'P”

AP”+BP”=A’'P”+PB”æA’'B”

AP”+BP”

A’'B”="√(1+1)¤ +(5+1)¤

='∂40

=2'∂10

O

P(0, b)

y

A'(-1, -1)A(1, -1)

x

B(1, 5)

-1-5

2

3-1

2

-5-(-1)

-1-3

y=ax+b

B(-1, -5)

A(3, -1)


432

유제 3-19A(7, 2) y=2x-2

A'(a, b) y=2x-2 AA'

AA' { , }

y=2x-2

=2{ }-2

2a-b=-8 yy

AA' y=2x-2

_2=-1

a+2b=11 yy

a=-1, b=6

A' A'(-1, 6)

AP”=A'P”

AP”+BP”=A'P”+PB”æA'B”

AP”+BP”

””A’'B”="√(4+1)¤ +(1-6)¤

='5å0

””A’'B”=5'2

3-1(0, 0) 1⁄ (0+1, 0-3) (1, -3)

(-1, 3) 1⁄ (-1+1, 3-3) (0, 0)

(2, 4) 1⁄ (2+1, 4-3) (3, 1)

3-2(2, b) x a y

-2 (2+a, b-2)

(-3, -5)

a=-5, b=-3

a+b=-8

O

yy=2x-2

A(7, 2)

A'(-1, 6)

P

B(4, 1)

-2x1

b-2

a-7

7+a

2

2+b

2

2+b

2

7+a

2

3-3y=3x x 1

y=3(x-1)

y=3x-3 yy

y=3x y k

y-k=3x

y=3x+k yy

k=-3

3-4x¤ +y¤ -6x+4y+12=0

(x-3)¤ +(y+2)¤ =1

(0, 0) (3 -2)

x¤ +y¤ =1

x 3

y -2

m=3, n=-2

m-n=5

3-5P (m, n) P x

(m, -n) (a, 3)

m=a, n=-3 yy

P y

(-m, n) (4, b)

m=-4, n=b yy

m=-4, n=-3

m+n=-7

3-6y=3x+k x

-y=3x+k

y=-3x-k

(1, 3)

3=-3-k

k=-6

3-7y=x¤ +1 x 3

y=(x-3)¤ +1

y

y

xO3

-2

(3, -2)



433

y=(-x-3)¤ +1

y=x¤ +6x+10

y=ax¤ +bx+c

a=1, b=6, c=10

a+b+c=17

3-8(a 3) (2 1)

(1 b) (a 3) (1 b)

(2 1)

=2, =1

a=3, b=-1

a+b=2

3-9B(1, 3) x B'

B'(1, -3)

B’P’=B’'P”

AP”+BP”=AP”+B'P”æAB'”

AP”+BP”

A’B'”="( √1+2)¤ +(-3-2)¤

='3å4

3-10A D

D x -2

E

3-11x¤ +y¤ =2 (0, 0) x a

y 1-a

(a, 1-a)

y=x+2

1-a=a+2

a=-;2!;

x

O

A(-2, 2)

B'(1, -3)

B(1, 3)

P

-2

y

3+b

2

a+1

2

3-125<2¥6 5<2¥5A(6, 5) 11⁄ (5, 5) 11⁄ (4, 5)

5<2¥4 5<2¥3 5>2¥211⁄ (3, 5) 11⁄ (2, 5) 11⁄ (2, 4)4=2¥2 B(2 4)

P 5

3-13ABC S

S= _AB”_BC”

S= _2a_2b

S=2ab=2

ab=1

3-14x¤ +y¤ =4 x a

(x-a)¤ +y¤ =4 yy

y=-x

(-y-a)¤ +(-x)¤ =4

x¤ +(y+a)¤ =4 yy

3x-4y+2=0

x¤ +(y+a)¤ =4 (0, -a)

3x-4y+2=0 2

=2

|4a+2|=10

⁄ 4a+2=10 a=2

¤ 4a+2=-10 a=-3

⁄ ¤ a -6

3-15x+y+1=0 h(x y)=0

f gh(x y)=0 1⁄ h(-y -x)=0 1⁄ h(y x)=0

f gh(x y)=0 1⁄ h(-x -y)=0 1⁄ h(x y)=0

4

99=4_24+3

h(x y)=0 99

h(-x -y)=0

|4a+2|

"√3¤ +(-4)¤

1

2

1

2

y

xO

C(-a, -b)

A(a, b)B(-a, b)


434

x+y+1=0 -x-y+1=0

y 1

3-16y= x-4 y

y=- x-4

2

2 (2, -1)

y+1=2(x-2)

y=2x-5

3-17y=a(x-2)¤ -3 (1, 2)

a(1-2)¤ -3=2 a=5

y=5(x-2)¤ -3

y=-f(-x) x -x y -y

y=-5(-x-2)¤ +3

(-1, k)

k=-5(1-2)¤ +3=-2

3-18x¤ +y¤ +4x-2y+k=0

(x+2)¤ +(y-1)¤ =5-k

x¤ +y¤ +6x-4y+9=0

(x+3)¤ +(y-2)¤ =4

P(a, b) P

(-2, 1) (-3, 2)

a= =- , b= =

P(a, b)

5-k=4 k=1

a+b+k={-;2%;}+;2#;+1=0

3-19A(4 3) y=2x

A'(a, b) y=2x AA'

AA' { , } y=2xb+3

2a+4

2

3

2

1+2

2

5

2

-2-3

2

1

2

1

2

=2 { }

2a-b=-5 yy

AA' y=2x

_2=-1

a+2b=10 yy

a=0 b=5

A' (0 5)

B x B'

B'(6 -1)

AP”=A’'P” QB”=Q’B'”

AP”+PQ”+QB”=A’'P”+PQ”+Q’B'”æA’'B'”

AP”+PQ”+QB”

A'B'”="√(6-0)¤ +√(-1-5)¤

=6'2

3-20A B

(0 0) (800 -620)

C C'

A y -20

A'(0 -20)

A’A'”=C’C'” A’A'” CC'”

AA'C'C

AC”=A’'C'”

20m

A'(0, -20) C

C'

B

O 800A(0, 0)

20

-620

x

y

Ox

yy=2x

A(4, 3)

P

Q

A'(0, 5)

B(6, 1)

B'(6, -1)

b-3

a-4

a+4

2b+3

2


435

AC”+C’C'”+C’'B”=A’'C'”+C’'B”+CC'”

=A’'B”+20

="√800¤ +600¤ +20

=1020 (̀m)

3-212 y C

(2 p) C l x+y-8=0

C(2 p) l

=2

|p-6|=2'2

p=6-2'2 p=6+2'2

C A p>8

p=6+2'2

2 x C'

(q 2) C' l

C'(q 2) l

=2

|q-6|=2'2

q=6-2'2 q=6+2'2

C' B q<8

q=6-2'2

C (2 6+2'2) C'

(6-2'2 2)

a=6-2'2-2=4-2'2

b=2-6-2'2=-4-2'2

ab=(4-2'2)¥(-4-2'2)

=(-16)+8=-8

3-22A, B x-y+1=0

H A(-1 3) x-y+1=0

AH”= =

AB”=2AH”=2¥ =3'2

AB x-y+1=0

-1 (-1, 3)

y-3=-1¥{x-(-1)} x+y-2=0

C(a, b) x+y-2=0 d

d= =

ABC

|a+b-2|'2

|a+b-2|

"√1¤ +1¤

3'22

3'22

|-1-3+1|

"√1¤ +(-1)¤

|q+2-8|

'2

|2+p-8|

'2

ABC=;2!;¥AB”¥d

ABC=;2!;¥3'2¥ =3

|a+b-2|=2 yy

C(a, b) x-y+1=0

a-b+1=0 yy

a=;2#;, b=;2%; ( a>0)

a+b=4

3-23y=-x¤ +4x+3

-y=-(-x)¤ +4(-x)+3

y=x¤ +4x-3 yy

y a

y-a=x¤ +4x-3

y=(x+2)¤ -7+a yy

x y>0

-7+a>0

a>7

a 8

3-24y=2x-1

x=0 y=-1 (0 -1)

x=1 y=1 (1 1)

⁄ (0 -1) y=-x+3

(a b)

⁄ { }

⁄ y=-x+3

⁄ =-;2A;+3 b-1=-a+6

⁄ a+b=7 yy

⁄ y=-x+3 (0 -1) (a b)

⁄ _(-1)=-1 b+1=a

⁄ a-b=1 yy

⁄

a=4 b=3

⁄ (0 -1) y=-x+3

(4 3)

b-(-1)

a-0

b-1

2

b-1

2

a+0

2

|a+b-2|'2


436

¤ (1 1) y=-x+3

(c d)

⁄ { }

⁄ y=-x+3

⁄ =- +3 1+d=-(1+c)+6

⁄ c+d=4 yy

⁄ y=-x+3 (1 1) (c d)

⁄ _(-1)=-1 d-1=c-1

⁄ c-d=0 yy

⁄

c=2 d=2

⁄ (1 1) y=-x+3

(2 2)

⁄ ¤ (4 3) (2 2)

y-2= (x-2)

x-2y+2=0

mx-2y+n=0

m=1 n=2

m+n=3

3-25OP”, OQ” C C', C"

AC”=AC'”, BC”=BC"”

ABC

AB”+BC”+CA”=AB”+BC"”+AC'”æC'C"”

AB”+BC”+CA” C'C"”

C'OC"

C'OC"=120˘

C'C"”=2¥10'3

C'C"=20'3 (m)

O

C'

C"10'3 m

20 m

20 m60˘

O

P

C'

C"

C Q

60˘A B

3-2

4-2

d-1

c-1

1+c

2

1+d

2

1+d

2

1+c

2

1

{1, 2, 3, y}, {5, 10, 15, y}

2A={2, 3, 5, 7}

A={x|x 10 }

B={1, 2, 3, y}

B={x|x }

3A={1, 2, 3, 4, y, 10} A

10

B={2, 3, 4, y} B

x<1 C=ΔC

4A={1 3 5 7}

B={1 2 3 4 5}

n(A)+n(B)=4+5=9

9

5A={2 4} B={1 2 4 8} C={2 4 6 8 y}

Δ,A

A,B

C.A

B¯C , , . ¯

1

5

3

7

1 2

4

3

5

A B

집합



437

6Δ {1} {2} {1 2}

{x|x 9 }={1 3 9}

Δ {1} {3} {9} {1 3} {1 9} {3 9} {1 3 9}

Δ {1} {2} {1 2}

Δ {1} {3} {9} {1 3}

{1 9} {3 9} {1 3 9}

7A={2, 3, 5, 7} 2

{2, 3}, {2, 5}, {2, 7}, {3, 5}, {3, 7}, {5, 7}

{2, 3}, {2, 5}, {2, 7},

{3, 5}, {3, 7}, {5, 7}

8{2, 3, a}={3, 7, b} a=7, b=2

a+b=9

9

9{x|x 25 }={1, 5, 25}

Δ, {1}, {5}, {25}, {1, 5}, {1, 25}, {5, 25}

Δ, {1}, {5}, {25}, {1, 5}, {1, 25}, {5, 25}

유제 4-1

③

유제 4-23 10 3, 6, 9

A={3, 6, 9}A={x|x는 10보다작은 3의배수}

유제 4-31, 3, 5, 15 15

A={1, 3, 5, 15}A={x|x는 15의약수}

A

3

9

6

유제 4-4S x

x=a+ab=a(1+b)

a<A={-1, 0, 2},

b<B={1, 3, 5}

a -1 0 2 1+b 2 4 6

a(1+b)

S={-6, -4, -2, 0, 4, 8, 12}

S

(-6)+(-4)+(-2)+0+4+8+12=12

유제 4-5A 0 n(A)=1

n(Δ)=0

n(A)=0

A

A=Δn(A)=1 n(B)=1

n(A)=n(B)

ㄴ

유제 4-6x‹ -3x¤ +2x=0

x(x¤ -3x+2)=0

x(x-1)(x-2)=0

x=0 x=1 x=2

A={0 1 2}

B

B={x|x=2n-1 n 6 }

={1 3 5 7 9 11}

n(A)+n(B)=3+6=9

유제 4-7a 7-a

aæ1 7-aæ1

1…a…6

A 1 2 3 4 5 6

a<A 7-a<A

유제 본문 p.100

1+ba -1 0 2

2 -2 0 4

4 -4 0 8

6 -6 0 12


438

유제 4-12

Δ,Δ

0≤Δ

y, - , -2, - , y

{x|x<0 }+Δ{x|x 2 }={2, 4, 6, 8, 10, 12, y}

{x|x 4 }={4, 8, 12, 16, y}

{x|x 2 }.{x|x 4 }

ㄱ, ㄹ

유제 4-13A,B A B

a<B, a+2<B

⁄ a=-1 a+2=1<B

A,B

¤ a=0 a+2=2≤B

A¯B

‹ a=1 a+2=3<B

A,B

› a=3 a+2=5≤B

A¯B

⁄~› a -1또는 1

유제 4-14B,A A, B

-3…k…4 4…-2k

-3…k…-2

유제 4-15A,B B,A

A=B

A, B

x-2=4 x=6

y=9

2x+y=12+9=21

x

B

A

4 -2kk-3

4

3

7

2

1<A 7-1=6<A 2<A 7-2=5<A

3<A 7-3=4<A 4<A 7-4=3<A

5<A 7-5=2<A 6<A 7-6=1<A

4

{1, 2, 5, 6}, {1, 3, 4, 6}, {2, 3, 4, 5}

유제 4-8

a :¡a§: a

16 1 2 4 8 16

1<X :¡1§:=16<X 2<X :¡2§:=8<X

4<X :¡4§:=4<X 8<X :¡8§:=2<X

16<X ;1!6^;=1<X

X

⁄ 1 {4}

¤ 2 {1 16} {2 8}

‹ 3 {1 4 16} {2 4 8}

› 4 {1 2 8 16}

fi 5 {1 2 4 8 16}

⁄~fi X 7

유제 4-92 {2, 5}, {2, 7},

{2, 10}, {5, 7}, {5, 10}, {7, 10}3 {2, 5, 7},

{2, 5, 10}, {2, 7, 10}, {5, 7, 10}{5}, {7}, {5, 7}

유제 4-10A={1, 3, 5, 7}

1 3

1 3 {5, 7} Δ {5}, {7}

1, 3

{1, 3}, {1, 3, 5}, {1, 3, 7}

유제 4-11A Δ, 1, {2, 3}

Δ<A, 1<A, {2, 3}<A

A

Δ,A, {Δ},A, {1},A, {{2, 3}},A,

{Δ, 1},A, {Δ, {2, 3}},A, {1, {2, 3}},A,

{Δ, 1, {2, 3}},A

④


439

유제 4-16A=B

{1 3 a ¤ +1}={2 a b-1}

a¤ +1=2 a¤ =1

a=-1 a=1

⁄ a=-1

A={1 3 2} B={2 -1 b-1}

-1≤A A+B

¤ a=1

A={1 3 2} B={2 1 b-1}

A=B

b-1=3

b=4

⁄ ¤

a=1 b=4

a…3x-1…5

a+1…3x…6

…x…2

A=B

=-1, 2=b

a=-4, b=2

유제 4-17A B

A

B

A={1 2 3 4 6 12} B={1 2 4} B

2fl —‹ =8

2fl -8=56

유제 4-18B {3, 4, 5, 6, 7}

B

2fi -1=31

유제 4-19X B 1 4 6

A 2

X

26-3-1

=2¤ =4

a+1

3

a+1

3

유제 4-20X B A

5 7

X

2n-2

2n-2

=32=2fi n=7

4-1

ㄱ, ㄹ

4-2

1<A, 2≤B, a<B

4-3A={1, 2, 5, 10}, B={2, 4, 6, 8}, C=Δn(A)=4, n(B)=4, n(C)=0

n(A)+n(B)+n(C)=8

4-41, 2, 4, 8, 16 16

A={x|x 16 }

③

4-5A,B -a<-4 a>5

a>5

a 6

4-6[반례] X={1} Y={2 3} n(X)<n(Y)

X¯Y

X=Y

n(X)=n(Y)

[반례]X={1 2} Y={2 3}

n(X)=n(Y) X+Y

ㄴ



440

4-7A,B B,A A=B

{4 a-2 9}={b+2 5 9}

a-2=5 b+2=4

a=7 b=2

a-b=5

4-8n(A)=a A

2å

2å =128=2‡

a=7

n(B)=b B

2∫ -1

2∫ -1=31

2∫ =32=2fi b=5

n(A)+n(B)=7+5=12

4-9A 2 3

A

2 3

2‡ -2‡ —¤ =128-32=96

4-10X {3 4 5 6} 5 6

{5 6} {3 5 6} {4 5 6} {3 4 5 6}

X ②

4-11

Δ,A

{5}<A {{5}},A

2<A 3<A

{2, 3},A

5 A

5≤A

2<A 3<A 5≤A

{2, 3, 5}¯A

④

4-12n({1 2 3})-n({1 2})=3-2=1+3

A,B A B

n(A)…n(B)

[반례]A=Δ n(Δ)=n(A)

[반례]A=B=C A,B,C

n(A)=n(C)

[반례] A={1} B={2 3} C={4 5 6}

n(A)…n(B) n(B)<n(C)

A¯C

②

4-13A x B y xy

C={2¤ 2‹ 2› y 2° }

7

4-14B,A a+4 3-a=5 b+5=5

⁄ 3-a=5 a=-2

⁄ A={1 5 b+5} B={5 -1}

b+5=-1

b=-6

a+b=-8

¤ b+5=5 b=0

A={1 3-a 5} B={5 a+1}

a+1=1 a+1=3-a

a=0 a=1

a+b=0 a+b=1

⁄ ¤ a+b M m

M=1 m=-8

M-m=9

4-15A,B,C A B C

A

B

C

-2 k 4 x

yx 2 2¤ 2‹ 2› 2fi

2 2¤ 2‹ 2› 2fi 2fl

2¤ 2‹ 2› 2fi 2fl 2‡

2‹ 2› 2fi 2fl 2‡ 2°


441

-2…k<4 k

-2 -1 0 1 2 3

k

(-2)+(-1)+0+1+2+3=3

4-16A,B B,A A=B

{-2 3}={a+1 a¤ -3a}

a+1=-2 a+1=3

⁄ a+1=-2 a=-3

a¤ -3a=(-3)¤ -3¥(-3)=18

B={-2 18}

A+B

¤ a+1=3 a=2

a¤ -3a=2¤ -3¥2=-2

B={-2 3}

A=B

⁄ ¤ a 2

4-17A a<A :•a¡:<A

A 81 1, 3, 9, 27, 81

1 81, 3 27

(1, 81), (3, 27), (9)

A

{9}, {1, 81}, {3, 27}, {1, 9, 81}, {3, 9, 27},

{1, 3, 27, 81}, {1, 3, 9, 27, 81} 7

4-18A a A

2å B b B

2∫

2å -2∫ =4

2∫ (2a-b

-1)=4

2∫ 2a-b

-1

2∫ =4 2a-b

-1=1

2∫ =2¤ 2a-b

=2

b=2 a=3

a+b=5

4-19A={2 4 6 y 18} B={4 8 12 16}

B,X,A X A

B

29-4

=2fi =32

4-20⁄ n(B)=0

B=Δ A=Δ(A B) 1

¤ n(B)=1

n(B)=1 B 3 A

1 2⁄ =2

(A B)

2_3=6

‹ n(B)=2

n(B)=2 B 3 A

2 2¤ =4

(A B)

4_3=12

› n(B)=3

n(B)=3 B 1 A

3 2‹ =8

(A B) 8

⁄~› (A B)

1+6+12+8=27

4-212<S =-1<S

-1<S =;2!;<S

;2!;<S =2<S

2<S =-1<S

-1 ;2!; 2 n(S)=3

S=[-1 ;2!; 2]

S

2+(-1)=1

4-221<A 1-1=0<A, 0-1=-1<A

1-(-1)=2<A

1

1-2

1

1-;2!;

1

1-(-1)

1

1-2


442

;2!;<A

;2!;-;2!;=0<A

0-;2!;=-;2!;<A

;2!;-{-;2!;}=1<A

-1<A 2<A

2-(-1)=3<A

3-(-1)=4<A

4-(-1)=5<A

A

;3!;<A

;3!;-;3!;=0<A,

0-;3!;=-;3!;<A,

;3!;-{-;3!;}=;3@;<A,

;3@;-{-;3!;}=1<A

A

a a<A

0-a=-a<A

A

ㄱ, ㄴ, ㄷ

4-23⁄ 5

5 1

¤ 4

4 1 2 3

25-1-3

=2⁄ =2

‹ 3

3 1 2

25-1-2

=2¤ =4

› 2

2 1

25-1-1

=2‹ =8

fi 1

1

25-1

=2› =16

⁄~fi

5¥1+4¥2+3¥4+2¥8+1¥16=57

4-24⁄ 1 3 5

26-2-1

=2‹ =8

1 3 8 2 4 6

2 4 6 4

¤ 1 5 3

26-3

=2‹ =8

1 5 8 2 4 6 4

‹ 3 5 1

26-3

=2‹ =8

3 5 8 2 4 6 4

⁄ ¤ ‹

S(X¡)+S(X™)+S(X£)+y+S(X«)=16¥(1+3+5)+12¥(2+4+6)

=144+144=288


443

1A={a b c d} B={a c e}

A;B={a c}

A'B={a b c d e}

A-B={b d}

B-A={e}

{a c}

{a b c d e}

{b d}

{e}

2U={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

A={1, 2, 4, 8}, B={1, 2, 3, 4, 6, 12}

A;B={1, 2, 4}

A'B={1, 2, 3, 4, 6, 8, 12}

AÇ ={3, 5, 6, 7, 9, 10, 11, 12}

A-B={8}

{1, 2, 4}

{1, 2, 3, 4, 6, 8, 12}

{3, 5, 6, 7, 9, 10, 11, 12}

{8}

3A'A=A

A;A=A

A;AÇ =ΔA;U=A

A B

A-B=ΔB;AÇ =B-A

4A'(A

C'B)

=(A'AC)'B

=U'B=U

(AC-B)Ç ;A

=(AC;BÇ )Ç ;A

=(A'B);A

=A

U A

5(A'B

C);B=(A;B)'(B

C;B)

=(A;B)'Δ=A;B

=A A,B

A

6n(A'B)=n(A)+n(B)-n(A;B)

n(A'B)=14+10-6=18

n(A-B)=n(A)-n(A;B)

n(A-B)=14-6=8

18 8

7n(A'B)=n(A)+n(B)-n(A;B)

12=7+n(B)-4

n(B)=12-3=9

9

유제 5-1

B={2, 5, 6, 7, 8}

유제 5-2A={2 3} B={x+y|x<A y<A}

C={xy|x<A y<A}

B={4 5 6} C={4 6 9}

B;C={4 6}

A'(B;C)={2 3}'{4 6}

={2, 3, 4, 6}

유제 5-3AÇ ;BÇ ={3, 7} ,

A B

1 2

3 8

65

7

A B

집합의연산


유제 본문 p.131


444

A;BÇ =A-B=A={1, 2, 4}

U A B

B={5, 6}

유제 5-4A B A

x¤ -3 B 5, 2x, 6 B

2x A 2, 3, x¤ -3

⁄ x¤ -3=5 x=—2'2

¤ x¤ -3=2x x=-1 x=3

‹ x¤ -3=6 x=—3

› 2x=2 x=1

fi 2x=3 x=;2#;

⁄~fi x

(-3)+(-1)+1+3=0

유제 5-5U={1 2 3 4 5 6 7 8 9 10 11 12}

A={1 2 3 4 6 12} B={3 6 9 12}

BC={1, 2, 4, 5, 7, 8, 10, 11}

A-B={1, 2, 4}A'B

C={1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12}

AÇ -B={5, 7, 8, 10, 11}

유제 5-6U={1 2 3 4 5 6 7 8 9}

A B

A;B={2 4}

A;B

2+4=6

유제 5-7A;B={5, b} 5<B a=5

A;B={1, 3, 5, 7};{2, 5, 7}

={5, 7}

A 6

2

9

713

54

8

BU

A 5

39

142

612

78

10

11

BU

B

1 2 5

64

3 7

UA

b=7

a+b=5+7=12

유제 5-8A;B={1} 1<A a=1

A;B={1} 1<B

b=1 b+1=1

⁄ b=1

A={1, 2, 3}, B={1, 2, 5}

A;B={1, 2}

¤ b+1=1 b=0

A={1, 2, 3}, B={0, 1, 5}

A;B={1}

⁄ ¤ a=1 b=0

A={1, 2, 3}, B={0, 1, 5}

B-A={0, 5}

유제 5-9A;B=B HjK B,A

A, B

A'B=A HjKB,A

AÇ ,BC HjKB,A

BC-A

C=Δ HjK BÇ ,AÇ HjKA,B

A'BC=U HjKB,A

(B-A)Ç =U HjK B-A=Δ HjKB,A

③

유제 5-10A-B=AHjKA;B=Δ A B

②

유제 5-11A;X=A A,X

(A'B)'X=A'B X,(A'B)

A,X,(A'B)

{a b f },X,{a b c d e f }

X {a b c d e f }

a b f

X

26-3

=2‹ =8

B

AU


445

유제 5-12(A-B)-X=Δ (A-B),X

(A'B);X=X X,(A'B)

(A-B),X,(A'B)

{1 2},X,{1 2 3 4 5 6}

X {1 2 3 4 5 6}

1 2

X

26-2

=2› =16

유제 5-13{(A'B);(A'BÇ )}'{(AÇ ;BÇ )'(B-A)}

={A'(B;BÇ )}'{(AÇ ;BÇ )'(B;AÇ )}

=(A'Δ)'{AÇ ;(BÇ 'B)}

=A'(AÇ ;U)=A'AÇ =U

유제 5-14B;(A;B)Ç

=B;(AÇ 'BÇ )

=(B;AÇ )'(B;BÇ )

=(B-A)'Δ=B-A

(A-B)-C

=(A;BÇ );CÇ

=A;(BÇ ;CÇ )

=A;(B'C)Ç

=A-(B'C)

유제 5-15(A-B)Ç =(A;BÇ )Ç =AÇ 'B

(A'B);(A-B)Ç =(A'B);(AÇ 'B)

=(A;AÇ )'B

=Δ'B=B

{(A'B);(A-B)Ç };A=B;A

B;A=B ②B,A

유제 5-16n(BÇ )=n(U)-n(B)

26=50-n(B)

n(B)=24

n(AÇ ;B)=n(B-A)

=n(B)-n(A;B)

=24-10

=14

유제 5-17n(A;B)=n(A)+n(B)-n(A'B)

=9+12-14=7

n(B;C)=n(B)+n(C)-n(B'C)

=12+8-20=0

n(C;A)=n(C)+n(A)-n(C'A)

=8+9-13=4

n(B;C)=0 B;C=ΔA;B;C=Δ n(A;B;C)=0

n(A'B'C)

=n(A)+n(B)+n(C)-n(A;B)

-n(B;C)-n(C;A)+n(A;B;C)

=9+12+8-7-0-4+0

=18

유제 5-18A;B=Δ A;C=Δn(A;B)=n(A;C)=0

n(AÇ ;BÇ ;CÇ )=n((A'B'C)Ç )=8

n(A'B'C)=n(U)-n((A'B'C)Ç )

=40-8=32

n(A)=n(A'B'C)-n(B'C)

=32-27=5

유제 5-19A,

B

n(A)=35, n(B)=30

50

n(A'B)=50

A;B

n(A;B)=n(A)+n(B)-n(A'B)

=35+30-50

=15

유제 5-20100 U 4

A 5 B A;B

A B C

U


446

20

n(U)=100 n(A)=25 n(B)=20

n(A;B)=5

n(A'B)=n(A)+n(B)-n(A;B)

=25+20-5

=40

4 5

AÇ ;BÇ

n(AÇ ;BÇ )=n((A'B)Ç )

=n(U)-n(A'B)

=100-40

=60

유제 5-21

A B C

n(A'B'C)=100

x y

27-x-y

(16-x)+16+30+x+(20-y)

+y+(27-x-y)=100

27-x-y=18

18

다른 풀이

A B C

n(A)=62 n(B)=66 n(C)=57 n(A;B)=46

n(A'B'C)=100

(A'B'C)-(A'B)

n(A'B'C)-n(A'B)

=n(A'B'C)-{n(A)+n(B)-n(A;B)}

=100-(62+66-46)

=18

B

A

3016

C

x

y

16-x

20-y 27-x-y

5-1A={2 4 5 8} B={1 3 5 7 9}

C={1 2 3 4 6 12}

B'C={1 2 3 4 5 6 7 9 12}

A;(B'C)={2, 4, 5}

다른 풀이

A;(B'C)=(A;B)'(A;C)

={5}'{2 4}

={2 4 5}

5-2A;B={1 3};Δ=ΔA;C={1 3};{-4 -3}=ΔA;D={1 3};{0 2 5}=ΔA;E={1 3};{1 2 5 10}={1}+ΔA;F={1 3};{2 4 6 y}=Δ

A ④

5-3U={1, 2, 3, 4, 5, 6, 7, 8, 9}

AÇ ;BÇ =(A'B)Ç ={5, 7}

A'B={1, 2, 3, 4, 6, 8, 9}

B-A={2, 4, 6}

A=(A'B)-(B-A)

A={1, 2, 3, 4, 6, 8, 9}-{2, 4, 6}

A={1, 3, 8, 9}

5-4A={1, a¤ -3a+1}, B={a-2, a+1}

A;B={5} a¤ -3a+1=5

a¤ -3a-4=0, (a+1)(a-4)=0

a=-1 a=4

⁄ a=-1

B={a-2, a+1}

⁄ a-2=-1-2=-3

⁄ a+1=-1+1=0

B={-3, 0}

A;B=Δ¤ a=4

B={a-2, a+1}

U

2

5 7

4

6

A B



447

⁄ a-2=4-2=2,

a+1=4+1=5

B={2, 5}

A;B={5}

⁄ ¤ a=4

5-5A;B={2, 3}

A'B={1, 2, 3, 4, 5}

{2, 3},X,{1, 2, 3, 4, 5}

X {1 2 3 4 5} 2 3

X

25-2

=2‹ =8

5-6

A-(B'C)

=A;(B'C)Ç

5-7X;(X-Y)Ç

=X;(X;YÇ )Ç

=X;(XÇ 'Y)

=(X;XÇ )'(X;Y)

=Δ'(X;Y)

=X;Y

5-8(A-B)'(B-A)=(A'B)-(A;B)

(A-B)'(B-A)={1, 3, 5}

A={1, 2, 3} 1, 3 A-B

2 A;B

5 B-A

B={2, 5}

B 2+5=7

5-9n(A)=11, n(B)=8, n(A'B)=17


1

32 5

A B

UA

B C

=11+8-17=2

n(A-B)=n(A)-n(A;B)

=11-2=9

5-10U

A B

n(U)=40, n(A)=18, n(A;B)=8,

n((A'B)Ç )=2 n(A'B)=38


38=18+n(B)-8 n(B)=28

n(B-A)=n(B)-n(A;B)=28-8=20

5-11x¤ +4x+3>0 (x+3)(x+1)>0

x<-3 x>-1

A={x|x<-3 x>-1}

A'B=R A;B={x|-1<x…4}

B={x|-3…x…4}

={x|(x+3)(x-4)…0}

={x|x¤ -x-12…0}

k=-12

5-12k

B={-2, -1, 0, 1, 2, 4}

B-A={2, 4}

B-A

2+4=6

5-13(A-B)'(B;AÇ )=Δ

AB

A

x-3 -1 4

y x -2 -1 0 1

-2 4 2 0 -2

-1 2 1 0 -1

0 0 0 0 0

1 -2 -1 0 1


448

(A-B)'(B-A)=ΔA-B=Δ B-A=Δ

A,B B,A A=B

A={4 6 a+1} B={6 7 b}

a+1=7 a=6 b=4

a+b=6+4=10

5-14A;X=X X,A (A-B)'X=X

(A-B),X (A-B),X,A

{2, 4, 5},X,{1, 2, 3, 4, 5}

X 2, 4, 5 A

X 2fi —‹ =2¤ =4

5-15(A-BÇ )'(BÇ -AÇ )

={A;(BÇ )Ç }'{BÇ ;(AÇ )Ç }

=(A;B)'(BÇ ;A)

=(A;B)'(A;BÇ )

=A;(B'BÇ )

=A;U=A=A'B

B,A

A'B=A

A;B=B

B-A=ΔAÇ 'B

AÇ 'B+U

A-(A'B)=A-A=Δ⑤

5-16A™ B™

A™={x|x 2 }={x|x }

B™={x|x 2 }={x|x }

A™'B™={x|x }

[반례] 6 5 6<A∞

6≤A™ 6≤A£ 6≤(A™'A£)

B™={x|x 2 } B£={x|x 3 }

B™;B£=B§

ㄱ

A

B

U

5-17A≠U=(A;U)'(A'U)Ç

=A'UÇ

=A'ΔA≠U=A

A≠B=(A;B)'(A'B)Ç

=(B;A)'(B'A)Ç

=B≠A

A≠Δ=(A;Δ)'(A'Δ)Ç

=Δ'AÇ

=AÇ

AÇ ≠BÇ =(AÇ ;BÇ )'(AÇ 'BÇ )Ç

=(A'B)Ç '(A;B)

=(A;B)'(A'B)Ç

=A≠B

A≠AÇ =(A;AÇ )'(A'AÇ )Ç

=Δ'UÇ

=Δ'ΔA≠AÇ =Δ

①

5-18A'X=B'X

3<A 3≤B 3<X yy

2<B 4<B 2≤A 4≤A

2<X 4<X yy

{2 3 4},X

{2 3 4},X,U

X 2 3 4

U X

210-3

=2‡ =128

5-19n(AÇ ;BÇ )=n((A'B)Ç )=20

n(A'B)=n(U)-n((A'B)Ç )

=32-20

n(A'B)=12


12=n(A)+n(B)-8

n(A)+n(B)=20

5-20U

A B


449

n(U)=40, n(A)=28, n(B)=16

n(A;B)…n(A), n(A;B)…n(B)

n(A;B)…16 yy


n(A'B)…n(U)

28+16-n(A;B)…40

n(A;B)æ4 yy

4…n(A;B)…16

M=16, m=4

M+m=16+4=20

5-21S

Δ {a} {b} {c} {a b} {b c} {a c} S

Δ<X S<X

Δ S X

{a} {b c} {b} {a c} {c} {a b}

X

{S Δ}

{{a} {b c} S Δ}

{{b} {a c} S Δ}

{{c} {a b} S Δ}

{{a} {b} {c} {a b} {b c} {a c} S Δ}

X 5

5-22A¢=A£'{A£}

A¢={Δ, {Δ}, {Δ, {Δ}}}

A«<A«≠¡A™={Δ}={A¡}

A£={Δ, {Δ}}={A¡, A™}

A¢={Δ, {Δ}, {Δ, {Δ}}}={A¡, A™, A£}

A«≠¡={A¡, A™, A£, y, A«}ㄱ, ㄴ, ㄷ

5-23(B-A),X,U X

B-A U

n(B-A)=a

X 29-a

=32

9-a=5 a=4

n(A)=3 n(B-A)=4

A;(B-A)=Δ A'(B-A)=A'B

n(A'B)=3+4=7

A,B n(B)=7

A;B=Δ n(B)=4

4…n(B)…7

B

B={3 4 5 6 7 8 9} S(B)

3+4+5+6+7+8+9=42

B

B={1 2 3 4} S(B)

1+2+3+4=10

S(B)

42+10=52

5-24

A B C

n(A;B;C)=x

n(A;B;C)…n(B;C)

0…x…3 yy

n(A'B'C)

=n(A)+n(B)+n(C)-n(A;B)

-n(B;C)-n(C;A)+n(A;B;C)

=22+18+11-10-3-8+x

=30+x

k

n(AÇ ;BÇ ;CÇ )=k

k=n((A'B'C)Ç )

=n(U)-n(A'B'C)

=50-(30+x)=20-x

17…k…20

17명

U

B C

A

x

k


450

19

x>0 x<0 x=0

2 6

26 3

4+5+10

'3

3P={-2, 2}

Q={1, 2, 4, 5, 10, 20}

4a

x+2+0

x+3 x+4

x…3 x>4

5x=1 x¤ =|x|=1

6

x x-3=0

x x-3+0

x |x|æx

x |x|<x

x x¤ <0

x x¤ æ0

7p 2⁄ q

[반례] a=1, b=3q 2⁄ p

p 2⁄ q

q 2⁄ p

8xy=1 x=1 y=1

[반례] x=2, y=;2!;

xy+1 x+1 y+1

A, B A,B

A;B=A

A, B A¯B

A;B+A

9p x=2 q x¤ =4 p jjK q

x=-2 q jjK p p q

p x<2 q x<1 q jjK p

x=;2#; p jjK q p q

A,B B,A A=B

유제 6-12 거짓

명제


유제 본문 p.163


451

거짓

p=3.14y 참

⑴, ⑶, ⑷

유제 6-21<x<2 x>1 x<2

x…1 xæ2

6 2 3 (거짓)

(참)

x…1 y…1

유제 6-3p P

U={1, 2, 3, 4, 5} 2, 3, 5

P={2, 3, 5}

유제 6-4p, q P, Q

p -2<x<3

P={-1, 0, 1, 2}

q x¤ -3x=0 x(x-3)=0

Q={0, 3}

p ~q

P;QC={-1, 0, 1, 2};{-2, -1, 1, 2}

P;QC={-1, 1, 2}

유제 6-5p x¤ -2x+1æ0 p

P

p x¤ -2x+1æ0 (x-1)¤ æ0

P={-1 0 1 2}

P=U

x=0 y=0 x¤ +y¤ =0

x=0 x¤ =0

q x+3>6 q Q

q x+3>6 x>3

P=Δ

ㄱ, ㄷ

유제 6-6x

x¤ -12x+kæ0

x¤ -12x+k=0 D

=(-6)¤ -k…0 kæ36

k 36

유제 6-7x¤ -3x+2=0 (x-1)(x-2)=0

x=1 x=2

p, q P, Q

P={1, 2}, Q={x|0<x<3}

P,Q p 2⁄ q

[반례] x=3, y=0 x¤ +y¤ =9+0

p 2⁄ q

[반례] x=-3, y=-2 xy=6>0

p 2⁄ q

p, q P, Q

P={1, 2, 4, 8}, Q={1, 2}

P¯Q p 2⁄ q

ㄱ

유제 6-8p 2⁄~q

P,QC

U

P Q

P'Q+U

P-Q=P

Q-P=Q

P;Q=ΔP

C'QC=(P;Q)

C=U

②

유제 6-9p 1<x…3 q a<x<a+3 p q

P Q

P={x|1<x…3} Q={x|a<x<a+3}

P,Q

a…1 a+3>3 0<a…1

x3 a+31a

P

Q

P Q

U

D

4


452

유제 6-10p q P Q

p |x-1|…a -a…x-1…a

-a+1…x…a+1

P={x|-a+1…x…a+1}

q x<-5 Q={x|x<-5}

QÇ ={x|xæ-5}

p 2⁄~q P,QÇ

-a+1æ-5 a…6

a>0 0<a…6

a 6

유제 6-11x¤ =x x=0 x=1

역：x=0또는 x=1이면x¤ =x이다. (참)

대우：x+0이고 x+1이면x¤ +x이다. (참)

xy>1 x>1 y>1

[반례] x=3, y= xy=;2#;>1

y<1

역：x>1이고 y>1이면xy>1이다. (참)

대우：x…1또는 y…1이면xy…1이다. (거짓)

[반례] x=-2, y=-1 x…1

y…1 xy=2>1

유제 6-12

A

유제 6-13p jjK q, q jjK~r p jjK~r

p jjK~r r jjK~p

~s jjK~q q jjK s

p jjK q, q jjK s p jjK s

ㄱ, ㄴ, ㄷ

NJ

1

2

-5

QÇ

P

a+1-a+1 x

유제 6-14p q

r

~p jjK q, ~r jjK~p

~r jjK~p, ~p jjK q~r jjK q~q jjK p, p jjK r ~q jjK r

ㄱ, ㄴ

유제 6-15P;Q=Δ Q,P

C

q jjK ~p

③

유제 6-16p 2⁄ q P,Q

r 2⁄~q q 2⁄~r

Q,RÇ

U P Q R

P-Q=ΔP,RÇ P;RÇ =P

P;R=Δ(P;R)Ç =Δ Ç =U

ㄱ, ㄴ

유제 6-17a=-b a¤ =b¤ p jjK qa=b a¤ =b¤ q jjK p

p q 필요조건

m+n m= n=

m= n= mn

p jjK qm=2, n=4 mn m+n

q jjK p

p q 충분조건

Q

PR

U


453

a a¤ æ0

a¤ …0 a=0 a=0 a¤ …0

p HjK q p q 필요충분조건

유제 6-18(P-Q)'(Q-R

C)=Δ

P-Q=Δ, Q-RC=Δ

P,Q, Q,RC

p jjK q, q jjK~r p jjK~r

p ~r

~r q

유제 6-19P'(Q-P)=P'(Q;P

C)

P'(Q-P)=(P'Q);(P'PC)

P'(Q-P)=(P'Q);U

P'(Q-P)=P'Q=Q

P,Q

p q q p

④

유제 6-20p, q P, Q

P={x|x…a}, Q={-1, 2}

p q p Hjj qP.Q aæ2

a 2

유제 6-21p q r P Q R

q p r p

Q.P, R,P

R,P,Q

P, Q, R

a<-1, bæ4

a M=-2

b m=4

M+m=-2+4=2

x3-1a 4 b

QP P

R

필요

충분

유제 6-22p q

p jjK q yy

~q ~r

~r jjK~q

q jjK r yy

p jjK rㄱ, ㄷ

6-1

x

x=0 x+2=2+3

x+2=3

ㄱ, ㄷ, ㄹ

6-2x<-2 xæ-2

xæ3 x<3

x<-2 xæ3

xæ-2 x<3

-2…x<3

6-32 P={2}

10 15 1, 3, 5

Q={1, 3, 5}

P'Q={1, 2, 3, 5}

P'Q 4

6-4x¤ -ax+6+0 x+1

x=1 x¤ -ax+6=0

x=1

x¤ -ax+6=0

1¤ -a¥1+6=0

a=7



454

6-5x p(x) U

x p(x)

P=U

x p(x) P

P+Δx p(x)

x ~p(x)

PC=U P=Δ

ㄱ, ㄴ, ㄷ

6-6P;Q=R R,P, R,Q

r 2⁄ p, r 2⁄ q

~p 2⁄ ~r, ~q 2⁄ ~r

③

6-712 6

[반례] 4, 12 12 6

6 12

6 12

a, b ab

ab a, b

[반례] a='2, b='2, ab=2

ab a b

ABC

ABC

ABC

ㄷ

6-8p x¤ -x-2=0 q x=2

q jjK p x=-1 p jjK qx¤ -x-2=0 x=2

충분

필요

A'B=A HjK B,A

A'B=A HjK A;B=B

A'B=A A;B=B

6-9|x|=2 HjK x=—2

x=2 jjK |x|=2

x…2 Hjj |x|=2

x¤ =4 HjK |x|=2

ㄱ

6-10x‹ +ax¤ +bx+0 x+1 x+3

x=1 x=3 x‹ +ax¤ +bx=0

x=1 1+a+b=0 yy

x=3 27+9a+3b=0 yy

a=-4, b=3

a-b=-7

6-11어떤실수x에대하여-1…x<1이다 ∴참

U={1, 2, 3, 4}를전체집합이라할때, 모든x에

대하여3x-10<0이다.

[반례] x=4 3x-10=2>0 거짓

6-12p |x+2|…3 q x…-2

P Q

P={x|-5…x…1} Q={x|x…-2}

{x|-2<x…1}=P;QC

P;QC p그리고~q

6-13p 0<x<4 q k-1<x<k+5 p

q P Q

P={x|0<x<4},

Q={x|k-1<x<k+5}

x1-2-5

P

P;QÇQ

필요충분


455

P,Q

k-1…0 k+5æ4

-1…k…1

6-14p |x-a|…2 q 1…x…3 p q

P Q

P={x||a-2…x…a+2}, Q={x|1…x…3}

P;Q+Δ⁄

a-2…3 a…5

¤

a+2æ1 aæ-1

⁄ ¤ -1…a…5

6-15p, q, r

P, Q, R

P-R=RC

ㄱ, ㄷ

6-16Q,P, Q,R

q 2⁄ p, q 2⁄ r

PC,Q

C, R

C,QC

~p 2⁄~q, ~r 2⁄~q

PC¯R

C ③~p 22⁄⁄ ~r

6-17(P;QÇ )'Q=(P'Q);(QÇ 'Q)

=(P'Q);U

=P'Q=P

Q,P

q jjK p ~p jjK~q

④

P

RÇ

Q=U

xa-2 31 a+2

PQ

x1 a-2 3 a+2

PQ

x

P

Q

4 k+50k-1

6-18p 2⁄ q [반례] x=-1

x¤ =1 x+1

q 2⁄ p x=1 x¤ =1

p q

p 2⁄ q [반례] x=1, y=-1

x+y=0 x¤ +y¤ +0

q 2⁄ p x¤ +y¤ =0 x=0, y=0

x+y=0

p q

p 2⁄ q [반례] x=-4

x<3 |x|>3

q 2⁄ p |x|<3

-3<x<3 x<3

p q

p 2⁄ q A-B=A A;B=Δq 2⁄ p A;B=Δ A-B=A

p q

⑤ p 2⁄ q x>0, y>0 xy>0

q 2⁄ p [반례] x=-2, y=-2 xy>0

x<0, y<0

p q

⑤

6-19P;Q=Δ, Q,R, P;R+Δ

P;Q=Δ PC.Q

Q,R RC,Q

C

P;Q=Δ P,QC

P¯RC

RC¯P

⑤

6-20a…x…5 Hjj 2…x…3

b…x…2 jjK -2…x…2

a…2, -2…b…2

a 2, b -2

2+(-2)=0

3 x x2 5a b 2-2


456

6-21

1 다운 2 3 4 새미

6-22⁄ x¤ +8æp x¤ æp-8 x

p-8…0

p…8

¤ x 2-x¤ æp

x

2-x¤ <p

2-x¤ <p x¤ >2-p x

2-p<0

p>2

⁄ ¤ 2<p…8 p

3 4 5 6 7 8 6

6-23p

q

r

s

p jjK q r jjK s

~q jjK ~r

~q jjK ~r r jjK q

r jjK s p jjK q s jjK p ~p jjK ~s

⑤

6-24⁄ A B=(A-B)'(B-A)=Δ

A-B=Δ B-A=ΔA,B B,A A=B

A B=Δ`jjKÀ=B

A=B

A B=(A-B)'(B-A)=Δ'Δ=ΔA=B`jjKÀ B=Δ

A B=Δ A=B

¤ A B=(A-B)'(B-A)

=(A'B)-(A;B)

필요충분

(A;B),A,(A'B)

A B=(A'B)-(A;B)=A

A'B=A A;B=ΔA'B=A B,A A;B=B=Δ

A B=A`jjK`B=ΔB=Δ

A B=(A-B)'(B-A)=A'Δ=A

B=Δ`jjKÀ B=A

A B=A B=Δ필요충분

A

B

C

1 2 3 4

_

_

_


457

유제 7-1a b

a=2m-1 b=2n-1 m n

a+b=(2m-1)+(2n-1)

=2m+2n-2

=2(m+n-1)

m+n-1 a b

a+b

ab=(2m-1)(2n-1)

=4mn-2m-2n+1

=2(2mn-m-n+1)-1

2mn-m-n+1 a

b ab

유제 7-2n n

n¤

n n=2k k

n¤ =(2k)¤ =4k¤ =2(2k¤ )

2k¤ n¤

n n¤ n

n n 3

n¤ 3

n 3

n=3k-1 n=3k-2 k

n¤ =(3k-1)¤ =3(3k¤ -2k+1)-2

n¤ =(3k-2)¤ =3(3k¤ -4k+2)-2

3k¤ -2k+1 3k¤ -4k+2

n¤ 3

n n¤ 3 n 3

유제 7-3b+0 a+b'2=0

'2=-;bA;

= b=0

b=0 a+b'2=0 a=0

a b a+b'2=0

a=b=0

1

p 2⁄ q

p

~p

답⃞ ~q 2⁄~p

2x¤ +5-4x=x¤ -4x+4+1

=(x-2)¤ +1>0

x¤ +5>4x

|x+2|æ0 |x|+2æ2>0

|x+2|¤ -(|x|+2)¤

=x¤ +4x+4-(x¤ +4|x|+4)

=4(x-|x|)…0

|x+2|…|x|+2

답⃞ x¤ +5>4x |x+2|…|x|+2

참고 a a¤ æ0

(x-2)¤ æ0

참고 (x-2)¤ +1æ1>0

3

a>0 >0

a+ æ2æ≠a¥ =2

, a=1 2

a, b >0, >0

+ æ2æ≠ ¥ =2

, a=b 2

답⃞ 2 2

a

b

b

a

a

b

b

a

a

b

b

a

1

a

1

a

1

a

~q 2⁄ ~p

명제의증명


유제 본문 p.196


458

유제 7-4a¤ +2ab+2b¤ =(a+b)¤ +b¤ æ0

a+b=0 b=0 a=b=0

4a¤ +3b¤ -4abæ0

4a¤ +3b¤ -4ab=(2a-b)¤ +2b¤ æ0

2a-b=0 b=0 a=b=0

4a¤ +3b¤ æ4ab

유제 7-55(x¤ +y¤ )-(x+2y)¤ æ0

5(x¤ +y¤ )-(x+2y)¤ =5x¤ +5y¤ -(x¤ +4xy+4y¤ )

=4x¤ -4xy+y¤

=(2x-y)¤ æ0

2x-y=0 y=2x

5(x¤ +y¤ )æ(x+2y)¤

유제 7-6⁄ |a|<|b| >0 <0

¤ |a|æ|b|

|a-b|æ0 |a|-|b|æ0

|a-b|¤ -(|a|-|b|)¤ æ0

|a-b|¤ -(|a|-|b|)¤

=a¤ -2ab+b¤ -(a¤ -2|a||b|+b¤ )

=2(|ab|-ab)

|ab|æab 2(|ab|-ab)æ0

|ab|=ab abæ0

⁄ ¤ |a-b|æ|a|-|b|

a>b>0

'ƒa-b>0 'a-'b>0 yy

('ƒa-b)¤ -('a-'b)¤

=a-b-(a-2'a åb+b)

=2'∂ab-2b=2'b('a-'b)>0

'ƒa-b>'a-'b

유제 7-7

a>0 ;a!;>0

a+;a!;æ2Æ…a¥;a!;=2

a=;a!; a=1

(a+b){;a!;+;b!;}=;aB;+;bA;+2

;aB;>0 ;bA;>0

;aB;+;bA;+2æ2æ≠;aB;¥;bA;+2=4

;aB;=;bA; a=b

유제 7-8x>0, 2y>0

x+2yæ2'ƒx¥2y

x+2y=2'∂2xy=2 { xy=;2!;}

x=2y

xy=;2!;

2y¤ =;2!; y¤ =;4!;

y=;2!; ( y>0), x=1

x+2y 2

유제 7-92x>0, 5y>0

2x+5yæ2'ƒ2x¥5y 8æ2'ƒ2x¥5y yy

2x=5y

('∂2x+'∂5y )¤ =8+2'ƒ2x¥5y

('∂2x+'∂5y )¤ …8+8 ( )

('∂2x+'∂5y )¤ =16

0<'∂2x+'∂5y …4 ( '∂2x>0, '∂5y>0)

'∂2x+'∂5y 4

유제 7-10

(a+2b){;a@;+;b!;}=2+;bA;+:¢aı:+2

=;bA;+:¢aı:+4

æ2æ≠;bA;¥:¢aı:+4

=8

{ = }

(a+2b){;a@;+;b!;} 8

4b

a

a

b


459

유제 7-11

{x+;]@;} {y+;[*;}=xy+;[!]^;+10

{x+;]@;} {y+;[*;}æ2Æ…xy¥;[!]^;+10

{x+;]@;} {y+;[*;}=18

{ xy=;[!]^; }

{x+;]@;} {y+;[*;}

xy=;[!]^; xy=4 18

a=4 b=18

a+b=22

유제 7-12x+y=(x+y)¥1

x+y=(x+y){;[@;+;]#;}

x+y=:£]”:+:™[’:+5

x+yæ2æ≠:£]”:¥:™[’:+5=5+2'6

{ = }

x+y 5+2'6

유제 7-13

2x+ =2(x-1)+ +2

2x+ æ2æ≠2(x-1)¥ +2

2x+ =6

{ 2(x-1)= }

2x+ 6

유제 7-14

a-;a!;+ = +

æ2æ≠ ¥

=2'2

{ = }

a-;a!;+ 2'22a

a¤ -1

2a

a¤ -1

a¤ -1

a

2a

a¤ -1

a¤ -1

a

2a

a¤ -1

a¤ -1

a

2a

a¤ -1

2

x-1

2

x-1

2

x-1

2

x-1

2

x-1

2y

x

3x

y

유제 7-15BP”=x CP”=2-x

ABPª QCP

AB” BP”=QC” CP”

2 x=QC” 2-x

QC”=

S=S¡+S™

S=;2!;x¥2+;2!;(2-x)¥

S=2x+;[$;-4

Sæ2æ≠2x¥;[$;-4=4('2-1)

{ 2x=;[$; }

S 4('2-1)

유제 7-16x y

(3¤ +4¤ )(x¤ +y¤ )æ(3x+4y)¤

{ ;3{;=;4}; }

x¤ +y¤ =1

25æ(3x+4y)¤

-5…3x+4y…5

유제 7-17a, b, x, y

(a¤ +b¤ )(x¤ +y¤ )æ(ax+by)¤

{ ;a{;=;b}; }

a¤ +b¤ =4 ax+by=6

4(x¤ +y¤ )æ36

x¤ +y¤ æ9

x¤ +y¤ 9

7-14x+8<0 x<-2

xæ-2

2(2-x)

x

2(2-x)

x

A

B

S¡ S™

x

2

2

C

D

P

Q



460

2x¤ >2x¤ -5

0>-5

x

-(x+1)¤ <0

(x+1)¤ >0

x=-1

x¤ +2x+5>0 (x+1)¤ +4æ4>0

x

ㄴ, ㄹ

7-2a¤ -ab+b¤ ={a-;2B;} ¤ +;4#;b¤ æ0

(|a|+|b|)¤ -|a+b|¤

=a¤ +2|ab|+b¤ -(a¤ +2ab+b¤ )

=2(|ab|-ab)

|ab|æab

2(|ab|-ab)æ0

|a|+|b|æ|a+b|

a¤ +a+1={a+;2!;}¤ +;4#;æ;4#;>0

('∂a+b)¤ -('a+'b)¤ =a+b-(a+2'a åb+b)

=-2'aåb…0

'∂a+b…'a+'b

a¤ +b¤ +c¤ -ab-bc-ca

=;2!;(2a¤ +2b¤ +2c¤ -2ab-2bc-2ca)

=;2!; {(a-b)¤ +(b-c)¤ +(c-a)¤ }æ0

a¤ +b¤ +c¤ æab+bc+ca

(a¤ +b¤ )(x¤ +y¤ )-(ax+by)¤

=a¤ x¤ +a¤ y¤ +b¤ x¤ +b¤ y¤ -(a¤ x¤ +2abxy+b¤ y¤ )

=a¤ y¤ -2abxy+b¤ x¤

=(bx-ay)¤ æ0

(a¤ +b¤ )(x¤ +y¤ )æ(ax+by)¤

ㄱ, ㄴ, ㅁ, ㅂ

7-3a+;a!;æ2Æ…a¥;a!;=2

b+;b$;æ2Æ…b¥;b$;=4

;2ıa;+:¢bÅ:æ2Æ…;2ıa;¥:¢bÅ:=2'2

ㄱ, ㄴ

7-4BC M AM

A’B”=A’C” B’M”=C’M” A’M”

ABM™ ACM SSS

B= yy

B’C”=

B= yy

A= B= C

C, A’C”, ∠A

7-5

a+bæ2'∂ab a=b

a+b=8

8æ2'∂ab '∂ab…4

0<ab…16 a>0 b>0

ab 16

4a+9bæ2'ƒ4a¥9b=2'ƒ36ab

4a=9b

ab=9

4a+9bæ36

4a+9b 36

7-6-'∂ab=

-'∂ab=

a, b

('a-'b )¤ `0 a=b

HjK -'∂ab `0

HjK æ'∂ab

('a-'b )¤ , æ

7-7(1+a){1+;a!;}=a+;a!;+2

(1+a){1+;a!;}æ2Æ…a¥;a!;+2=4

{ a=;a!; }

(1+a){a+;a!;} 4

a+b

2

æa+b

2

æ

('a-'b )¤

2

('a)¤ +('b)¤ -2'a'b2

a+b

2

A

A’C”

C


461

7-8x>3 x-3>0

x+ =x-3+ +3

æ2æ≠(x-3)¥ +3=9

{ x-3= }

x+ x=6 9

a=6 m=9

a+m=15

7-9a, b, x, y

(a¤ +b¤ )(x¤ +y¤ )æ(ax+by)¤

{ ;a{;=;b}; }

3¥8æ(ax+by)¤

24æ(ax+by)¤

-2'6…ax+by…2'6

ax+by 2'6

7-104x¤ +4ax+a+6=0 D

=(2a)¤ -4(a+6)<0

4a¤ -4a-24<0 a¤ -a-6<0

(a+2)(a-3)<0

-2<a<3

7-11ABC AB AC

O O D

E F

O’A”= =O’C’ O’B’=O’C’

O’E’ OEB= =90˘

OBE™ OCE

BE”=

OE” BC”

O

O’B’, OA”, ∠OEC, RHS, CE”,수직이등분선

C’E’

RHS

OEC

O’A”O’B’

D

4

9

x-3

9

x-3

9

x-3

9

x-3

9

x-3

7-12a, b a+0 b+0

a¤ +b¤ +0

⁄ a+0, b=0 a¤ >0, b¤ =0, a¤ +b¤ >0

⁄ a¤ +b¤ +0

¤ a=0, b+0 ⁄ a¤ +b¤ +0

‹ a+0, b+0 a¤ >0, b¤ >0 a¤ +b¤ +0

⁄ ¤ ‹ a¤ +b¤ +0

a, b

a¤ +b¤ =0 a=0, b=0

7-13'2

'2=;pQ; p+0, p, q

'2p=q

2p¤ =q¤

q¤ q

q=2k (k 2p¤ =4k¤

p¤ =2k¤

p p, q

'2

서로소 짝수 짝수

7-14x+;[$;æ2æ≠x¥;[$; =4 { x=;[$; }

y+;](;æ2æ≠y¥;](; =6 { y=;](; }

x+y+;[$;+;](;æ4+6=10

x+y+;[$;+;](; 10

7-15+ +

=;aB;+;aC;+;bC;+;bA;+;cA;+;cB;

={;aB;+;bA;}+{;bC;+;cB;}+{;aC;+;cA;}

æ2æ≠;aB;¥;bA; +2æ≠;bC;¥;cB; +2æ≠;aC;¥;cA;

a=b=c

=2+2+2=6

+ + 6a+b

c

c+a

b

b+c

a

a+b

c

c+a

b

b+c

a


462

7-16(a b) y=;[*;

b=;a*; ab=8

2a>0 b>0

2a+bæ2'∂2ab=8 2a=b

2a+b 8

7-17

a, b (a>0, b>0) a+b=20

ab

a+bæ2'a åb a=b

20æ2'a åb

0<ab…100 ab>0

100

7-18a b

[{;3!;}¤ +{;4!;} ¤ ](a¤ +b¤ )æ{;3A;+;4B;}¤

3a=4b

;1™4∞4;(a¤ +b¤ )æ25

a¤ +b¤ æ144

a¤ +b¤ 144

7-19f(x)>g(x)kx¤ +6x>3x¤ +2x-k

(k-3)x¤ +4x+k>0

x

k>3 yy

(k-3)x¤ +4x+k=0 D

=2¤ -k(k-3)<0

k¤ -3k-4>0

(k+1)(k-4)>0

k<-1 k>4 yy

k>4

D

4

7-20a(x¤ -2x+2)>4x-1

ax¤ -2(a+2)x+2a+1>0

⁄ a+0 x

ax¤ -2(a+2)x+2a+1>0

a>0

ax¤ -2(a+2)x+2a+1=0 D

={-(a+2)} ¤ -a(2a+1)<0

a¤ -3a-4>0

(a+1)(a-4)>0

a<-1 a>4

a>4

¤ a=0 4x-1<0

¤ x<;4!;

¤ a+0

⁄ ¤ a>4

7-21a‹ +b‹ +c‹ -3abc

=(a+b+c)(a¤ +b¤ +c¤ -ab-bc-ca)

a>0 b>0 c>0

a+b+c>0

a¤ +b¤ +c¤ -ab-bc-ca

=;2!; {(a-b)¤ +(b-c)¤ +(c-a)¤ }æ0

a=b=c

a‹ +b‹ +c‹ -3abcæ0

a‹ +b‹ +c‹ æ3abc

a+b>0 b+c>0 c+a>0

a+bæ2'aåb

b+cæ2'båc

c+aæ2'c åa

(a+b)(b+c)(c+a)æ2'aåb¥2'b åc¥2'c åa

=8abc

a=b=c

('a+'b)¤ -{'ƒ2(a+b)}¤

=a+2'a åb+b-2(a+b)

=2'aåb-a-b

=-('a-'b)¤ …0 a=b

'a+'b…'ƒ2(a+b)

D

4


463

7-22(a+b+c){;a!;+ }= + +2

(a+b+c){;a!;+ }æ2æ≠ ¥ +2=4

a=b+c

(a+b+c){;a!;+ } 4

7-23

(a¤ +b¤ )(x¤ +y¤ )æ(ax+by)¤

(ax+by)¤ …64

-8…ax+by…8 yy

a¤ +b¤ æ2øπ(ab)¤ =2|ab|

|ab|…8

-8…ab…8

-2…xy…2

-10…ab+xy…10 yy

a=-8, b=8, c=-10, d=10

ac+bd=80+80=160

7-24

a b c a>0 b>0 c>0

"√a¤ +b¤ +c¤ ='∂12 a¤ +b¤ +c¤ =12

(1+1+1)(a¤ +b¤ +c¤ )æ(a+b+c)¤

a=b=c

36æ(a+b+c)¤

0<a+b+c…6 a+b+c>0

a+b+c 6

4(a+b+c)=24

1

b+c

b+c

a

a

b+c

b+c

a

a

b+c

1

b+c

1X 4 Y

X 2 Y

{1 2 3 4}

{a b c} {a b}

{1 2 3 4}

{a b c} {a b}

2{1, 2, 3, 6}

{1, 2, 3, 4, 5, 6, 7, 8}

f(3)=3+2=5

f(1)=3, f(2)=4, f(3)=5, f(6)=8

{3, 4, 5, 8}

{1, 2, 3, 6}

{1, 2, 3, 4, 5, 6, 7, 8}

5

{3, 4, 5, 8}

3x

x y

x

y

y=x x

x

x y -a

y

8

543

xO 1 2 3 6

함수



464

4

유제 8-1

-2

0

1

2

0

1

2

3

4

X Yh

-2

0

1

2

0

1

2

3

4

X Yg

-2

0

1

2

0

1

2

3

4

X Yf

y

xO

4y=4

y

xO

-2y=-x-2

y

xO

1

y=x+1y

xO

y=x¤

X 0 Y

X 1 Y

ㄱ, ㄴ

유제 8-2X {-2, -1, 0, 1, 2}

Y {1, 2, 3, 4, 5}f(-2)=2, f(1)=2

a=-2또는 a=1

f(-2)=2, f(-1)=1, f(0)=4, f(1)=2,

f(2)=4 {1, 2, 4}

유제 8-32 f(2)=2'2+3

'2 f('2)=-2'2+3

f(2)+f('2)=2'2+3+(-2'2)+3=6

유제 8-43⁄ =3 3¤ =9 3‹ =27 3› =81 3fi =243 y

3≈ 3 9 7 1

f(6)+f(13)=9+3=12

유제 8-5f(xy)=f(x)+f(y) yy

x=2, y=2

f(4)=f(2)+f(2)=5+5=10 f(2)=5

x=4, y=2

f(8)=f(4)+f(2)=10+5=15

유제 8-6f(x)f(y)=f(x+y)+f(x-y) yy

x=1 y=0

f(1)f(0)=f(1)+f(1)

f(0)=2 f(1)=1

x=1 y=1

f(1)f(1)=f(2)+f(0)

-2

0

1

2

0

1

2

3

4

X Yk

유제 본문 p.225


465

f(2)=1-2=-1 f(1)=1 f(0)=2

x=2 y=1

f(2)f(1)=f(3)+f(1)

(-1)¥1=f(3)+1 f(1)=1 f(2)=-1

f(3)=-1-1=-2

유제 8-7f=g f(-1)=g(-1)

(-1)¤ +b=-2-1

b=-4

f(a)=g(a)a¤ -4=2a-1 a¤ -2a-3=0

(a+1)(a-3)=0

a=3 a+-1

a+b=3+(-4)=-1

유제 8-8f=g 2x+3=x¤ -x-1

x¤ -3x-4=0 (x+1)(x-4)=0

x=-1 x=4

X -1 4

X={-1}또는X={4}또는X={-1, 4}

유제 8-9k x

x=k

O k x

y

x=k

O k x

y

x=k

O k x

y

x=k

kO x

y

x=k

kO x

y

x=k

O k

x=k

x

y

x x=k

x=k 2

ㄱ, ㄴ, ㅁ, ㅂ

유제 8-10k y=k

ㄱ, ㄴ, ㄹ은 일

대일함수 k

y=k

ㄴ, ㄹ은일대일대응

ㄴ은항등함수 ㄷ은상수함수

유제 8-11a>0 f(x)=ax+b

f(-1)=-a+b=0 yy`

f(1)=a+b=1 yy`

a=;2!;, b=;2!;

f(x)=;2!;x+;2!;

f {;2!;}=;2!;¥;2!;+;2!;=;4#;

유제 8-12f(x)

f(x) X X

f(k)=2k+5=k

k=-5

유제 8-13f

y=f(x)

y=3x+b

(0 1)

b=1

x<0 xæ0

y=ax+1

a>0

y=f(x)

O

1

x

y

y=f(x)

k

k

O

y

x


466

a b a+b a=1 b=1

a+b 2

유제 8-14

f(x)=[x¤ 1…x…3

ax+b x<1 x>3

f(x)

x 1, 3

y=x¤ , y=ax+b

1=a+b yy

9=3a+b yy

a=4, b=-3

a-b=7

유제 8-15f f(x)=x

f(3)=3, f(5)=5 yy

g f(5)=g(3)=k

k=5

g(x)=5 g(2)=5

=;5#;

유제 8-16f(x) f(x)=x

⁄ x<-2

¤ f(x)=-3 f(x)=x x

¤ -3

¤ -2…x…1

¤ f(x)=2x-1

¤ 2x-1=x x=1

‹ x>1

f(x)=2 f(x)=x x

2

⁄ ¤ ‹

X={a, b, c}={-3, 1, 2}

a+b+c=0

유제 8-17X X

f(3)

g(2)

y

x1

1

O 3

9

1 1 2 3 3

2 1

2

3 1 2

1

a=3¥2¥1=6

X Y

1 1 2 3 4 4

2 1 3

3 1 2

2

b=4¥3¥2=24

a+b=6+24=30

유제 8-18X Y n«

n

f(n)=n« -n

X Y

n(n-1)_y_1

x¡=y¡

(n-1)(n-2)_y_1

g(n)=n(n-1)_y_1-(n-1)(n-2)_y_1

=(n-1){(n-1)(n-2)_y_1}

f(3)+g(4)=3‹ -3+(4-1){(4-1)(4-2)(4-3)}

=27-3+3_6

=42

8-1f(-1)=3, f(0)=3, f(1)=3

g(-1)=1, g(0)=2, g(1)=3

h(-1)=4, h(0)=3, h(1)=4

k(-1)=0, k(0)=1, k(1)=4

k(-1)=0 0

X Y ㄱ ㄴ ㄷ

8-2f(3)=2

f(4)=5 a=4

{1, 2, 3, 4}



467

{1, 2, 3, 5}

8-3f(x)=

f(2)=2+1=3, f(3)=1

f(2)+f(3)=4

8-4f(-2)=2_(-2)-1=-5, f(1)=2_1-1=1

f(2)=2_2-1=3, f(a)=2a-1

f {-5, 1, 3, 2a-1}

{-5, 1, 7, b} {-5, 1, 3, 2a-1}

2a-1=7, b=3

a=4, b=3

a+b=7

8-5f(x)=g(x) f(1)=g(1), f(2)=g(2)

1¤ -1=a+b a+b=0 yy

2¤ -2=2a+b 2a+b=2 yy

a=2, b=-2

a-b=4

8-6a>0 f(x)=ax+b x

f(x)

f(0)=b=-4 f(4)=4a+b=12

a=4, b=-4

a+b=0

8-7x y 2

x¡+x™ f(x¡)=f(x™) x¡, x™

④

8-8f f(x)=x

f(10)=10 f(30)=30

f(10)+g(20)=50

x+1 x…2

+1 x=3[

10+g(20)=50 g(20)=40

g g(x)=40

g(40)=40

f(30)+g(40)=30+40=70

8-9f(x)=2x+1

f(1)=2¥1+1=a

a=3

f(b)=2¥b+1=7

b=3

a+b=6

8-10X X

4_3_2_1=24 a=24

X X

f(x)=1, f(x)=2, f(x)=3, f(x)=4 4

b=4

X X f(x)=x 1

c=1

a+b+c=24+4+1=29

8-11X

f(-1)=-(-1)¤ +2=1

f(0)=0¤ +2=2

f(1)=3

f(2)=2-1=1

f(3)=3-1=2

f {1, 2, 3}

8-120…x…4

f(x)=x+1

f(3)=3+1=4

x>4

f(x)=f(x-4)

f(27)=f(23)=f(19)=y=f(7)=f(3)=4

f(3)+f(27)=4+4=8

8-132 1, 2 2 f(2)=2

7

3

31

y

xO

y=f(x)


468

3 1, 3 2 4 1, 2, 4

3

f(3)+f(4)=2+3=5

f(n)=2 n

2 10 n

2, 3, 5, 7 4

ㄱ, ㄷ

8-14f(x+y)-f(y)=f(x)+xy

f(x+y)=f(x)+f(y)+xy yy

x=1, y=1

f(2)=f(1)+f(1)+1

f(2)=2f(1)+1=3 f(1)=1

x=2, y=1

f(3)=f(2)+f(1)+2

f(3)=6 f(2)=3, f(1)=1

x=3, y=2

f(5)=f(3)+f(2)+6

f(3)=15 f(3)=6, f(2)=3

8-15X x f(x)=g(x)

2x¤ =x‹ -3x x‹ -2x¤ -3x=0

x(x¤ -2x-3)=0

x(x+1)(x-3)=0

x=-1 x=0 x=3

X

{-1 0 3}

2‹ -1=7

8-16g(x) g(3)=3

f(2)=h(6)=3

f(x) f(2)=3

f(2)f(3)=f(6) f(3)=2

h(x) h(2)=3

f(3)+h(2)=2+3=5

8-17f(x)=a|x-2|+x-1

⁄ x<2

f(x)=a(-x+2)+x-1=(1-a)x+2a-1

¤ xæ2

f(x)=a(x-2)+x-1=(a+1)x-2a-1

⁄ ¤ (2 1)

f(x)

(1-a)(a+1)>0

(a-1)(a+1)<0

-1<a<1

8-18f(x)=2x¤ -4x+k=2(x-1)¤ +k-2

xæ3 x f(x)

f f(3)=3

2¥3¤ -4¥3+k=3

k=-3

8-19f

y=f(x)

y=ax-b

(1, 3)

a-b=3 yy

xæ1 x<1

a>0 a=b+3>0

b>-3

8-20A B f f(-x)=f(x)

⁄ -1 1

-2, -1, 0, 1, 2 5

¤ ⁄ 0

-2, -1, 0, 1, 2 5

⁄ ¤ f

5_5=25

8-211000 1000=2‹ _5‹

f(1000)=f(2‹ _5‹ )=f(2‹ )+f(5‹ )

f(1000)=f(2)+f(2¤ )+f(5)+f(5¤ )

f(1000)=f(2)+f(2)+f(2)+f(5)+f(5)+f(5)

f(1000)=3{ f(2)+f(5)}

f(1000)=3(2+5) f(2)=2, f(5)=5

f(1000)=21

y

xO

3

y=f(x)

1


469

8-22⁄ A

A 3_2_1=6

n(A)=6

¤ B f(1)=2

2 2 3 4 3

3 2 3 4 3

n(B)=3_3=9

‹ A;B f(1)=2

2 1

2

3 1 2

1

n(A;B)=2_1=2

⁄ ¤ ‹


n(A'B)=6+9-2=13

8-23f(x)=f(x+4) f(x) 4

f(x)=;n!;x

[

⁄ 2…n…4 3

¤ n=5 5

‹ 6…n…8 7

› n=9 9

fi 10…n…12 11

fl n=13 13

27 n 26…n…28

n 26

8-24f(a+b)=f(a)+f(b)+1

a=0, b=0

y=f(x)

y=;n!;x

y

x

y= x1

6

y= x1

2

n=2

y y

n=6

O-1

y=f(x)

1

2 4 6-2-4

-6

f(0)=f(0)+f(0)+1

f(0)=-1

a=x, b=-x

f(x-x)=f(x)+f(-x)+1

f(0)=f(x)+f(-x)+1

-1=f(x)+f(-x)+1

f(x)=-f(-x)-2 yy

y=f(x) (0, -1)

x -x, y

-2-y

-2-y=f(-x)

y=-f(-x)-2

y=f(x)

y=f(x) (0, -1)

ㄱ, ㄷ

참고 y=f(x) (a b)

2b-y=f(2a-x)


470

1(g Á f)(1)=g(f(1))=g(2)=4

(g Á f)(2)=g(f(2))=g(2)=4

(g Á f)(1)=4, (g Á f)(2)=4,

(g Á f)(3)=g(f(3))=g(4)=5

g Á f {4, 5}

4 4 {4, 5}

2(f Á g)(x)=f(g(x))=f(3x)

(f Á g)(x)=(3x)¤ -1

(f Á g)(x)=9x¤ -1

(g Á f)(x)=g(f(x))=g(x¤ -1)

(g Á f)(x)=3(x¤ -1)

(g Á f)(x)=3x¤ -3

(f Á g)(x)=9x¤ -1

(g Á f)(x)=3x¤ -3

3g(2)=2+1=3

( fΩg)(2)=f(g(2))=f(3)=3¤ =9

f(2)=2¤ =4

(gΩf )(2)=g( f(2))=g(4)=4+1=5

h(-1)=2¥(-1)-3=-5

g(-5)=-5+1=-4

( fΩ(gΩh))(-1)=f((gΩh)(-1))

=f(g(-5))

=f(-4)=(-4)¤ =16

h(-1)=-5 g(-5)=-4

(( fΩg)Ωh)(-1)=(fΩg)(h(-1))

=(fΩg)(-5)

=f(g(-5))

=f(-4)=(-4)¤ =16

9 5 16 16

4f(1)=3 f —⁄ (3)=1

f —⁄ (2)=2, f —⁄ (3)=1, f —⁄ (4)=3

f —⁄ {1, 2, 3}

1 {1, 2, 3}

5

y=3x-9 x= = y+3

x y y= x+3

f —⁄ (x)= x+3

y=4x+2 x= = y-

x y y= x-

g—⁄ (x)= x-

f —⁄ (x)=;3!;x+3

g—⁄ (x)=;4!;x-;2!;

6f —⁄ (-3)=k f(k)=-3

k-2=-3

k=-1

f —⁄ (-3)=-1

(g Á f)—⁄ (4)=(f —⁄ Á g—⁄ )(4)=f —⁄ (g—⁄ (4))=f —⁄ (5)

f —⁄ (5)=k f(k)=5

k-2=5

k=7

(g Á f )—⁄ (4)=7

-1 7

7x y

유제 9-1(g Á f )(2)=g(f(2))=g(5)=7

(g Á f )(3)=g(f(3))=g(6)=8

1

2

1

4

1

2

1

4

1

2

1

4

y-2

4

1

3

1

3

1

3

y+9

3

유제 본문 p.251

합성함수와역함수



471

(g Á f )(2)+(g Á f )(3)=7+8=15

(g Á f)(1)=g(f(1))=g(6)=8

(g Á f)(2)=7 (g Á f)(3)=8

(g Á f)(4)=g(f(4))=g(6)=8

g Á f {7, 8}

유제 9-2f(2)=-1

( fΩf)(2)=f( f(2))=f(-1)=2

f(-1)=2

(gΩf)(-1)=g( f(-1))=g(2)=7

(hΩg)(x)=h(g(x))=-(2x+3)¤ +2

((hΩg)Ωf)(x)=(hΩg)( f(x))=(hΩg)(-x+1)

=-{2(-x+1)+3}¤ +2

=-(-2x+5)¤ +2

=-4x¤ +20x-23

유제 9-3fΩ(gΩh)=(fΩg)Ωh

( fΩ(gΩh))(4)=(( fΩg)Ωh)(4)

=(fΩg)(h(4))=(fΩg)(1) h(4)=4-3=1

=2¥1¤ -5

=-3

유제 9-4(gΩf)(-2)+(fΩg)(3)=g( f(-2))+f(g(3))=g(-3)+f(8)

=-18+17=-1

유제 9-5f(x)=ax+3, g(x)=2x-1

( fΩg)(x)=f(2x-1)

=a(2x-1)+3

=2ax+3-a yy

(gΩf)(x)=g(ax+3)

=2(ax+3)-1

=2ax+5 yy

2ax+3-a=2ax+5

3-a=5

a=-2

유제 9-6f(x)=ax+b g(x)=2x¤ +3x+1

( fΩg)(x)=f(g(x))=a(2x¤ +3x+1)+b

=2ax¤ +3ax+a+b yy

(gΩf)(x)=g( f(x))=2(ax+b)¤ +3(ax+b)+1

=2a¤ x¤ +(4ab+3a)x+2b¤ +3b+1

yy

x

2a=2a¤ , 3a=4ab+3a, a+b=2b¤ +3b+1

a=1, b=0 a+0

f(x)=x

f(1)+f(2)+f(3)+y+f(10)=1+2+3+y+10

=55

유제 9-7(h Á f )(x)=h(f(x))=h(3x+4)=g(x)

3x+4=t x=

h(t)=g { }=6¥ -3=2t-11

t x

h(x)=2x-11

(h Á g Á f )(x)=h(g(f(x)))=h(18x+21)

=g(x)18x+21=t

x=

h(t)=g { }=6¥ -3

h(t)=;3!;t-10

t x

h(x)=;3!;x-10

유제 9-8

f { }=6x+4

=t x=

f(t)=6¥ +4=4t+6

t x

f(x)=4x+6

2t+1

3

2t+1

3

3x-1

2

3x-1

2

t-2118

t-2118

t-2118

t-43

t-43

t-43


472

유제 9-9f ⁄ (x)=f(x)=x-1

f ¤ (x)=(fΩf)(x)=f( f(x))=f(x-1)=x-2

f ‹ (x)=(fΩf ¤ )(x)=f( f ¤ (x))=f(x-2)=x-3

f « (x)=x-n

f fi ‚ (x)=x-50

f fi ‚ (a)=a-50=50

a=100

유제 9-10f 4

f › (x)=x

f › « (x)=x f4n+1

(x)=f(x) f4n+2

(x)=f ¤ (x)

f4n+3

(x)=f ‹ (x)

f102(1)=f ¤ (1)=f( f(1))=f(4)=3

f103(4)=f ‹ (4)=f( f( f(4)))=f( f(3))=f(2)=1

f102(1)+f

103(4)=3+1=4

유제 9-11

f {;4!;}=b f(b)=c f(c)=d

( fΩfΩf){;4!;}=f { f { f {;4!;}}}=f( f(b))

( fΩfΩf){;4!;}=f(c)=d

f(x)=t

( fΩf)(x)=f( f(x))=f(t)=c

f(b)=c t=b

f(x)=b x

x=;4!;

ab

b

c

c

=

d

d

e

e

1

1

yy=x

y=f(x)

xO 1

4

a

f f f fX

1

2

3

4

X

1

2

3

4

X

1

2

3

4

X

1

2

3

4

X

1

2

3

4

유제 9-12

f(x)=

y=(fΩg)(x)=f(g(x))g(x)=-x+1 f(x) x

f(g(x))=

f(g(x))=

f(g(x))=

y=f(g(x))

유제 9-13f —⁄ (1)=2 f(2)=1

f(2)=2¥2+k=1 k=-3

f(x)=2x-3

f(1)=2¥1-3=-1

유제 9-14xæ0 f(x)æ3, x<0 f(x)<3

f —⁄ (2)=k f(k)=2

k<0 f(k)=-k¤ +3=2

k¤ =1 k=-1 k<0

f —⁄ (2)+f —⁄ (a)=5 -1+f —⁄ (a)=5

f —⁄ (a)=6

a=f(6)=2¥6+3=15

유제 9-15f f

O

1

3

y=f(g(x))

1-2 -1 x

y

0 x>1

-x+1 0<x…1

1 -1<x…0

-2x-1 x…-1

(\{\9

0 -x+1<0

-x+1 0…-x+1<1

1 1…-x+1<2

2(-x+1)-3 -x+1æ2

(\{\9

0 g(x)<0

g(x) 0…g(x)<1

1 1…g(x)<2

2g(x)-3 g(x)æ2

(\{\9

0 x<0

x 0…x<1

1 1…x<2

2x-3 xæ2

(\{\9


473

y=f(x)

f(a)=-2a+5=-3 a=4

f(1)=-2¥1+5=b b=3

a+b=7

유제 9-16

f(x)=|x-2|+ax-5=

f f

xæ2 x<2

(a+1)(a-1)>0

a<-1 또는 a>1

유제 9-17y=;2!;x+3 R R

y=;2!;x+3 x

x=2y-6

x y

y=2x-6

y=-2x+4 xæ1

y=-2x+4 x

x=-;2!;y+2

x y y=-;2!;x+2

{y|y…2}

{x|x…2}

y=-;2!;x+2, 정의역：{x|x…2}

유제 9-18g—⁄ Á f —⁄ =(f Á g)—⁄(f Á g)(x)=f(g(x))=f(3x-5)

=4(3x-5)-3

=12x-23

(g—⁄ Á f —⁄ )(a)=(f Á g)—⁄ (a)=3

(f Á g)(3)=a

a=12¥3-23=13

f Á (g Á f)—⁄ Á f=f Á (f —⁄ Á g—⁄ ) Á f=(f Á f —⁄ )Ω(g—⁄ Á f)=IÁ g—⁄ Á f I

(a+1)x-7 xæ2

(a-1)x-3 x<2[

=g—⁄ Á f(f Á (g Á f)—⁄ Á f)(2)=(g—⁄ Á f)(2)

=g—⁄ (f(2))=g—⁄ (5)

g—⁄ (5)=k g(k)=5

3k-5=5 k=:¡3º:

(f Á (g Á f)—⁄ Á f )(2)=:¡3º:

다른 풀이

(g—⁄ Á f —⁄ )(a)=3 g—⁄ (f —⁄ (a))=3

f —⁄ (a)=g(3)a=f(g(3))=f(4)=13

유제 9-19(( f —⁄ Ωg)—⁄ Ωf)(5)=(g—⁄ ΩfΩf)(5)

=g—⁄ ( f( f(5))=g—⁄ ( f(-3))

=g—⁄ (11)g—⁄ (11)=k g(k)=11

2k-3=11 k=7

(( f —⁄ Ωg)—⁄ Ωf)(5)=7

유제 9-20(f Á g)—⁄ =g—⁄ Á f —⁄f —⁄ (b)=k f(k)=b k=a

f —⁄ (b)=a

g—⁄ (a)=l g(l)=a l=b

g—⁄ (a)=b

(f Á g)—⁄ (b)=(g—⁄ Á f —⁄ )(b)=g—⁄ (f —⁄ (b))=g—⁄ (a)=b

유제 9-21y=f(x) y=f—⁄ (x)

y=f(x) y=x

;2!;x+3=x x=6

(6, 6)

유제 9-22y=f(x) y=f—⁄ (x) P

y=f(x) y=x

P y=x P

(t, t) OP”=2'2


474

"√t¤ +t¤ =2'2, 2t¤ =8

t¤ =4 t=2 tæ1

P(2, 2) f(x)=x¤ -2x+k

f(2)=2¤ -2¥2+k=2 k=2

9-1(f Á g)(x)=f(g(x))=f(x¤ +1)

=-3(x¤ +1)+1=-3x¤ -2

(g Á f )(x)=g(f(x))=g(-3x+1)

=(-3x+1)¤ +1=9x¤ -6x+2

9-2f(x)=x+2, g(x)=2x-1, h(x)=3x-2

((hΩg)Ωf)(2)=(hΩg)( f(2))=(hΩg)(4)=h(g(4))=h(7)=19

다른 풀이

((hΩg)Ωf)(x)=(hΩg)( f(x))=h(g(x+2))

=h(2(x+2)-1)=h(2x+3)

=3(2x+3)-2=6x+7

((hΩg)Ωf)(2)=6¥2+7=19

9-3(gΩf)(1)=g( f(1))=g(5)=9

(gΩf)(3)=g( f(3))=g(6)=7

(gΩf)(1)+(gΩf)(3)=9+7=16

9-4( fΩg)(x)=f(g(x))=f(2x-3)

=(2x-3)¤ -2(2x-3)+1

=4x¤ -12x+9-4x+6+1

=4x¤ -16x+16=4(x-2)¤

fΩg 1…x…2

( fΩg)(1)=4, ( fΩg)(2)=0

fΩg {y|0…y…4}

a+b=0+4=4

9-5f ⁄ (x)=f(x)=;3{;

f ¤ (x)=(fΩf)(x)=f( f(x))=f {;3{;}

f ¤ (x)=;3!;¥;3{;=

f ‹ (x)=(fΩf ¤ )(x)=f( f ¤ (x))=f { }

f ‹ (x)=;3!;¥ =

f « (x)=

f fi (x)=

f fi (a)= =3

a=3fl =729

9-6f-1(1)=2 f(2)=1

f(2)=3¥2+k=1 k=-5

f(x)=3x-5

f(10)=3¥10-5=25

9-7y=f—⁄ (x) (4 0) (0 2)

y=-;2!;x+2

x

x=-2y+

x y f—⁄ (x)

( f—⁄ )—⁄ (x)=-2x+4

( f—⁄ )—⁄ (x)=f(x)

f(x)=-2x+4

f(7)=-2¥7+4=-10

9-8(f Á (g Á f)—⁄ Á f)(2)=(f Á f —⁄ Ág—⁄ Á f)(2)(f Á (g Á f)—⁄ Á f)(2)=(IΩg—⁄ Á f)(2)

I

(f Á (g Á f)—⁄ Á f)(2)=(g—⁄ Á f)(2)(f Á (g Á f)—⁄ Á f)(2)=g—⁄ (f(2))

(f Á (g Á f)—⁄ Á f)(2)=g—⁄ {;3@;}

, g—⁄ {;3@;}=k g(k)=;3@;

g(k)=-3k+2=;3@; k=;9$;

(f Á (g Á f)—⁄ Á f)(2)=;9$;

a

3fi

x

3fi

a

3«

x

3‹

x

3¤

x

3¤

x

3¤



475

9-9f(x)=x+a, g(x)=bx-4

( fΩg)(x)=x-2 f(g(x))=x-2

f(bx-4)=x-2

(bx-4)+a=x-2

a=2, b=1

f(x)=x+2, g(x)=x-4

f-1(4)=k f(k)=4

f(k)=k+2=4 k=2

f-1(4)=2

g-1(-4)=l g(l)=-4

g(l)=l-4=-4 l=0

g—⁄ (-4)=0

f —⁄ (4)+g—⁄ (-4)

f-1(4)+g-1

(-4)=2+0=2

9-10y=f(x) y=f—⁄ (x)

y=f(x) y=x

x¤ -6x=x x¤ -7x=0

x(x-7)=0 x=7 xæ3

(7 7)

10ab=10¥7¥7=490

9-11h(x)=ax+b

(hΩf)(x)=h( f(x))=h(2x+3)

=a(2x+3)+b=2ax+3a+b

(hΩf)(x)=g(x)2a=-4, 3a+b=-5

a=-2, b=1 h(x)=-2x+1

(hΩg)(-2)=h(g(-2))=h(3)=-5

다른 풀이

(hΩf)(x)=g(x) h( f(x))=g(x)f(x)=2x+3 h(2x+3)=g(x)

(hΩg)(-2)=h(g(-2))=h(3)

h(2x+3)=h(3)

x=0

h(3)=g(0)=-5

9-12(f Á f)(x)=f(f(x))=f(2x-1)

=2(2x-1)-1=4x-3

g(x)=(f Á f Á f )(x)=f((f Á f )(x))

=f(4x-3)=2(4x-3)-1

=8x-7

y=8x-7

1…x…3

x=3 M=8¥3-7=17

x=1 m=8¥1-7=1

M+m=18

9-13f ⁄ {;2!;}=f{;2!;}=;2!;+1=;2#;

f ¤ {;2!;}=f{f{;2!;}}=f{;2#;}=;2#;-1=;2!;

f ‹ {;2!;}=f{f ¤ {;2!;}}=f{;2!;}=;2!;+1=;2#;

f ⁄ {;2!;}+f ¤ {;2!;}+y+f ⁄ ‚ {;2!;}=5_;2#;+5_;2!;

=10

9-14f(x)=[ , g(x)=-x+1

⁄ xæ1

⁄ (gΩf)(x)=g( f(x))=g(2)=-1

¤ x<1

⁄ (gΩf)(x)=g( f(x))=g(x+1)

=-(x+1)+1=-x

⁄ ¤ (gΩf)(x)=[

y=(gΩf)(x) ②

9-15h—⁄ (1)=k h(k)=1=f(3k+1)

f

f —⁄ (1)=4 f(4)=1

3k+1=4

k=1

h—⁄ (1)=1

9-16f(b)=a

f(c)=b f —⁄ (b)=c

g(b)=c

f(b)+g(b)=a+c

f —⁄ (a)=k f(k)=a

-1 xæ1

-x x<1

2 xæ1

x+1 x<1


476

f(b)=a k=b

(f Á f)—⁄ (a)=(f —⁄ Á f —⁄ )(a)=f —⁄ (f —⁄ (a))

(f Á f)—⁄ (a)=f —⁄ (b)

(f Á f)—⁄ (a)=c f(c)=b

9-17f(x)=x¤ -2x-10=(x-1)¤ -11

f

aæ1

f

f

f(a)=a

a¤ -2a-10=a a¤ -3a-10=0

(a+2)(a-5)=0

a=5 aæ1

9-18f(g(1))=2 f(1)=2 g(1)=1

g(2)=5 g(3)=2 g(4)=3 g(5)=4

(gΩf)—⁄ (1)=f —⁄ (g—⁄ (1)) g—⁄ (1)=k

g(k)=1 k=1

f —⁄ (1)=l f(l)=1 l=5

g(2)+(gΩf)—⁄ (1)=5+5=10

9-19f ‹ =I

f(1)=3 f(3)=2 f(2)=1

f g g ‹ =I

g ⁄ ‚ =g g ⁄ ⁄ =g ¤

g ⁄ ‚ (2)+g ⁄ ⁄ (3)=g(2)+g ¤ (3)=3+2=5

g gX

1

2

3

X

1

2

3

X

1

2

3

Xf

1

2

3

X

1

2

3

9-20y=f(x) y=f—⁄ (x)

y=x

A B D C y=x

A A(1, 2)

B B(2, 1)

D D(a, 1) a<0

C C(1, a)

f(1)-f —⁄ (1)=AC”=|2-a|=7

a=-5 a<0

ADB

;2!;_7_(2-1)=;2&;

9-21h(x)=(f Á g)(x)⁄ x

⁄ h(x)=f(g(x))=f(2)=2

¤ x

⁄ h(x)=f(g(x))=f(3)=1

h(x)=‡

h(1)+h(2)+y+h(10)

=2+1+2+1+y+1

=5¥2+5¥1=15

9-22x

(f Á g)(x)=f(g(x))æ0

{g(x)}¤ -g(x)-6æ0

{g(x)+2} {g(x)-3}æ0

g(x)…-2 g(x)æ3

g(x)=x¤ -ax+4…-2 x

x

g(x)=x¤ -ax+4æ3

x¤ -ax+1æ0

x¤ -ax+1=0 D

D…0

D=a¤ -4=(a+2)(a-2)…0

-2…a…2

9-23( fΩf)(x)=f(x) f(x)=t

f(t)=t

2 x

1 x


477

(t-a)¤ +a=t (t-a){t-(a+1)}=0

t=a t=a+1

⁄ t=a f(x)=a

⁄ (x-a)¤ +a=a, (x-a)¤ =0

⁄ x=a

¤ t=a+1 f(x)=a+1

⁄ (x-a)¤ +a=a+1 (x-a)¤ =1

⁄ x=a+1 x=a-1

⁄, ¤ a+(a+1)+(a-1)=3

a=1

9-24f

A B

A'B=S, A;B=Δ A, B

2 A B

A

{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} 6

A={1, 2} B={3, 4}

f 2

6¥2=12

9-25x<1 y=x-2 xæ1

y=x¤ +nx+m (1, -1)

x y

y=x¤ +nx+m={x+;2N;} 2+m-

x=-;2N; -;2N;…1

næ-2 yy

(1, -1)

f(1)=1+n+m=-1

m=-n-2 yy

m=-n-2 næ-2

m n nm

③

n¤4

1

2

3

4

Af

B

1

2

3

4

Af

B

1

-3, , x-3

, ,

2

= ¥ ¥ ¥ =

= =

3

,

,

4

- = -

- = =

+

= +

=x+2+x-2

=2x

2x

5

_

= _ =1

(x+2)¤

x

(x-2)(x+2)

x-2

x(x+2)

x

x¤ -4

x-2

x¤ +2x

1

x

(x-2)(x+1)

x+1

(x-1)(x+2)

x-1

x¤ -x-2

x+1

x¤ +x-2

x-1

1

x

x-1

x(x-1)

1

x(x-1)

x

x(x-1)

1

x(x-1)

1

x-1

x(x-2)

(x-1)(x-2)

3(x-1)

(x-1)(x-2)

y¤

x¤ y

3x

x¤ y

x-2

x+1

2y

xz¤

x-2

x+1

(x-1)(x-2)

(x-1)(x+1)

(x-1)(x-2)

x¤ -1

2y

xz¤

1

z¤

y‹

y¤

x

x¤

6

3

6xy‹

3x¤ y¤ z¤

x¤ +2x

x-3

x-1

x+1

2

x

2x

5

유리함수



478

÷

= _ =

6

= =xy

=

xy

7

8

= = +1

= = +4

+1 +4

9

y=- -1 y=-

y -1

{x|x x+0 }

{y|y y+-1 }

y

O-1

2y=- -1x

x

2

x

2

x

7

x-3

5

x-3

7

x-3

4(x-3)+7

x-3

4x-5

x-3

5

x-3

(x-3)+5

x-3

x+2

x-3

y

-1

-2

21

O

2y=-x

x

y

1-3

-1

3 xO

3y= x

2(x+1)

x(x-1)

2(x+1)

x(x-1)

2

x-1

x

x+1

x¤ y

x

y

x

1

x¤

x‹

(x+1)¤¤

1

(x+2)¤

x‹

(x+1)¤¤

x(x-1)

x+1

x¤

(x-1)(x+1)

x+1

x(x-1)

x¤

x¤ -1y= +1 y=

x 3 y 1

{x|x x+3 }

{y|y y+1 }

10

y= = = +2

y= y= x

1 y 2

x=1, y=2

y= = = -1

y= y= x

2 y -1

x=2, y=-1

y

O -1

2

y=x-4

2-x

x

2

x

x-4

2-x

2

x-2

-(2-x)-2

2-x

x-4

2-x

y

O 1

2

y=2x

x-1

x

2

x

2x

x-1

2

x-1

2(x-1)+2

x-1

2x

x-1

y

O

1

3

y=2

+1x-3

x

2

x

2

x-3


479

유제 10-1

+ = +

+ =

+ =

+ =

-

= -

=

=

=

=

유제 10-2

-;[!;- +

= +

=

=

- - +

={1+ }-{1+ }

-{1+ }+{1+ }

= - - +

= -

=

=6x

(x-2)(x-1)(x+1)(x+2)

(x+1)(x+2)-(x-2)(x-1)

(x-2)(x-1)(x+1)(x+2)

1

(x+1)(x+2)

1

(x-2)(x-1)

1

x+2

1

x+1

1

x-1

1

x-2

1

x+2

1

x+1

1

x-1

1

x-2

x+3

x+2

x+2

x+1

x

x-1

x-1

x-2

6(x¤ -x-1)

x(x-1)(x+2)(x-3)

(x+2)(x-3)+5x(x-1)

x(x-1)(x+2)(x-3)

5

(x+2)(x-3)

1

x(x-1)

1

x-3

1

x+2

1

x-1

2a-9

2a-1

(2a-9)(a+3)

(2a-1)(a+3)

2a¤ -3a-27

(2a-1)(a+3)

a(2a-1)-(2a+27)

(2a-1)(a+3)

2a+27

(2a-1)(a+3)

a

a+3

2a+27

2a¤ +5a-3

a

a+3

1a+b

a-b

(a-b)(a+b)

a-b

a¤ -b¤

-b

a¤ -b¤

a

a¤ -b¤

b

b¤ -a¤

a

a¤ -b¤

유제 10-3

_

= _

=

÷

= _

=

유제 10-4

_ ÷

= _ _

=;[!;

유제 10-5

=1+

=1+

=1+

=1+

=

= [{ - }+{ - }

+{ - }]

= { - }

=3

x(x+6)

1

x+6

1

x

1

2

1

x+6

1

x+4

1

x+4

1

x+2

1

x+2

1

x

1

2

3x+22x+1

x+1

2x+1

1

2x+12211x+1

1

x1+2211

x+1

1

11+2211

x+1112x

x+4

(x-1)(x-3)

x-1

x+2

(x+2)(x-3)

x(x+4)

x¤ -4x+3

x+4

x-1

x+2

x¤ -x-6

x¤ +4x

x-2y

3x

(x+y)(x-2y)

3x¤

x(x+y)

(x+y)¤

3x¤

x¤ -xy-2y¤

x¤ +xy

x¤ +2xy+y¤

x+2

x-5

(x-2)(x+2)

x(x+3)(x-5)

x(x+3)

x-2

x¤ -4

x‹ -2x¤ -15x

x¤ +3x

x-2

유제 본문 p.283


480

유제 10-6

y= = = +2

y=

x 1 y 2

정의역：{x|x는 x+1인실수}치역：{y|y는 y+2인실수}점근선의방정식：x=1, y=2

y= = =- -3

y=-

x 2 y -3

정의역：{x|x는 x+2인실수}치역：{y|y는 y+-3인실수}점근선의방정식：x=2, y=-3

유제 10-7

y= +c

x=2 y=-1

a=-2 c=-1

y= -1

0= -1 b=-2

abc=(-2)¥(-2)¥(-1)=-4

b

0-2

b

x-2

b

x+a

y

O2

-3

x

4

x

4

x-2

-3(2-x)+4

2-x

3x-2

2-x

y

O 1

2

x

3

x

3

x-1

2(x-1)+3

x-1

2x+1

x-1

유제 10-8x=1 y=2

y= +2

(2, 4)

4= +2 k=2

k=2

y= +2=

y= =

a=2, b=0, c=-1

다른 풀이

x=1 y=2

x=-;1C;=1 c=-1

y=;1A;=2 a=2

y= (2 4)

4= b=0

유제 10-9

y= = +2 y=;[!;

x 1 y 2

y= = -3

y=;[!; x -1 y

-3

y= x -2

y -5

y=

m=-2, n=-5

m+n=-7

유제 10-10

y=- =- = -2

2-c=-3 c=5

2-c

x+1

2(x+1)+c-2

x+1

2x+c

x+1

-3x-2

x+1

2x-1

x-1

1

x+1

-3x-2

x+1

1

x-1

2x-1

x-1

4+b

2-1

2x+b

x-1

ax+b

x+c

2x

x-1

2+2(x-1)

x-1

2

x-1

k

2-1

k

x-1


481

유제 10-11

y= = = +3

x=-3 y=3

(-3, 3) 1 -1

a>0 1 (-3, 3)

y-3=x+3

y=x+6

a=1, b=6

a+b=7

유제 10-12

y= = = +;3@;

x=;3!;, y=;3@;

{;3!; ;3@;}

1 -1

y-;3@;=x-;3!;

y=x+;3!;

y-;3@;=-{x-;3!;}

y=-x+1

a=;3!;, b=-1, c=1

abc=-;3!;

y

x1

3

2

3

O

5

3(3x-1)

;3@;(3x-1)+;3%;11111113x-1

2x+1

3x-1

y

x

y=-x

y=x+6

O

3

-3

1

x+3

3(x+3)+1

x+3

3x+10

x+3

유제 10-13

f(x)= =

f(x)= +3

x…-2 xæ3

y=f(x)

x=3 M=6

x=-2

m=1

M+m=7

유제 10-14

y= x=-1, y=2

a=1, b=2

y= =

y=- +2

1…x…2

x=2 ;3%;

x=1 ;2#;

;3%;+;2#;=:¡6ª:

참고 y=

참고 x=-;cD;, y=;cA;

유제 10-15

y=f—⁄ (x)= x

-xy+y=4x-3, (y+4)x=y+3

x=

x y y=x+3

x+4

y+3

y+4

4x-3

-x+1

ax+b

cx+d

O

2

1-1 2 x

y

3

5

2

3

1

x+1

2(x+1)-1

x+1

2x+1

x+1

bx+a

x+a

O

6

3

1

31-2 x

y

6

x-1

3(x-1)+6

x-1

3x+3

x-1


482

f(x)= =

a=1, b=3, c=4

a+b+c=8

참고 (f—⁄ )—⁄ (x)=f(x) f—⁄ (x)

f(x)

유제 10-16

-;[@;+2=2x+k -2+2x=2x¤ +kx

2x¤ +(k-2)x+2=0 yy

y= =-;[@;+2 y=2x+k

D

D=(k-2)¤ -16=0, k¤ -4k-12=0

(k+2)(k-6)=0 k=-2 k=6

k

-2+6=4

유제 10-17

y= = =- +2

x=-1, y=2

{x|0…x…2}

(0, -1), (2, 1)

y=m(x+2) m

(-2, 0)

⁄ (0, -1) -1=2m, m=-;2!;

¤ (2, 1) 1=4m, m=;4!;

⁄ ¤ m

-;2!;…m…;4!;

a=-;2!;, b=;4!;

8ab=8¥{-;2!;}¥;4!;=-1

y

x

-1

-1 2-2

1

2

O

3

x+1

2(x+1)-3

x+1

2x-1

x+1

2x-2

x

ax+b

x+c

x+3

x+4

10-1+

= +

=

=

- - +

={1+ }-{1+ }-{1+ }

+{1+ }

=;[@;- - +

= +

= +

=

=

_

= _

=1

= _

_

=x

10-2y= -1 y+1=

x

x+2=

x= -23

y+1

3

y+1

3

x+2

3

x+2

(x+3)(x-2)

x+3

x(x+2)

(x+2)(x+3)

(x+3)(x+1)

(x+1)(x-2)

x¤ -x+1

x-1

(x-1)(x+1)

(x+1)(x¤ -x+1)

x¤ -x+1

x-1

x¤ -1

x‹ +1

2(5x¤ -3x+12)

x(x+1)(x-3)(x-4)

2(x-3)(x-4)+8x(x+1)

x(x+1)(x-3)(x-4)

8

(x-3)(x-4)

2

x(x+1)

-8(x-4)+8(x-3)

(x-3)(x-4)

2(x+1)-2x

x(x+1)

8

x-4

8

x-3

2

x+1

8

x-4

8

x-3

2

x+1

2

x

x+4

x-4

x+5

x-3

x+3

x+1

x+2

x

3x-4

(x+2)(x-3)

3(x-3)+5

(x+2)(x-3)

5

(x+2)(x-3)

3

x+2

5

x¤ -x-6

3

x+2



483

y= -2

x=-1, y=-2

a=-1 b=-2

a+b=-3

다른 풀이

y= -1 {x|x+-2 }

{y|y+-1 }

{x|x+-1 } {y|y+-2 }

x=-1, y=-2

a=-1 b=-2

a+b=-3

10-3y= y=;[#;

x -1 0

-1

x=-1, y=0

1, 2, 3 y=x+1

ㄴ, ㄷ, ㄹ

10-4y= =

y= +5

x=1, y=5

(1, 5)

a=1, b=5

a+b=6

10-5y=

y=

y= -2

y=;[!; x 1

y -2

제2사분면

1

x-1

-2(x-1)+1

x-1

y

x

-2

1O

-2x+3

x-1

8

x-1

5(x-1)+8

x-1

5x+3

x-1

3

x+1

3

x+2

3

x+110-6y= = +2

x=1, y=2

(1, 2)

(1, 2)

—1

y=x+n x=1, y=2

2=1+n

n=1

10-7y= +q

x=2 y=-2

p=2, q=-2

y= -2

0= -2

k=-4

10-8y= = =- +1

y= y=-;[!;

x -1 y

1

y= = = +1

y= y=;[@;

x 1 y

1

y= = = +2

y= y=;[@;

x -2 y

2

y=;[@;

ㄴ, ㄷ

2x+6

x+2

2

x+2

2(x+2)+2

x+2

2x+6

x+2

x+1

x-1

2

x-1

(x-1)+2

x-1

x+1

x-1

x

x+1

1

x+1

(x+1)-1

x+1

x

x+1

k

0-2

k

x-2

k

x-p

1

2

y

xO

3

x-1

2x+1

x-1


484

10-9y=;[K; x 2

y 1

y= +1 yy

(4, 10)

10= +1

k=18

10-10( fΩg)(x)=x g(x) f(x)

y= x

xy-2y=x+3 (y-1)x=2y+3 x=

g(x)=

g(3)= =;2(;

10-11y= +3

x=1,

y=3

x=0 y

+3<0 a>3

10-12(1, -2)

x=1, y=-2

y= -2 k+0 yy

(2, -4)

-4= -2 k=-2

y= -2= =

a=-2, b=0, c=-1

a+b+c=-3

ax+b

x+c

-2x

x-1

-2

x-1

k

2-1

k

x-1

a

0-1

1

3

y

x

O

a

x-1

2¥3+3

3-1

2x+3

x-1

2y+3

y-1

x+3

x-2

k

4-2

k

x-2

10-13y=f(x) x=2, y=2

y= +2 k+0

y= +2= =

a=2, b=2, k=5

f(x)= = +2

f(-x)=-f(x) y=f(x)

y=f(x)

(2, 2)

y= x

xy-2y=2x+1 x=

f —⁄ (x)= =f(x)

y=f(x)

y=-x

ㄴ

10-14y= = = +1

3…x…5

y= y=ax a>0

y=ax (3, 4) a

M=;3$; (5, 2) a

m=;5@;

3M+5m=4+2=6

1

4

2

y

xO 2 3 5

x+1

x-2

3

x-2

(x-2)+3

x-2

x+1

x-2

y

x1

2

O

2

2-

2x+1

x-2

2y+1

y-2

2x+1

x-2

5

x-2

2x+1

x-2

ax+1

x-b

2x+k-4

x-2

k

x-2

k

x-2


485

10-15f(x)= = =- +1

x a y b

f(x)=- +1+b yy

y= x

xy+y=x-3

(y-1)x=-y-3

x=

f—⁄ (x)= =

f—⁄ (x)=- -1 yy

1-a=-1

1+b=-1

a=2, b=-2

ab=-4

10-16y= y=

y=x

y= x

2xy-6y=ax+1

(2y-a)x=6y+1

x=

y= =

a=-6 b=6

b-a=12

다른 풀이

y=x

y=x

y= x=3 y=;2A;

{3 ;2A;}

ax+1

2x-6

bx+1

2x+6

6x+1

2x-a

6y+1

2y-a

ax+1

2x-6

bx+1

2x+6

ax+1

2x-6

4

x-1

-(x-1)-4

x-1

-x-3

x-1

-y-3

y-1

x-3

x+1

4

x+1-a

4

x+1

(x+1)-4

x+1

x-3

x+1

y= x=-3 y=;2B;

{-3 ;2B;}

{3 ;2A;} {-3 ;2B;} y=x

3=;2B; ;2A;=-3 a=-6 b=6

b-a=12

10-17y=;[!; A

A {p ;p!;} B C

B {kp ;p!;} C {p ;pK;}

AB”=kp-p=(k-1)p

AC”=;pK;-;p!;=

ABC 50

;2!;_AB”_AC”=;2!;_(k-1)p_ =50

(k-1)¤ =100

k=11 k>1

10-18f(x)=

f ¤ (x)=(f Á f)(x)=f(f(x))=

f ¤ (x)= =-

f ‹ (x)=f(f ¤ (x))= =x

f › (x)=f(f ‹ (x))= =f(x)

, - , x

50=3_16+2

f fi ‚ (x)=f ¤ (x)=-

f fi ‚ (2)=- =12

1-2

2

1-x

x

1-x

x

1

1-x

1

1-x

1

1-x1+112

x

1-x

x

1

11-112

1-x

1

1-f(x)

1

1-x

k-1

p

k-1

p

bx+1

2x+6


486

10-19y= = +2 yy

y=m(x-1)+2 yy

x=1, y=2

(1, 2) m

m

m…0

10-20P P {a } Q R

Q(a 0) R {0 }

PQ”+PR”= +a

PQ”+PR= +a-1+1

PQ”+PRæ2æ≠ _(a-1)+1=7

{ =a-1 }

PQ”+PR” 7

10-21y= = =- +1

y |2x+3| 4

2x+3=—1, —2, —4

x

2x+3=—1

x=-1 x=-2

(-1, -3), (-2, 5)

a+b+c+d=(-1)+(-3)+(-2)+5

=-1

4

2x+3

2x+3-4

2x+3

2x-1

2x+3

9

a-1

9

a-1

9

a-1

9

a-1

9

a-1

9

a-1

y

xO 1

2

1

x-1

2x-1

x-1

10-22y= x+-1

⁄ x<-1 y= =1

¤ -1<x<0 y= =-1

‹ xæ0 y= =- +1

⁄ ¤ ‹

y=mx+2m-2

y=m(x+2)-2

m

(-2, -2)

(-1, -1)

-1=m-2 m=1

(-1, 1)

1=m-2 m=3

1…m…3 m

3+1=4

10-23y= =

y= +1 yy

y=mx+1 y=nx+1

2…x…3

2…x…3 mx+1… …nx+1

y=nx+1

y=mx+1

x+1

x-1

y

O

3

21

321 x

y=nx+1

y=mx+1

x+1

x-1y=

2

x-1

(x-1)+2

x-1

x+1

x-1

y

x

O

-2

-2-1

-1

1

1

2

x+1

x-1

x+1

-x-1

x+1

-x-1

-(x+1)

|x|-1

|x+1|


487

⁄ y=nx+1 (2 3)

3=2n+1 n=1

y=nx+1

næ1

¤ y=mx+1 (3 2)

¤ 2=3m+1 m=;3!;

¤ y=mx+1

¤ m…;3!;

⁄ ¤ m ;3!; n 1

m n ;3!;

10-24Q R

QR”="√{0-(-1)}¤ √+(-4-0)¤ ='1å7

Q R

+ =1 4x+y+4=0

y=;[!; x>0 P P {a ;a!;}

P 4x+y+4=0 h

h= =

a>0

4a+;a!;æ2Æ…4a¥;a!;=4

{ 4a=;a!; }

h =

PQR

;2!;¥'1å7¥ =48

'1å7

8

'1å7

|4+4|

'1å7

|4a+;a!;+4|

'1å7

|4a+;a!;+4|

"√4¤ +1¤

y

-4

x

-1

1'ƒx-3

x-3æ0 xæ3

'ƒ2-x+'ƒx+2

2-xæ0, x+2æ0 -2…x…2

x+3>0 x>-3

'ƒx+1+

x+1æ0, x>0 x>0

xæ3 -2…x…2

x>-3 x>0

2x>2 x-2>0, x+2>0

"√x¤ -4x+4+|x+2|="√(x-2)¤ +|x+2|

"√x¤ -4x+4+|x+2|=|x-2|+|x+2|

"√x¤ -4x+4+|x+2|=(x-2)+(x+2)

"√x¤ -4x+4+|x+2|=2x

x>2 x-2>0

('ƒx-2)¤ =x-2

2x x-2

3

=

=

=

=

=

=2'ƒx-1

2'ƒx-1'ƒx+2+'x

2

2(x-1)'ƒx-1

x-1

(2x-2)'ƒx-1

'ƒx-1'ƒx-1

2x-2

'ƒx-1

'ƒx+2+'x

2

'ƒx+2+'x

x+2-x

'ƒx+2+'x

('ƒx+2-'x)('ƒx+2+'x)

1

'ƒx+2-'x

1

'x

1

'ƒx+3

무리함수



488

4

5y='ƒx-2 y='x

x 2

{x|xæ2} {y|yæ0}

y='ƒ-x+3

y='ƒ-x y 3

O 2 x

y

x-2y=

-3

-3 Ox

y

-3xy=-

O

-3

3

x

y

3xy=-

O

3

-3 x

y

-3xy=

O

3

3 x

y

3xy=

{x|x…0} {y|yæ3}

y=-'ƒx-1+2

y=-'x x 1 y

2

{x|xæ1} {y|y…2}

y=-'∂-x+1

y=-'∂-x y 1

{x|x…0} {y|y…1}

{x|xæ2} {y|yæ0}

{x|x…0} {y|yæ3}

{x|xæ1} {y|y…2}

{x|x…0} {y|y…1}

6y=-'aåx (-2 -2)

-2=-'ƒ-2a

'ƒ-2a=2

-2a=4

a=-2

-2

O

1-1

x

y

-x+1y=-

O

2

1

5 x

y

x-1+2y=-

O

3

x

y

-x+3y=


489

유제 11-1'ƒ2-3x+2'ƒx+1

2-3xæ0 x+1æ0

-1…x…;3@;

'ƒ3x-2-

3x-2æ0 1-x¤ >0

xæ;3@; -1<x<1

;3@;…x<1

x¤ -3x+2æ0 -x¤ -x+12>0

x¤ -3x+2æ0 (x-1)(x-2)æ0

x…1 xæ2 yy

-x¤ -x+12>0 (x+4)(x-3)<0

-4<x<3 yy

-4<x…1또는 2…x<3

유제 11-2"√x¤ -2kx-3k+4

x¤ -2kx-3k+4æ0

x¤ -2kx-3k+4=0 D

D…0

=k¤ +3k-4…0

(k+4)(k-1)…0 -4…k…1

유제 11-3

=

=

= =x¤

=

+

=(1+x)-2'ƒ1+x'ƒ1-x+(1-x)

(1+x)-(1-x)

('ƒ1+x+'ƒ1-x)¤

('ƒ1+x-'ƒ1-x)('ƒ1+x+'ƒ1-x)

('ƒ1+x-'ƒ1-x)¤

('ƒ1+x+'ƒ1-x)('ƒ1+x-'ƒ1-x)

3x¤

x+3-x

3x¤

('ƒx+3)¤ -('x)¤

3x('x)¤

('ƒx+3-'x)('ƒx+3+'x)

D

4

"√x¤ -3x+2

"√-x¤ -x+12

1

"√1-x¤

+

= = =;[@;

유제 11-4

=

= ='ƒx+2-'ƒx+1

='ƒx+3-'ƒx+2

='ƒx+4-'ƒx+3

='ƒx+2-'ƒx+1+'ƒx+3-'ƒx+2

+'ƒx+4-'ƒx+3

='ƒx+4-'ƒx+1

유제 11-5y=-'ß2x+4=-"√2(x+2)

y=-'ß2x x -2

정의역：{x|xæ-2}, 치역：{y|y…0}y=-'ß-ßx+2-2=-"√-(x-2)-2

y=-'ß-x x

2 y -2

정의역：{x|x…2}, 치역：{y|y…-2}

y

-2

2Ox

y=-'∂ -x

y=-'∂ -x+2-2

y

-2

-2

Ox

y=-'∂ 2x

y=-'∂ 2x+4

1

'ƒx+3+'ƒx+4

1

'ƒx+2+'ƒx+3

'ƒx+1-'ƒx+2

(x+1)-(x+2)

'ƒx+1-'ƒx+2

('ƒx+1+'ƒx+2)('ƒx+1-'ƒx+2)

1

'ƒx+1+'ƒx+2

4

2x

4

(1+x)-(1-x)

(1+x)+2'ƒ1+x'ƒ1-x+(1-x)

(1+x)-(1-x)

유제 본문 p.310


490

유제 11-6

y='ƒ-2x+1+4=æ≠-2 {x-;2!;}+4

y='ƒ-2x

x ;2!;

y 4

y=7 x=-4

y=5 x=0

{x|-4…x…0}

a=-4, b=0

a+b=-4

유제 11-7y='ƒ5-x+3='ƒ-(x-5)+3 1…x…4

x=1

M=5

x=4

m=4

M+m=9

유제 11-8f(x)='ƒax+b+c

f(x)=æ≠a{x+;aB;}+c

f(x) x=-;aB;

c

-;aB;=-1 c=3 b=a c=3 yy

f(1)=5

'ƒa+b+c=5

'2åa=2

a=2

a=2 b=2 c=3

a+b+c=7

유제 11-9y='ƒ3x+6-1='ƒ3(x+2)-1

y='∂3x x

-2 y -1

y

xO 1

345

4 5

y

xO-4

54

7

1;2;

a=3, b=-2, c=-1

a+b+c=0

유제 11-10y=-'ƒ-2x x 5

y -5

y=-'ƒ-2(x-5)-5

=-'ƒ-2x+10-5

y

y=-'ƒ-2¥(-x)+10-5

=-'ƒ2x+10-5

g(x)=-'ƒ2x+10-5

g(3)=-'ƒ2¥3+10-5=-4-5=-9

유제 11-11y=a'x x

2 y 1

y=a'ƒx-2+1

(3 0)

0=a'ƒ3-2+1 a=-1

y=-'ƒx-2+1

y=a'ƒx+b+c

a+b+c=-1+(-2)+1=-2

유제 11-12y=-'a åx x

4 y 3

y=-'ƒa(x-4)+3

(0 -1)

-1=-'ƒa(0-4)+3 'ƒ-4a=4

-4a=16 a=-4

y=-'ƒ-4(x-4)+3

=-'ƒ-4x+16+3

y=-'ƒax+b+c

2a+b-c=2¥(-4)+16-3=5

유제 11-13y=ax¤ +bx+c a<0, b>0,

c>0 f(x)=a'ƒ-x+b-c

y

xO

y=f(x)


491

유제 11-14y=-'ßx+1 xæ-1, y…0

y¤ =x+1

x x=y¤ -1

x y y=x¤ -1 (̀x…0)

정의역：{x|x…0}, 치역：{y|yæ-1}y="√-3(x+1)-2

y+2="√-3(x+1) x…-1, yæ-2

(y+2)¤ =-3(x+1)

y¤ +4y+4=-3x-3

x

x=-;3!;(y¤ +4y+7)

x y

y=-;3!;(x¤ +4x+7)

y=-;3!;(x+2)¤ -1 (̀xæ-2)

정의역：{x|xæ-2}, 치역：{y|y…-1}

유제 11-15y=-'ƒ2x-4+1

'ƒ2x-4=1-y xæ2 y…1

x

2x-4=(1-y)¤

x=;2!;y¤ -y+;2%;

x y g(x)

y=;2!;x¤ -x+;2%;

g(x)=;2!;x¤ -x+;2%; x…1

g(x)=;2!;x¤ +ax+b x…c

a=-1, b=;2%;, c=1

a+b+c=;2%;

유제 11-16y=f(x) y=f—⁄ (x)

y=f(x) y=x

'ƒ2x-3+1=x, 'ƒ2x-3=x-1

2x-3=x¤ -2x+1, x¤ -4x+4=0

(x-2)¤ =0 x=2

(2, 2)

유제 11-17y='a åx x

-6

y='ƒa(x+6)

(0 '6)

'6='6a a=1

f(x)='ƒx+6

P y=f(x) y=x

'ƒx+6=x, x¤ -x-6=0

(x+2)(x-3)=0

x=3 xæ0

P(3 3) ABP

ABP=2¥ AOP+ ABO

ABP=2 {;2!;¥6¥3}+;2!;¥6¥6=36

유제 11-18y=x+k y=-'ƒ4-2x

k

-'ß4-2x=x+k

x¤ +2(k+1)x+k¤ -4=0

D

=(k+1)¤ -(k¤ -4)=0 2k+5=0

k=-;2%;

유제 11-19y=x+k y='ƒx+3

D

4

y

y=x+k

O 2

-2

x

y=-'∂ 4-2x

y

xO-6

-6

6'P

B

y=x

y=f(x)

y=f-1(x)

A


492

'ƒx+3=x+k

x¤ +(2k-1)x+k¤ -3=0

D

D=(2k-1)¤ -4(k¤ -3)=0 -4k+13=0

k=:¡4£:

n(A;B)=0 A;B=Δy=x+k y='ƒx+3

k>:¡4£:

11-1'ƒ2-x+'ƒx+3

2-xæ0 x+3æ0

-3…x…2

2x-3æ0 x¤ -x-6>0

xæ;2#; x<-2 x>3

x>3

11-2

=

=

=

=x+"√x¤ -1

x='3

x+"√x¤ -1='3+'ƒ3-1='3+'2

2x+2"√x¤ -1

2

(x+1)+2"√x¤ -1+(x-1)

(x+1)-(x-1)

('ƒx+1+'ƒx-1)¤

('ƒx+1-'ƒx-1)('ƒx+1+'ƒx-1)

'ƒx+1+'ƒx-1

'ƒx+1-'ƒx-1

'ƒ2x-3

"√x¤ -x-6

y

xO-3

y=x+k

y='∂ x+3

11-3

x y=-'∂ax

axæ0 x

a<0 {x|x…0}

y=-'∂ax a>0 4

y=-'∂2x

y=-'x y

|a| y

ㄱ, ㄴ

11-42-xæ0 x…2

{x|x…2}

a=2

-'ƒ2-x…0 {y|y…-3}

b=-3 ab=-6

11-5y=-'2åx x m

y n

y=-'ƒ2(x-m)+n=-'ƒ2x-2m+n

y=-'ƒ2x-5+2

2m=5 n=2 m=;2%; n=2

mn=5

11-6y='a åx x

-2 y -1

y='ƒa(x+2)-1

(-1 0)

0='ƒa(-1+2)-1 a=1

y='ƒx+2-1

y='ƒax+b+c

a+b+c=1+2+(-1)=2

y

x

O

y=-'

1

-1

x-' 2

y=-'∂ 2x



493

11-7y=3-'ƒ2x+3=-æ≠2 {x+;2#;}+3

-1…x…11

x=-1 M=2

x=11 m=-2

M+m=2+(-2)=0

11-8y=(x-2)¤ +5 (x-2)¤ =y-5

x…2 x-2…0

x-2=-'ƒy-5

x y y-2=-'ƒx-5

f —⁄ (x)=-'ƒx-5+2

f —⁄ (x)=-'ƒx+a+b

a=-5, b=2

a+b=-3

11-9a='1=1, b=2'1=2

b='ßd d=b¤ =4

c=2'ßd=2'4=4

c='e e=c¤ =16

a+b+c+d+e=1+2+4+4+16=27

11-10y=x+a y='x

x+a='x

x¤ +2ax+a¤ =x

x¤ +(2a-1)x+a¤ =0

D

D=(2a-1)¤ -4a¤ =0, -4a+1=0

a=;4!;

11-11y='ƒ-2x x a

y b

y='ƒ-2(x-a)+b

y

xO-2-1

y=3-'∂ 2x+3

32

113;2;-

y

y='ƒ-2(-x-a)+b='ƒ2x+2a+b

y='ƒcx-6+5

a+b+c=-3+5+2=4

11-12y='ƒa-2x+b=æ≠-2 {x-;2A;}+b

x=-6 x=0

'ƒa+12+b=5 yy

'a+b=3 yy

a=(5-b)¤ -12

a=(3-b)¤

(5-b)¤ -12=(3-b)¤

b=1, a=4

a¤ +b¤ =17

11-13y='a åx x

1 y -2

y='ƒa(x-1)-2

(2 0)

0='ƒa(2-1)-2 a=4

y='ƒ4(x-1)-2='ƒ4x-4-2

y='ƒax+b+c

a=4 b=-4 c=-2

y=

x=2, y=4

11-14y=x

y=a(x-2)¤ -1 xæ2

(3, 0)

0=a(3-2)¤ -1 a=1

y

x

O

-1 32

y=a(x-b)¤ +c

4x-4

x-2


494

y=(x-2)¤ -1 xæ2

y=a(x-b)¤ +c xæb

a+b+c=1+2+(-1)=2

11-15y=f(x) y=f—⁄ (x)

y=f(x) y=x

'ƒ2x+2-1=x 'ƒ2x+2=x+1

2x+2=x¤ +2x+1, x¤ =1

x=-1 x=1

P(-1, -1) Q(1, 1)

PQ”=øπ(1+1)¤ +(1+1)¤ =2'2

11-16f(x)='ƒax+b f(2)=1

'ƒ2a+b=1 2a+b=1 yy

f —⁄ (2)=1 f(1)=2

'ƒa+b=2 a+b=4 yy

a=-3, b=7

11-17f —⁄ (4)=a f(a)=4

f(a)= =4 a+2=4(a-1)

a=2

( fΩ(gΩf)—⁄ )(2)=(fΩf —⁄ Ωg—⁄ )(2)=(IΩg—⁄ )(2) I

=g—⁄ (2)=b

g(b)=2 g(b)='ƒ2b-1=2

2b=5 b=;2%;

ab=2¥;2%;=5

11-18

O

Q˚

P˚

k-1 1 x

y

x+1y=

k+1

k-1 x-1y=

a+2

a-1

P˚Q ”̊='ƒk+1-'ƒk-1

P¡Q¡”+P™Q™”+P£Q£”+y+PªªQªª”

=('2-0)+('3-1)+('4-'2)+y

+('9å9-'9å7)+('∂100-'9å8)

=-1+'9å9+'∂100

=9+3'1å1

=a+b'1å1

a=9 b=3

a+b=12

11-19y=x+k 1

y='ƒx-2 y=x+k

⁄ y=x+k y='ƒx-2

'ƒx-2=x+k

⁄

⁄ x¤ +(2k-1)x+k¤ +2=0

D

⁄ D=(2k-1)¤ -4(k¤ +2)=0, -4k=7

⁄ k=-;4&;

¤ y=x+k (2, 0)

⁄ 0=2+k k=-2

⁄ ¤ -2…k<-;4&;

ab=(-2)¥{-;4&;}=;2&;

11-20y=mx+1 m (0 1)

y='ƒ2x-3 y=mx+1

y

xO

1

2

3;2;

y=mx+1

y=mx+1

y='∂ 2x-3

y

x

O

2

7;4;k=- k=-2


495

⁄ y=mx+1 y='ƒ2x-3

'ƒ2x-3=mx+1

m¤ x¤ +2(m-1)x+4=0

D

⁄ =(m-1)¤ -4m¤ =0

⁄ 3m¤ +2m-1=0

⁄ (m+1)(3m-1)=0

⁄ m=-1 m=;3!;

⁄ m=;3!;

¤ y=mx+1 {;2#; 0}

¤ 0=;2#;m+1 m=-;3@;

⁄ ¤ -;3@;…m…;3!;

a-b=;3!;-{-;3@;}=1

11-21

x<1 y=-2(x-1)¤ +b

xæ1 y='ƒa(1-x)+2

y='ƒa(1-x)+2='ƒ-a(x-1)+2

-a>0 a<0

f(1)=b=2

a<0, b=2

11-22

y=2'ƒ|2x-1|=

2'ƒ2x-1 {xæ;2!;}

2'ƒ-2x+1 {x<;2!;}

({9

O

y

x

y='ƒa(1-x)+2

¤

1

2

D

4 y=kx y=2'ƒ2x-1 k

kx=2'ƒ2x-1

k¤ x¤ -8x+4=0

D

=16-4k¤ =0

(k+2)(k-2)=0

k=-2 k=2

k=2

y=kx y=2'ƒ|2x-1|

y=kx x

y=2x

0<k<2

k 1

11-23y='ƒmx+1 y='∂mx

x -

y='ƒmx+1 m>0 yy

⁄ (3, 2)

⁄ 2='ƒ3m+1 4=3m+1

¤ m=1

¤ (2, 3)

⁄ 3='ƒ2m+1 9=2m+1

m=4

⁄ ¤ m 1…m…4

a+b=1+4=5

O1;m;-

y

x

2

32

3(2, 3)

(3, 2)

y='∂mx+1

1

m

D

4

O

y

x12

y=2'∂ 2x-1

y=2'∂-2x+1y=kx


496

11-24

⁄ n=1

¤ 1<n…13

1 n-1

f(13)=12

‹ 13<n…33

1

(n-1)+(n-13)=2n-14

‹ '2

n-13

‹ 2

n-14

‹

f(n)=2n-14+n-13+n-14=4n-41

n=33 f(n)=91 n>33

› 33<n…61

1

(n-1)+(n-13)+(n-33)=3n-47

› '2

(n-13)+(n-33)=2n-46

› 2

(n-14)+(n-34)=2n-48

› '5

2(n-34)+1=2n-67

›

f(n)=3n-47+2n-46+2n-48+2n-67

=9n-208

›

f(n)…300

9n-208…300 9n…508

n…56.4___

n 56

O 13

14

15 33

34

35 61

62

63-3 1

1

2

2

x

yx+3

2y=

' 2 ' 5

y yy

1+

=3+2=5

5

2⁄ 6

(1 5) (2 4) (3 3) (4 2) (5 1)

5

¤ 7

(1 6) (2 5) (3 4) (4 3) (5 2) (6 1)

6

5+6=11

11

3

4_3=12

12

4(x+y)(a+b+c) x y a b c

2_3=6

6

5A B

3

B C

4

A B C

3_4=12

12

경우의수



497

유제 12-1⁄ 6

(1 6) (2 3) (3 2) (6 1) 4

¤ 12

(2 6) (3 4) (4 3) (6 2) 4

4+4=8

10 10 11 12

⁄ 10

(4 6) (5 5) (6 4) 3

¤ 11

(5 6) (6 5) 2

‹ 12

(6 6) 1

3+2+1=6

유제 12-2⁄ 3

(1 1 3) (1 3 1) (3 1 1) 3

¤ 4

(1 1 4) (1 4 1) (4 1 1) (1 2 2)

(2 1 2) (2 2 1) 6

3+6=9

유제 12-34 4

A 7 7

B

n(A)=25

n(B)=14

A;B 4 7 28

n(A;B)=3


=25+14-3

=36

유제 12-475=3_5¤ 75 3

5

75 3 25

5 15

3 5 15 5

3 5

25+15-5=35

75-35=40

유제 12-54 A B C D

a b c d

4

9

유제 12-6a™+2

4

유제 12-7⁄ c=0 a+3b=10

a b (a b)

(10 0) (7 1) (4 2) (1 3) 4

¤ c=1 a+3b=5

a b (a b) (5 0) (2 1) 2

‹ c=2 a+3b=0

a b (a b) (0 0) 1

a¡ a™ a£

1 3 2

21 3

3 1

3 1 2

A B C D

a d c

b c d a

d a c

a d b

c

da b

b a

a b c

dc

a b

b a

유제 본문 p.338


498

⁄ ¤ ‹ (a b c)

4+2+1=7

유제 12-8100 x 200 y 500

z

100x+200y+500z=1500 xæ1 yæ1 zæ1

x+2y+5z=15

⁄ z=1 x+2y=10 x y

(x y)

(2 4) (4 3) (6 2) (8 1) 4

¤ z=2 x+2y=5 x y

(x y) (1 2) (3 1) 2

⁄ ¤

4+2=6

유제 12-9x y

⁄ x=1 y…4 y=1 2 3 4 4

¤ x=2 y…3 y=1 2 3 3

‹ x=3 y…2 y=1 2 2

› x=4 y…1 y=1 1

⁄ › (x y)

4+3+2+1=10

유제 12-10x¤ +2ax+3b=0 D

D<0

=a¤ -3b<0 b>

⁄ a=1 b>;3!; b=1 2 3 4 5 6 6

¤ a=2 b>;3$; b=2 3 4 5 6 5

‹ a=3 b>3 b=4 5 6 3

› a=4 b>:¡3§: b=6 1

fi a=5 6

⁄ fi

6+5+3+1=15

유제 12-112

2 4 6 8 4

1 3 5 7 9 5

4_5=20

a¤

3

D

4

유제 12-12

3

4_3=12

유제 12-13(x¤ +1)(y¤ +y+x) x¤ 1 y¤

y x

2_3=6

(a¤ +a+1)(b‹ +b¤ +b+1) a¤ a 1

b‹ b¤ b 1

3_4=12

(a+b)¤ (x+y)+(c+d+e)(z+w)

=(a¤ +2ab+b¤ )(x+y)+(c+d+e)(z+w)

(a¤ +2ab+b¤ )(x+y)

a¤ 2ab b¤ x y

3_2=6

(c+d+e)(z+w) c d e

z w

3_2=6

6+6=12

유제 12-14

⁄

3_2=6

¤

2_2=4

⁄ ¤

6+4=10(가지)

유제 12-15540

540=2¤ _3‹ _5

(2+1)(3+1)(1+1)=24

540

(1+2+2¤ )(1+3+3¤ +3‹ )(1+5)

=7_40_6=1680


499

3 2¤ _5

(2+1)(1+1)=6

3

24-6=18

유제 12-1610 2

0 1 2 3

50 3

0 1 2 3 4

100 4

0 1 2 3 4 5

0

3_4_5-1=59

100 4 50 8

10 2

50 11

10 2

0 10 20 3

50 11

0 50 100 150 200 250 300

350 400 450 500 550 12

0

3_12-1=35

유제 12-175

4

3

3

5_4_3_3=180

유제 12-18⁄ B D B=D

⁄ A 5

⁄ B A 4

⁄ C A B 3

⁄ D B 1

⁄ E A B D 3

⁄ 5_4_3_1_3=180

¤ B D B+D

⁄ A 5

⁄ B A 4

⁄ C A B 3

⁄ D A B C 2

⁄ E A B D 2

⁄ 5_4_3_2_2=240

⁄ ¤

180+240=420

유제 12-19

999-99=900

2

0 2 8

2 9

8_9_9=648

2

900-648=252

유제 12-201 999 0

1 001 10 010 98 098

3 6 9

7 0

7_7_7-1=342

999-342=657

12-13 3 6 9 3 4

4 8 2

3+2=5

12-2A

B

A={2 4 6} B={2 3 5} A;B={2}


=3+3-1

=5



500

12-3⁄ A C B

A⁄C ⁄B 2_3=6

A⁄D ⁄B 2_2=4

6_4=24

¤ A D B

24

⁄ ¤

24+24=48

12-4

2_2_2_2=16

4

16-1=15

12-52̊ 3¤ (k+1)(2+1)

3(k+1)=21 k+1=7

k=6

12-6a a=1 2 3 4

b b=1 2 3 4 5 6

ab 6

12 18 24

⁄ 12 (a b)

(2 6) (3 4) (4 3) 3

¤ 18 (a b) (3 6) 1

‹ 24 (a b) (4 6) 1

⁄ ¤ ‹

3+1+1=5

12-72 A

5 B

n(A)=50 n(B)=20

2 5 A'B

A;B 2 5 10

n(A;B)=10


=50+20-10=60

100-60=40

12-8x y

⁄ y=1 x…5 x=1 2 3 4 5 5

¤ y=2 x…4 x=1 2 3 4 4

‹ y=3 x…3 x=1 2 3 3

› y=4 x…2 x=1 2 2

⁄~› (x y)

5+4+3+2=14

12-9

P Q R S

P 4

Q P 3

R P Q 2

S P R 2

4_3_2_2=48

12-10

0 1 2 y 9 10

0 1 2 y 9 10

1 3 5 7 9 5

10_10_5=500

12-11a b

⁄ |a-b|=1

(1 2) (2 3) (3 4) (4 5)

(5 6) (6 7) (7 8)

7

P

R

S

Q


501

¤ |a-b|=2

(1 3) (2 4) (3 5) (4 6) (5 7) (6 8)

6

‹ |a-b|=3

(1 4) (2 5) (3 6) (4 7) (5 8)

5

⁄ ¤ ‹

7+6+5=18

12-12A

D

⁄ A⁄ B ⁄ C⁄ D 2_1_2=4

¤ A⁄ C ⁄ B⁄ D 3_1_3=9

⁄ ¤ m

m=4+9=13

A

D

‹ A⁄ B ⁄ D 2_3=6

› A⁄ C ⁄ D 3_2=6

‹ › n

n=6+6=12

m+n=25

12-13(a¡-1)(a™-2)(a£-3)(a¢-4)+0

a¡+1 a™+2 a£+3 a¢+4

a¡ a™ a£ a¢

1 4 3

2 3 4 1

4 1 3

1 4 2

3

41 2

2 1

1 2 3

4

31 2

2 1

9

12-14100 2 50 4

500 3 50

7 10 4

500 3

0 500 1000 1500 4

50 7

0 50 100 150 200 250 300

350 8

10 4

0 10 20 30 40 5

0

4_8_5-1=159

12-15x¤ +2ax+3b=0 D

Dæ0

=a¤ -3bæ0

b…

⁄ a=0 b…0 b=0 1

¤ a=1 b…;3!; b=0 1

‹ a=2 b…;3$; b=0 1 2

› a=3 b…3 b=0 1 2 3

⁄ ›

1+1+2+3=7

12-16

⁄ A B

⁄ 2_6=12

¤ A B

⁄ 6_5=30

⁄ ¤

12+30=42

12-170 0

1 2 3 4 1

0 2 4 1

0 4_4=16

2 3_3=9

4 3_3=9

a¤

3

D

4


502

16+9+9=34

12-18

N N

N=1_3π _5œ _7®

=3π _5œ _7® p=0 1 2 q=0 1 2 3 r=0 1

2

3¤ _5‹ _7 1

(2+1)(3+1)(1+1)-1=23

12-19f(1) f(2) f(3)=0

f(1)=0 f(2)=0 f(3)=0

f(1) f(2) f(3) 0

f X 2⁄ Y f(1) f(2) f(3)

0

⁄ f X 2⁄ Y

5_5_5=125

¤ f(1) f(2) f(3) 0

4_4_4=64

⁄ ¤ f

125-64=61

12-20

a a

(1 3) (1 5) (1 6) (2 4) (2 6)

(3 4) (3 5) (4 6)

8

a, a b 4

a, a, b c 3

a, a, b, c d 2

a, a, b, c, d e 1

8_4_3_2_1=192

1 2 3

4 5 6

12-21y=;aB;x y=x¤

x

;aB;x=x¤ x {x-;aB;}=0

x=0 x=;aB;

P {;aB; {;aB;}¤ } H {;aB; 0}

OPH

;2!;_;aB;_{;aB;}¤ =;2!; {;aB;}‹

OPH ;aB;

2 (a b) (1 2)

(1 4) (1 6) (2 4) (3 6) 5

12-222

x 3

y 5 z

⁄ 0

2x+3y+5z=25

xæ1 yæ1 5z…20

z=1 2 3 4

⁄ z=1 2x+3y=20

(x y) (7 2) (4 4) (1 6)

3

⁄ z=2 2x+3y=15

(x y) (6 1) (3 3) 2

⁄ z=3 2x+3y=10

(x y) (2 2) 1

⁄ z=4 2x+3y=5

(x y) (1 1) 1

¤ 1

2x+1+3y+1+5z+1=25

xæ1 yæ1 5z…17

z=1 2 3

⁄ z=1 2x+3y=17

(x y) (7 1) (4 3) (1 5)

3

⁄ z=2 2x+3y=12

(x y) (3 2) 1

⁄ z=3 2x+3y=7

(x y) (2 1) 1


503

⁄ ¤

3+2+1+1+3+1+1=12

12-2318 4

0 1 2 3

A˚ k=0 1 2 3

Aº={4 8 12 16}

A¡={1 5 9 13 17}

A™={2 6 10 14 18}

A£={3 7 11 15}

a+b 4

⁄ a=4k b=4k'

a b Aº

4_4=16

¤ a=4k+2 b=4k'+2

a b A™

5_5=25

‹ a=4k+1 b=4k'+3

a A¡ b A£

5_4=20

› a=4k+3 b=4k'+1

a A£ b A¡

4_5=20

⁄~› (a b)

16+25+20+20=81

12-24x ax¤ +bx+c>0

a>0 D<0

D=b¤ -4ac<0 b¤ <4ac

(a b c) b¤ æ4ac

⁄ (a b c)

4_3_2=24

¤ b¤ æ4ac (a b c)

(1 5 3) (1 7 3) (1 7 5) (3 5 1)

(3 7 1) (5 7 1) 6

⁄ ¤ (a b c)

24-6=18

1§P™=6_5=30

£P£=3!=3_2_1=6

∞Pº=1

30 6 1

24!=4_3_2_1=24

= =120

24 120

3

¶P¢= =

=3

= =ªP¢

=4

3 4

4«P™=n(n-1)=30

n¤ -n-30=0 (n+5)(n-6)=0

n=-5 n=6

næ2 n=6

∞P®=60

60=5_4_3

∞P®=5_4_3=∞P£

r=3

n=6 r=3

5

3!_2!=12

2!_™–¡≠™P™=2!_£P™=2_6=12

12 12

9!

(9-4)!

9!

5!

7!

3!

7!

(7-4)!

6¥5¥4¥3¥2¥1

3¥2¥1

6!

3!

순열



504

다른 풀이

4 4!

3!_2!=12

4!-12=12

유제 13-1«≠™P™=56 (n+2)(n+1)=56

n¤ +3n-54=0 (n-6)(n+9)=0

n=6 næ0

1440=(5_4_3)_4!

∞P®_4!=1440 4!

∞P®=5_4_3

r=3

«P™+«≠¡P™=n(n-1)+(n+1)n

n(n-1)+(n+1)n=98

2n¤ =98 n¤ =49

n=—7

næ2 n=7

«P£ : «–¡P£=5 : 4

4¥ «P£=5¥«–¡P£4n(n-1)(n-2)=5(n-1)(n-2)(n-3)

næ4 (n-1)(n-2)

4n=5(n-3)

n=15

유제 13-2«–¡P®+r¥«–¡P®–¡

= +r¥

= +r¥

=

= =«P®

«P®=«–¡P®+r¥«–¡P®–¡«P¬¥«–¬P®–¬

= _

= =«P®

«P¬¥«–¬P®–¬=«P®

n!

(n-r)!

(n-l)!

{(n-l)-(r-l)}!n!

(n-l)!

n!

(n-r)!

(n-1)!{(n-r)+r}(n-r)!

(n-1)!

(n-r)!

(n-r)¥(n-1)!

(n-r)!

(n-1)!

{(n-1)-(r-1)}!(n-1)!

(n-1-r)!

유제 13-35 3

∞P£=5_4_3=60

유제 13-4n 3

«P£=120

n(n-1)(n-2)=6_5_4

n=6

유제 13-56

1

6 2

§P™=6_5=30

유제 13-6f(x¡)=f(x™) x¡=x™ f

f(1)=2 f(3)=6 Y 1 3

4 5 7 5 3

∞P£=5_4_3=60

유제 13-7f 2 3 5 7

2 3 5 7

1 4 6 1 4 6

¢P¢_£P£=4!_3!=144

유제 13-83 5

5!=120

3

3!=6

120_6=720

4 4

4!=24

4

유제 본문 p.366


505

4!=24

24_24=576

유제 13-9n

4

4!=24

n!

24_n!=17280

n!=720=6!

n=6

유제 13-105

5!=120

6 3

§P£=120

120_120=14400

유제 13-11A B AB C F

3!=6

D E D E

AB C F 4

¢P™=12

A B

2!=2

6_12_2=144

유제 13-124

3

4

4!=24

3!=6

24_6=144

유제 13-13

4

2!_2!_2=8

5

∞P¡=5

8_5=40

유제 13-14d u 6 2

§P™=30

유제 13-15o l 6

6!

o l 2!

6!_2!=720_2=1440

i t 2

§P™

i t 5

5 5!

i t 2!

§P™_5!_2!=30_120_2=7200

8! o i e a 4

¢P™_6!

8!-¢P™_6!=40320-(12_720)=31680

유제 13-160

1 ¢P£

2 ¢P£

3 ¢P£

4 ¢P£

4_¢P£=4_24=96


506

n¤ -n-110=0

(n+10)(n-11)=0

n=11 næ2

™«P£=100¥«P™2n(2n-1)(2n-2)=100n(n-1)

4n(2n-1)(n-1)=100n(n-1)

næ2 n(n-1)

4(2n-1)=100

2n-1=25

n=13

13-25

5!=120

13-3a 5 3

a=∞P£=60

b A 4

b=¢P™=12

a+b=60+12

=72

13-4A 1 2

3 B

B a b c d 3

¢P£=24

13-55 2

∞P™=20

13-6

3!=6

3 2 2

3!_2!_2!=24

6_24=144

2100

⁄ 2

21 £P™

23 £P™

24 £P™

3_£P™=3_6=18

¤ 3 4

3 ¢P£

4 ¢P£

2_¢P£=2_24=48

⁄ ¤

18+48=66

⁄ 0

0 ¢P£=24

¤ 2 4

2 £P¡_£P™

4 £P¡_£P™

2_£P¡_£P™=36

⁄ ¤

24+36=60

유제 13-1742000

1 4!=24

2 4!=24

3 4!=24

41 3!=6

24+24+24+6=78

유제 13-18a ：5!=120

b ：5!=120

ca ：4!=24

cb ：4!=24

cda ：3!=6

cdb ：3!=6

120+120+24+24+6+6=300

300 cdb

cdbfea

13-1«P™=n(n-1)=110



507

13-74 4!

5 3

∞P£

4!_∞P£=24_60=1440

13-8a o 2 ∞P™

a o 4

4!

a o 2!

∞P™_4!_2!=20_24_2=960

13-9

2

7!

2 ™P™_5!

7!-™P™_5!=5040-(2_120)=4800

13-10

2 4 6

£P™

∞P™

£P™_∞P™=6_20=120

13-11§P® ¢P®=5 : 2

5_¢P®=2_§P®

5_ =2_

=

(6-r)(5-r)=12

r¤ -11r+18=0 (r-2)(r-9)=0

5_4!

(4-r)!

2_6_5_4!

(6-r)(5-r)_(4-r)!

5_4!

(4-r)!

6!

(6-r)!

4!

(4-r)!

남남남남

r=2 r=9

§P® ¢P® 0<r…4

r=2

5(«P£+«≠¡P¢)=12¥«≠¡P£5{n(n-1)(n-2)+(n+1)n(n-1)(n-2)}

=12(n+1)n(n-1)

5n(n-1)(n-2){1+(n+1)}

=12n(n-1)(n+1)

næ3 n(n-1)

5(n-2)(n+2)=12(n+1)

5n¤ -12n-32=0

(n-4)(5n+8)=0

n=4 næ3

13-124 4!

2

4!_2=48

13-13A B B C

C D E F F G 5

5

5 5!

2!

5_5!_2!=5_120_2=1200

13-142 4

1 3 6

2 4 5

1 3 6 3!

4 2

2 4 5 ¢P™

2 4

2!

3!_¢P™_2!=6_12_2=144

13-15b c d 2

£P™

3 3!


508

m=£P™_3!=6_6=36

a e 2 b c d 3

3!

2!

n=3!_2!=12

m+n=36+12=48

13-163 2

£P™

4 4!

£P™_4!=6_24=144

13-17⁄ 1 2 3 4 5 6 6

§P£=120

¤

£P™_¢P¡=24

⁄ ¤

120-24=96

13-18⁄ 6

6!

¤ A 5!

‹ B 5!

› A B

4!

⁄ ›

6!-(5!+5!-4!)=720-(120+120-24)=504

13-19

8 3

1 7 2 6 3 5 5 3 4

4

5 3

∞P£=60

4_60=240

13-20k o r e a a e

k o r

a 4!=24

e 4!=24

ka 3!=6

kea 2!=2

24+24+6+2=56 keoar 57번째

13-211 1 2 3 3!

1 2 3 1 1

2!

3!_2!=12

13-22⁄ 4

¤

4_¢P¡=16

‹

4_¢P™=48

› 1 2

¢P£=24

fi 30 31 32

£P™=6

fl 340 341 2

‡ 3420 1

⁄ ‡ 3421

4+16+48+2_24+3_6+2_2+1=139

13-234 4 A B C D

4!=24

A B

3 C D

작성한주제 A B C D

검토하는주제 B A D C

검토하는주제 B C D A

검토하는주제 B D A C


509

B 3

3_3=9

24_9=216

13-242abc0

a+b+c=

a b c 0 9

a+b+c=

⁄

∞P£=60

¤

3_∞P™_∞P¡=300

⁄ ¤

60+300=360

1

¢C¡= = =4

∞C™= = = =10

¡¡Cª= = = =55

4 10 55

2

§C™= = = =

=2

¡ºP¢= !_¡ºC¢ ¡ºC¢=

¡ºC¢= = =

=4

2 4

3

«C™= = =21

n(n-1)=42=7_6n=7

¡ºC¢=¡ºC¡º–¢=¡ºC§

r=6

«C£=«C«–£=¶C¢

n=7

§C®=§C§–®=§C®≠™6-r=r+2 2r=4

r=2

•C∞=•–¡C∞–¡+•–¡C∞=¶C¢+¶C∞n=7 r=5

ªC£+ªC¢=¡º–¡C¢–¡+¡º–¡C¢=¡ºC¢n=10 r=4

n=7 r=6 n=7 r=2

n=7 r=5 n=10 r=4

n(n-1)2!

«P™2!

¡ºP¢4!

10_9_8_74!

10!4!6!

¡ºP¢!

6_5!

6_52_1

6!2!4!

6!2!(6-2)!

11_10

2_1

11!

9!2!

11!

9!(11-9)!

5_4

2_1

5!

2!3!

5!

2!(5-2)!

4!

1!3!

4!

1!(4-1)!

조합



510

유제 14-1«C£=«–¡C™

=

næ3 (n-1)(n-2)

=

n=3

⁄ r-2=2r-2 r=0

r-2æ0

¤ r-2+2r-2

•C®–™=•C8-(r-2)=•C¡º–®=•C™®–™10-r=2r-2 r=4

⁄ ¤ r=4

«C™+«C£=2¥™«C¡

+ =2¥2n

næ3 n

+ =4

n¤ -25=0 (n+5)(n-5)=0

n=5 næ3

«P£+6¥«C£=2¥«P™

n(n-1)(n-2)+6¥

=2n(n-1)

næ3 n(n-1)

(n-2)+(n-2)=2 2n=6

n=3

유제 14-26 4

§C¢=§C™= = =15

1 3 5 7 9 5

3

∞C£= = =10

10<a<b<c<20 S 11 12

13 y 19 9 3

5_4_3

3_2_1

∞P£3!

6_5

2_1

§P™2!

n(n-1)(n-2)

3!

(n-1)(n-2)

6

n-1

2

n(n-1)(n-2)

3!

n(n-1)

2!

1

2

n

6

(n-1)(n-2)

2!

n(n-1)(n-2)

3!

ªC£= = =84

n

780

«C™= =780

n(n-1)=1560=40_39

n=40

유제 14-35 3

∞C£= =10

6 2

§C™= =15

∞C£_§C™=10_15=150

5 3

∞C£= =10

n 3

«C£∞C£+«C£=10+«C£=66 «C£=56

=56

n(n-1)(n-2)=336=8_7_6

n=8

유제 14-4

⁄

1 3 5 7 9 3

∞C£=10

¤

1 3 5 7 9 1 2 4 6 8 10

2

∞C¡_∞C™=5_10=50

⁄ ¤

10+50=60

유제 14-51 4 3

2 5

n(n-1)(n-2)

3!

5_4_3

3_2_1

6_5

2_1

5_4_3

3_2_1

n(n-1)

2!

9_8_7

3_2_1

ªP£3!

유제 본문 p.393


511

3

¢C£_∞C£=4_10=40

유제 14-6A B C 7 3 A

¶C£=35

유제 14-76 2

§C™=15

2 4 6 3 2

£C™=3

§C™-£C™=15-3=12

유제 14-810 4

¡ºC¢=210

⁄ C

⁄ A B 4

⁄ ∞C¢=5

¤ C 1

⁄ C 1 A B

3

⁄ ∞C¡_∞C£=5_10=50

⁄ ¤

210-(5+50)=155

유제 14-95 3 5 2

∞C£_∞C™=10_10=100

3

3 3

3!_3!=36

∞C£_∞C™_3!_3!=100_36=3600

유제 14-103 5 7 3

¶C£=35

3 3 4

4!=24

¶C£_4!=35_24=840

유제 14-11f(2)=6 x¡<x™ f(x¡)>f(x™) 1

Y 7 8

f(3)>f(4)>f(5) Y 1 2 3 4 5

3 f(3) f(4)

f(5)

2_∞C£=2_10=20

유제 14-12

2

a=∞C™=10

3

b=∞C£=10

a+b=20

유제 14-1310 ¡ºC™

5 2

∞C™

1

¡ºC™-∞C™+1=45-10+1

=36

10 ¡ºC£

5 3

5 3 ∞C£

¡ºC£-∞C£=120-10

=110

유제 14-145 2 4 2

∞C™_¢C™=10_6=60


512

유제 14-158 4

•C¢=70

2

4

4 2

¢C™=6

유제 14-16m μC™-m=27

-m=27 m¤ -3m-54=0

(m+6)(m-9)=0

m=9 mæ4

n «C™-n=44

-n=44 n¤ -3n-88=0

(n+8)(n-11)=0

n=11 næ4

m+n=20

유제 14-17

4

¡™C¢=495

14-1«C∞=«C«–∞=«C¢n-5=4

n=9

«C®= «P®r!

n(n-1)

2

m(m-1)

2

™ºC§=

™ºP§=6!™ºC§

n=6

«P™=«C™+36«P™-«C™=36

n(n-1)- =36

n(n-1)=72=9_8

n=9

14-210 2

¡ºC™= =45

14-3

3

a

a=¡ºP£=10_9_8=720

3 b

b=¡ºC£= =120

a+b=840

14-4n 3

20

«C£=20 =20

n(n-1)(n-2)=120=6_5_4

n=6

6종류

14-57 3

¶C£= =35

5 2

∞C™= =105_4

2_1

7_6_5

3_2_1

n(n-1)(n-2)

3!

10_9_8

3_2_1

10_92_1

n(n-1)

2!

™ºP§6!



513

¶C£_∞C™=35_10=350

14-6A B C 3 2

£C™ ¢C™ ∞C™ 2

£C™+¢C™+∞C™=3+6+10=19

14-72 8 2

•C™= =28

2 8

3

™C¡_•C£=2_ =112

2 8 4

•C¢= =70

14-810 3

¡ºC£= =120

2 8 3

•C£= =56

¡ºC£-•C£=120-56=64

14-91 10 10

¡ºC™= =45

5

∞C™= =105_4

2_1

10_9

2_1

8_7_6

3_2_1

10_9_83_2_1

8_7_6_54_3_2_1

8_7_63_2_1

8_72_1

¡ºC™-∞C™=45-10=35

14-108

•C™= =28

14-11«≠™C«=«≠™C«≠™–«=«≠™C™=15

=15

n¤ +3n-28=0 (n+7)(n-4)=0

n=4 næ0

«C£-«C™=4(n-1)

- =4(n-1)

næ3 n-1

- =4

n¤ -5n-24=0 (n+3)(n-8)=0

n=8 næ3

⁄ r+2=2r-2 r=4

¤ r-2+2r-2

¤ ¡∞C®≠™=¡∞C15-(r+2)=¡∞C¡£–®=¡∞C™®–™¤ 13-r=2r-2

¤ r=5

⁄ ¤ r=4또는 r=5

14-12

ªC¡=9

8 4

•C¢=70 4

ªC¡_•C¢=9_70=630

14-13b

a 9 b

a a a a a a a a

⁄ a

b 8

n

2

n(n-2)

6

n(n-1)

2!

n(n-1)(n-2)

3!

(n+2)(n+1)

2!

8_7

2_1


514

•C¢=70

¤ b

a

3 b 7

¶C£=35

⁄ ¤

70+35=105

14-14⁄ 2

⁄ 5 2 3

2

∞C™_£C™=10_3=30

2

3!_2!=6_2=12

30_12=360

¤ 3

5 1 3

3

∞C¡_£C£=5_1=5

3

2!_3!=2_6=12

5_12=60

⁄ ¤

360+60=420

14-152

™C¡_™P¡_£P¡=12

3 3

3!=6

12_6=72

다른 풀이

5 5

5!=120

2 3

2!_3!=12

2 3

3

£P™_3!=36

120-(12+36)=72

14-16n

3

¡∞C£-«C£=455-«C£=420

«C£= =35

n(n-1)(n-2)=210=7_6_5

n=7

7

14-17f(1)<f(2)<f(3)<f(4)<f(5)

∞C∞=1

f(1)=f(2)<f(3)<f(4)<f(5)

∞C¢=5

f(1)<f(2)<f(3)<f(4)=f(5)

∞C¢=5

f(1)=f(2)<f(3)<f(4)=f(5)

∞C£=10

1+5+5+10=21

14-18x y

1

1

∞C™_£C™=10_3=30

14-198 3

a=•C£=56

4

1 6

O x

y

n(n-1)(n-2)

6


515

b=4_6=24

a-b=56-24=32

14-2012 4

¡™C¢=495

4 3

4

4

¢C¢=1

4 3

8 1

¢C£_•C¡=4_8=32

3 3

9 1

£C£_ªC¡=1_9=9

¡™C¢-(3_¢C¢+3_¢C£_•C¡+8_£C£_ªC¡)

=495-(3+96+72)=324

14-211

⁄ 1

1

9

2

£C¡_ªP™=3_72=216

¤ 1

0 2 3 y 8 9

2

8

ªC¡_£C™_•C¡=9_3_8=216

⁄ ¤

216+216=432

14-22n(A)=3 n(B)=3

n(A'B)=5


=3+3-5=1

U 6 5

A-B 4 1

A;B 3 1

A-B 2 B-A

§C∞_¢C¡_£C¡=6_4_3=72

14-23

10 3

¡ºC£=120

3

3_£C£+¢C£+∞C£=3+4+10=17

120-17=103

14-24a£ 4 3

3 2

a£=™C¡_ª¶C™

a¡º=ªC¡_ªºC™ aªº=•ªC¡_¡ºC™

a¡º+aªº

a˚=a™+a£+a¢+y+aª• 1 100

4

¡ººC¢

ㄱ ㄷ

98

¡

k=2

O

3

-3 3

-3

x

yy=|x|

x¤+y¤ =9

x¤ +y¤ …9

yæ|x|g


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