고등수학Ⅱ기본서해(001~035)-ok

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2 0 I. 0 A={4, 8,, 6, 0} 4 A 03 {3, 5, 7, 9} {3, 5, 7, 9} {3, 5, 7} {, 3, 5, 7, 9} {, 3, 5, 7} 04 A={, 4, 6, 8} B a, 4, 6, 8, 3_a- 3_-=4, 3_4-=0, 3_6-=6, 3_8-= B={4, 0, 6, } {4, 0, 6, } 05 B»A B A B»A={-, -, 0,, } A»(B»A) A B»A A»(B»A)={-, -, 0,, } (-)+(-)+0++= {4, 8,, y} {, 3, 5, y} <x<3 =, x=_= =, x=_=4 =3, x=_3=6 {, 4, 6, y} 07 A={0} (A)= ({, 3})= ({0,, })-({, })=3-= ({3})=, ({5})= ({3})=({5}) ({0})+()+({})+({0, }) =+0++=4 08 B, 5A B A A={, 3, 5, y}, B={, 4, 6, y} A B A,B A={,, 3, y}, B={, 3, 5, 7, y} B,A A,B

3 09,A a<a {a, c},a c<a 0 A,, {}, {, } {, }A {, }<A, A {, },A A <A {}, A {{}, },A {}A {{}, {}} A S, 0, {0},, S<S,S 0S 0<S {0}S {{0}},S 0, {0}S {0, {0}},S 0 {a}, {b}, {c}, {d} {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d} 3 {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d} 4{a, b, c, d}, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d} A, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d} =6 -=5 3 7, 9 A 7, 9 {, 3, 5} fi = =8 8 4, 6 «=6= -=4 =6 6 5 A={,, 3, 4, 6, }, 3 fl -=6-=5 5 6 <X, 3<X k X, 3 4 X,3<X k X4 X, 3, 4 S X fl = =8 8 7, 3, 5 - =6-=4 4 8 A {0,, 5, 9} 0, 3

4 A = =4 A {0, }, {0,, 5}, {0,, 9}, {0,, 5, 9} 9 B= A=B A={, 4, 6, y}, B={4, 8,, y} A+B A={,, 3, y, 9}, B={,, 3, y, 9} A=B A={, 4, 6, y}, B={, 3, 5, y} A+B B={, 4, 6} A+B 0 A=B a-3=3b+=5 a=6, b=3 a-b=3 3 A,B B,A A=B 4<A 4<B a -3a=4a -3a-4=0 (a-4)(a+)=0 a=4 a=- a=4, A={, 4, 5}, B={, 4, 5} A=B a=-, A={-3, 0, 4}, B={, 4, 5} A+B A=B a4 4 A,B A B -<A -<B a-=- -a=- a-=-, a= A={-, 0}, B={-, 0, } A,B -a=-, a= A={-, 3}, B={-, 0, } A B A,B a 3 B;C={, 3, 4} A'(B;C)={,, 3}'{, 3, 4} ={,, 3, 4} A'B={,, 3, 4} (A'B);C={,, 3, 4};{, 3, 4, 5} ={, 3, 4} A B B={, 3, 4} 3 4 A B b g d e A={b, d, g} 4 (A'B).A A'B=A B,A A;(A'B)=A (A;)'(B')='B=B 5 A;B={4, 8} 8<A a+3=8 a=5 4<B b+=4 b+=4 b+=4, b=3 4

5 A={, 4, 8}B={4, 5, 8} A'B={, 4, 5, 8} b+=4, b= A={, 4, 8}B={3, 4, 8} A'B={, 3, 4, 8}, a=5, b=3 a+b=8 A;B={, 5} 5<A a +=5 a= a= A={, 4, 5}, B={, 3, 5} A;B={, 5} a=- A={, 4, 5}, B={-, -3, 3} A;B=, a= 8 6 {, 3, 5};X={, 3, 5} {, 3, 5},X A'X=A X,A {, 3, 5},X,{,, 3, 4, 5} X fi = =4 4 7 A={,, 3, 6}B={, 3, 9, 7} (A;B)'X=X (A;B),X (A'B);X=X X,(A'B) (A;B),X,(A'B) {, 3},X,{,, 3, 6, 9, 7} X fl = =6 6 8 (B-A)'B={5, 6}'{3, 4, 5, 6} (B-A)'B={3, 4, 5, 6} AÇ=U-A={,, 3, y}-{,, 3, 4} AÇ={5, 6, 7, y} {(B-A)'B};AÇ={5, 6} 9 {5, 6} BÇ ;A=A-B={, 6} 30 A-AÇ =A 3 B-A={} <B b= B={8, 9, } B-A={8, 9, }-{a, 7, 9}={} a=8 A={7, 8, 9} {, 6} A-B={7} {7} 3 U A 6 A-B={, 3} <A a += a= a= A={,, 3, 5}, B={0,, 7} A-B={, 3, 5} a=- A={,, 3, 5}, B={-,, 5} A-B={, 3} a=- B={-,, 5} 7 B {-,, 5} 5

6 33 A;B=A A,B A,B HjK A'B=B A,B HjK A-B= A,B HjK A;BÇ= A,B HjK AÇ 'B=U A,B HjK BÇ,AÇ U B A U A B U A B U A B ; = B A-B B, A-B U U U A B A B A B ; = 34 A-(B'C) A B C (B;C)-A A B A-(B;C) A B C (B'C)-A A B B (A;B)'(A-B) A;B B, (A;B)'(A-B) 36 (A'B);(AÇ 'BÇ )=(A'B);(A;B)Ç =(A'B)-(A;B) ={, 3, 6} (A'B)-(A;B)=(A-B)'(B-A) C C A B A-B A;B B-A 35 U A B U U A B A B ; = (A-B)'(B-A)={, 3, 6} A={,, 3, 4} A, B A-B A-B, B-A U U U A B A B A B ; = A;B A;B, A;BÇ U U U A B A B A B ; = A B-A A;BÇ (A;B)Ç A-B A, (A;B)Ç B={, 4, 6} {, 4, 6} 37 AÇ -BÇ =AÇ ;(BÇ )Ç =AÇ ;B AÇ -BÇ =B-A={4, 6} (A'B);B=B={, 3, 4, 6} AÇ ;BÇ =(A'B)Ç ={5, 7} U={,, 3, 4, 5, 6, 7} A B

7 A, B U A B A ++3= (A-B)'(A-BÇ ) =(A; BÇ )'{A ; (BÇ )Ç } =A;( BÇ 'B) =A; U = A BÇ, ;, BÇ,U,A 39 (A'B);(B-A)Ç =(A'B);(B;AÇ )Ç =(A'B);(BÇ 'A) =(A'B);(A'BÇ ) =A'(B;BÇ ) =A' =A A=A;B A,B 40 A;(AÇ'B)=(A;AÇ)'(A;B) A;(AÇ 'B)=A;B A;AÇ= B;(B'C)=B {A;(AÇ 'B)}'{B;(B'C)} =(A;B)'B=B B 4 (A'B)=(A)+(B)-(A;B) 0=8+5-(A;B) (A;B)=3 3 4 A, B (A)=, (B)=, (A'B)=30 (A;B) (A;B)=(A)+(B)-(A'B) =+-30= A B A;B, A'B (A)=5, (A;B)=8, (A'B)=3 (B)=(A'B)-(A)+(A;B) =3-5+8 = (A'B)=(A)+(B)-(A;B) (A'B)=0+3-8=5 (A-B)'(B-A) =(A'B)-(A;B) (A;B),(A'B) ((A-B)'(B-A)) =((A'B)-(A;B)) =(A'B)-(A;B) =5-8=7 A B 7 45 (A'C)=(A)+(C)-(A;C) 7=5+3-(A;C) (A;C)= (B'C)=(B)+(C)-(B;C) 5=4+3-(B;C) (B;C)= 7

8 A;B= A;B;C= (A'B'C)=(A)+(B)+(C) =-(A;B)-(B;C) =-(C;A)+(A;B;C) = = (A'B'C)=(A)+(B)+(C) =-(A;B)-(B;C) =-(C;A)+(A;B;C) (A;B;C)=(A'B'C)-(A) =-(B)-(C)+(A;B) =+(B;C)+(C;A) = =3 3 47, A, B, 80 (A'B)=(U)=80 (A)=5, (B)=45 (A'B)=(A)+(B)-(A;B) (A;B)=(A)+(B)-(A'B) = =7 A-B (A-B)=(A)-(A;B) =5-7= A(5) B(45) U, A, B (U)=40, (A)=0, (B)=5 8 (B;AÇ )=(B-A)=8 (B-A)=(B)-(A;B) 8=5-(A;B) (A;B)=7 (A-B) (A-B)=(A)-(A;B) =0-7= UA AB B (U)=40, (A)=5, (B)=0, ((A'B)Ç )=5 (A'B)=(U)-((A'B)Ç ) =40-5=35 (A;B)=(A)+(B)-(A'B) =5+0-35=0 A A-B (A-B)=(A)-(A;B)=5-0= A 'A =A A ;(A 'A )=A ;A =A (A 'A );(A 'A ) =A '(A ;A ) =A 'A =A A A 5 (AB)A=(BA)A =B(AA) =B{(A-A)'(A-A)} =B(')=B =(B-)'(-B) =B'=B 8

9 5 x x 53 a<0, b<0 ab>0 x=-, x =(-) =4 x x<- <x 5x<- (x> x 5) x<- <x 5 xæ- (x x>5) 55 f(x)g(x)=0 f(x)=0 g(x)=0 f(x)g(x)+0 f(x)+0 g(x)+0 AÇ ;BÇ 56 ab=0a=0 b=0 ab+0a+0 b+0 a, b0 57 p, q P, Q P={, 4, 6, 8, 0}, Q={3, 6, 9} PÇ ;Q PÇ ;Q={, 3, 5, 7, 9};{3, 6, 9} PÇ ;Q={3, 9} {3, 9} 58 pq PQ P={x - x<}q={x x - xæ4} ~q QÇ QÇ ={x -<x<4} p ~q QÇ P -- 4 x P;QÇ ={x -<x<} {x -<x<} 59 p, q P={x xæ3}, Q={x x<-} Q P - 3 x - x<3 {x - x<3} - x x>5-5 x - x x>5 QÇ ;PÇ =(P'Q)Ç - 3 x 9

10 60 p8q4 p, q P, Q P={,, 4, 8}, Q={,, 4} P.Q () () x=-3 x =(-3) =9 x =(-3) =-7() px -x-=0, q- x p, q P, Q P={x x -x-=0}={-, } Q={x - x } P,Q() () x=0, y= xy=0 x=0 y+0() x y x+y 4() () x=-, y=- xy> x< y<() x + y =0 x =0, y =0 x=0 y=0() (),() () 6 P;R=Q U P R P, Q, R Q Q,P, Q,R q p, q r PÇ,QÇ,RÇ,QÇ ~p ~q, ~r ~q 63 p, q P, Q P={x a- x a+} Q={x 0 x 5} p q P,Q Q P a-æ0, a+ 5 0 a- a+ 5 x ;!; a 3 a,, px<- x>3 ~p- x 3 p, q P, Q ~p q PÇ,Q 6 p ~q P,QÇ U QÇ P Q PÇ -a<-, a+6>3 -a - 3 a+6 x a> a> Q,PÇ 65, 3, 5, 7, y, 3, 5, 7, y 0

11 66 x x >0 x=0x =0 x x -x+4 0 x -x+4={x-;!;} +;; 4 ;;>0 67 x =x x -x=0, x(x-)(x+)=0 x=- x=0 x= x=0 x= x =x() x+0 x+ x +x() () x=- x> y> xy>() x y xy () () x=-, y=- () () 5, 4 0, 0 () x +y >0 xy<0() () x=, y= x +y >0 xy>0 x +y 0 xyæ0() x=0 y=0 x + y =0() x+0 y+0 x + y +0 () 68 ~p ~q~q ~p p q 69 x a y x+y 3 x a, y x+y a+ a+ 3 a a 70 ~p q ~q p p ~r r ~p ~q p p ~r ~q ~r ~q p~p q r ~p ~p q r q q rr q r ~pp ~r 7 x, y x=0 y=0 xy=0 x=0 y=0 xy=0 x, y x< y< x+y<

12 x< x-<0, y< y-<0 (x-)+(y-)<0 x+y-<0 x+y< m m m m m=k(k) m =(k) =4k = k m 7 '3 '3 '3= 5 (m, ) yy m 3= 55 =3m yy m 3 3 =3k(k) (3k) =3m m =3k m 3 m 3 m, 3 m, '3 -'3 -'3 -'3 - (-'3)+(-)=-'3 '3 -'3 =k- (k) yy =(k-) =4k -4k+=(k -k)+ a+0 b+0 a >0 b >0 a +b >0 a +b =0 a=0 b=0 a, b a +b =0 a=0 b=0 '+'3 '+'3 '+'3=a(a 0) yy '=a-'3 =(a-'3) a + =a -'3a+3 '3= 5 a a + a 5 a '3 '+'3 73 p, q P, Q P={x -3<x<3}, Q={x -<x<} P.Q P={-, }, Q={-, } P=Q P={x x<0}, Q={x x 0} P,Q P={,, 3, 6}, Q={,, 3, 6, 9, 8} P,Q

13 px -x-6=0 (x+)(x-3)=0 p x=- x=3 P={-, 3} qx -3x-0=0 (x+)(x-5)=0 p x=- x=5 Q={-, 5} P, Q p q P={x 0 x 3}, Q={x 0 x 3} P=Q 74 ( ) xy> _ x> y> ( ) x=-, y=- x+y x, y ( ) x=, y=3 xy=0 x + y =0 ( ) x=0, y= (a-b)(b-c)=0 a=b=c ( ) a=, b=, c= pa;c=c C,A q(a;b)'c,a (A;B),A C,A A;C=C x, y (A;B)'C,A xy ( ) x=', y=' 75 _ _ _ x +kx-k+0 x++0 x +kx-k+0 x++0 x+=0 x +kx-k=0 x=- x +kx-k=0 (-) +k (-)-k=0 k= x-5+0 x -3x+k+0 x -3x+k+0 x x-5=0 x -3x+k=0 x=5 x -3x+k= k=0 k=-0-0 p, q P, Q p q P.Q -;6K;æ-, kæ4 4 k k 6 x>a -<x< x>3 {x x>a}.{x -<x< x>3} a - x>b -<x< x>3 {x x>b},{x -<x< x>3} bæ3 a-, b 3 -_3= ~q p ~q Hjj p P,QÇ 79 p jjk r Q - k 4-6 a- 3 x - 3 b x ~q jjk ~r r jjk q P k x 3

14 p Hjj q q jjk r, r jjk q q r p Hjj q r 80 a b a(+b)-b(+a) = 333 +a +b (+a)(+b) (a-b) - = 335 (+a)(+b) a>b>0 a-b>0 (a-b) 335 >0 (+a)(+b) a b 33 > 33 +a +b A =('+'7) =9+' 4 B =('3+'6) =9+' 8 C =(+'5) =9+' 0 ' 4<' 8<' 0 A <B <C A, B, C A<B<C ={ 3 } ={;@7%;} < <3 ={ } ={ } < <5, <5 < >, 3 > 5 { } 5 3 >, 3 > 5> 8 a +ab+b ={a+;!;b} +;4#;b {a+;!;b} æ0, ;4#;b æ0 jjk jjk {a+;!;b} +;4#;b æ0 a +ab+b æ0 a+;!;b=0, b=0, a=b=0 ('a+'b) -('ƒa+b ) =a+' ab+b-(a+b) =' ab>0 3 a>0, b>0 ('a+'b) >('ƒa+b ) 'a+'b>0, 'ƒa+b>0 'a+'b>'ƒa+b a - b a-b a a-b + b a æ0, a-b + b æ0 a ( a-b + b ) ( a-b + b ) - a =a -ab+b + (a-b)b +b -a ={ ab-b -(ab-b )} ab-b æab-b { ab-b -(ab-b )}æ0 a ( a-b + b ) a a-b + b, a - b a-b abæ0 83 (a+b){;a!;+;b!;}-4=+;ba;+;ab;+-4 (a+b){;a!;+;b!;}-4=;ab;+;ba;- (a+b){;a!;+;b!;}-4={æ;ab;-æ;ba; }æ0 (a+b){;a!;+;b!;}æ4 (, a=b) a+b>0, b+c>0, c+a>0 ()æ() a+bæ' ab, b+cæ' bc, c+aæ' ca 4

15 (a+b)(b+c)(c+a) æ' ab ' bc ' ca=8abc (a+b)(b+c)(c+a)æ8abc (, a=b=c) 84 a>0, b>0 ()æ() 3a+4bæ' 3a 4b=' ab ab=3 3a+4bæ' 3= (, 3a=4b) 3a+4b 85 a>0, b>0 ()æ() 9a +b æ" 9a b =6ab 9a +b =36 36æ6ab ab 6 (, 3a=b) ab x >0, y >0 ()æ() x +y æ" x y = xy x +y =8 8æ xy xy 9 (, x =y =9) -9 xy 9 xy x>- x+>0 ()æ() 3 3 3x =3(x+) x+ x+ 3 3x+5+ ææ 3(x+) 33 - x+ 3x+5+ = 3-=5 3 {, 3(x+)= 33, x=- x+ {} a b (3a+4b){;a#;+;b!;}= b a 3a b æ3+æ b a =3+ 6=5 3a b {, 33 = 333, a=b} b a x, y (4 +3 )(x +y )æ(4x+3y) x +y =4 00æ(4x+3y) -0 4x+3y 0 {, ;4{;=;3};} -0 4x+3y 0 90 a, b, x, y (a +b )(x +y )æ(ax+by) 3æ(ax+by) (ax+by) 6 -'6 ax+by '6 ax+by '6 '6 9 a, b, c ( + +3 )(a +b +c )æ(a+b+3c) a +b +c = 4 æ(a+b+3c) 5

16 -'7 a+b+3c '7 {, a=;b;=;3c;} a+b+3c -'7 -'7 9 k=0,0 x +x+0<0 x k+0, kx +(k+)x+k<0 k<0 yy kx +(k+)x+k=0 D D=(k+) -4k <0, 3k -k->0 (3k+)(k-)>0 k<-;3!; k> yy, k<-;3!;, k<-;3!; k<-;3! 93 y cm x cm, x cm y cm 40 cm x+5y=40 x>0, y>0 ()æ() x+5yæ'ƒx 5y='ƒ0xy 40æ'ƒ0xy 'ƒ0xy 0 ( x=5y) xy 0xy 400 xy cm x=5y x=5y x+5y=40 x+x=40 x=0 0 cm 40 cm, 0 cm II. 94, X Y X- Y 95 x= -+=3-a a= -'<, '> f(-')=3(-')-=4-3' f(')=-'+ f(-')-f(') =4-3'-(-'+) =-' -' 96 X Y X Y X Y X f(0)=0, f()=, f()=4, f(3)=4, f(4)=, f(5)=0 X Y X Y

17 f {0,, 4} {0,, 4} 97, f=g f {x x+-}, g f+g f(-)=g(-)= f(0)=g(0)=0 f()=g()= f=g f g 98 f()=g() -4+6=a+b a+b=3 yy f()=g() 4-8+6=a+b a+b= yy, a=-, b=4 ab= y x=a(a<( )) y O y O a x=a a x=a x x y a O x x=a y a x=a O x 00 y y 4, y=k(k) x,,, x 0 O y O y O y a x=a - O x f(x)=-x+b = x x x y O O a x=a x y - O x y O x a x 7

18 (-, 7), (a, -) f(-)=7 +b=7 yy f(a)=- -a+b=- yy a=3, b=5 a-b=- - 0 f(x)=x +x+a y f(x)=(x+) +a- 3 xæy=f(x) - O x = (, 3) f()=3 4+4+a=3 a= xæ f(x)=-x+3 x< y y=ax+b y=ax+b a O x y=-x+3 a<0 = y=ax+b(, ) =a+b f ab a<0a+b= a<0a+b= 04 f f(x)=x f(7)=7 g f(5)=5 g(5)=5 g(x)=5 g(7)=5 f(7)+g(7)=7+5= 05 X Y,, 3, 4 4 4_4_4_4=4 =56 a,, 3, 4 4 b a 3 c a, b d a, b, c 4_3 =4 X Y,, 3, , 4 06 Y,, 3, 4, 5 5 Y 4 3Y 3 X Y 5_4_3= f()= ( fωf )()=f( f())=f()=3 (fωfωf )()=( fωf )( f()) =( fωf )()=f( f()) =f(3)=4 f()+( fωf )()+( fωfωf )() =+3+4=8 8 8

19 08 (gωf )(3)=g( f(3))=g(_3+) =g(7)=7 -=47 (fωg)(-)=f(g(-)) =f((-) -)=f() =_+=5 (gωg)(-)=g(g(-)) =g((-) -) =g(-)=(-) - = (gωf)(x)=g( f(x))=g(x+) (gωf)(x)=(x+) -3 =x +4x+ (gωf)(x)=-3 x +4x+=-3 x +4x+4=0, (x+) =0 x=- - 0 (fωg)()+(gωf )(-) =f(g())+g(f(-)) =f(-+3)+g(_(-)-) =f()+g(-3) =(_-)+5 =6 6 (hωf )()=h( f())=h()= (hωgωf)(-)=h(g( f(-))) =h(g(-))=h(4) =6 (gωgωg)(x)=g(g(g(x))) =g(g(-x)) =g(4x) =-8x 6-8x (hωg)(x)=x-, f(x)=x+ (hω(gωf ))()=((hωg)ωf )() =(hωg)( f()) =(hωg)(4) =_4-=7 7 3 ( fω(gωf))(x)=f(g( f(x)))=f(g(x-)) (fω(gωf))(x)=f((x-) +4) (fω(gωf))(x)=f(x -x+5) (fω(gωf))(x)=(x -x+5)- (fω(gωf))(x)=x -x+4 (fω(gωf))(x)=8 x -x+4=8, x -x-4=0 x 4 (fωg)(x)=f(g(x))=f(x+k) (fωg)(x)=(x+k)+3 (fωg)(x)=x+k+3 (gωf)(x)=g( f(x))=g(x+3) (fωg)(x)=x+3+k fωg=gωf x+k+3=x+3+k k+3=3+k k=0 g(x)=x g(-)=- - 5 f (x)=x+ f (x)=( fωf)(x)=f( f(x))=f(x+)=x+4 f (x)=(fωf )(x)=f(f (x))=f(x+4)=x+6 f (x)=(fωf )(x)=f(f (x))=f(x+6)=x+8 f «(x)=x+ f (x)=x+_009=x+408 f ()=+408=

20 6 f(x-)=x -4x+ f() x-= x= f()=-4+=- - f(x-)=x -4x+ t+ x-=tx= t+ f(t)={ } t f()={ } =- 7 x- f{ }=x +3x+ x+ x- f()+f(3) = x+ x-=x+ x=-3 f()=(-3) +3 (-3)+= x- =3 x+ x-=3x+3 x=- f(3)=(-) +3 (-)+=- f()+f(3)=0 0 x- x- f { }=x +3x+ =t x+ x+ x-=t(x+), (-t)x=t+ t+ x= -t t+ t+ f(t)={ }+3 + -t -t + + f()={ } f()=9-9+ = f(3)={ } f()=4-6+ =- f()+f(3)=0 8 (fωh)(x)=f(h(x))=h(x)-, (fωh)(x)=f(x) h(x)-=x- h(x)=x (hωgωf)(x) g(x-) =h(g(f(x))) =-3(x-)+4 =h(g(x-)) =-6x+7 =h(-6x+7), (hωgωf)(x)=g(x) h(-6x+7)=-3x+4-6x+7=tx=-;6!;t+;6&; h(t)=-3{-;6!;t+;6&;}+4=;!;t+;!; t x h(x)=;!;x+;!; h(x)=x h(x)=;!;x+;!; 9 (fωf)(3)=f( f(3))=f()=0 0 0 y=x y y=x y=f(x) y d y f(d)=c, f(c)=b, f(b)=a c b a O a b c d e x (fωfωf)(d) =f(f(f(d))) =f(f(c)) =f(b) =a f(x)=t ( fωf)(x)=f( f(x))=f(t)=c 0

21 f(t)=c tt=d f(x)=d xx=e a e f(x) 0 x (0-)(0) f(x)=x-(0 x ) g(x) - x<0 (00) (-), 0 x (00)() -x (- x<0) g(x)= [ x (0 x ) (gωf)(x)=g( f(x)) -f(x) (- f(x)<0) (gωf)(x)=[ f(x) (0 f(x) ) -(x-) (- x-<0) (gωf)(x)=[ x- (0 x- ) -x+ (0 x<) (gωf)(x)=[ x- ( x ) y=(gωf)(x) y O x f()= f ()= g (3)= g()=3 f ()+g()=+3=5 5 3 f (-)= f()=- +a=- a=-3 g (4)=- g(-)=4 4+b=4 b=0 a+b=-3+0= f(6)= ;3@;_6-k= 4-k= k= f(x)=;3@;x- f (8)=a f(a)=8 ;3@;a-=8 a=5 f (8)=5 5 5 f(x) g(x) f (x)=g(x) g(8)=k f(k)=8 -k+6=8 k=- g(8)=- f (x)=g(x) g (x)=f(x) g (3)=f(3)=- 3+6=0 g(8)+g (3)=-+0=- - 6 (f Ωg)(a)=f (g(a))= g(a)=f() a-= a=3 3 7 (gωf )()=g(f ()) f ()=k f(k)= 3k-= k= f ()= (gωf )()=g()=+3-=4 4 8 f(x)=3x+ f(x) y=f(x) f(0)=a f()=b a=b=5 a+b=7 7

22 9 y=;!;x+ R R y=;!;x+ x x=y- x y y=x- y=x - (xæ0, yæ-) {x xæ0}, {y yæ-} y=x - x x =y+ x='ƒy+ ( xæ0) x y y='ƒx+ (xæ-) y=x -x (xæ, yæ-) {x xæ}, {y yæ-} y=x -x+-=(x-) - x (x-) =y+ x-='ƒy+ ( xæ) x=+'ƒy+ x y y=+'ƒx+ (xæ-) 30 y=;3!;x+ x 3y=x+6 x=3y-6 x y y=3x-6 3x-6=ax+b a=3, b=-6 a+b= x+ x+ f{ 3 }=-4x+ 3 =t 3 3 x+=3t x= 3t- 3 3t- f(t)=-4 3 +=-6t+3 f(x)=-6x+3 y=-6x+3 6x=-y+3 x=-;6!;y+;!; x y y=-;6!;x+;!; f (x)=-;6!;x+;!; f (-3)=;!;+;!;= f(x)=-6x+3 f (-3)=k f(k)=-3-6k+3=-3 k= f (-3)= 3 (f Ωg) (3)=(g Ωf )(3) =g ( f(3)) =g (5) g (5)=k g(k)=5 ;!;k-=5 k= (f Ωg) (3)= 33 (fω(gωf) Ωf)(x) =( fωf Ωg Ωf)(x) (gωf) =f Ωg =(g Ωf)(x) fωf =I g(x)=x+4 y=g(x) y=x+4 x=y-4 x y y=x-4 g (x)=x-4 (g Ωf)(x)=g ( f(x)) =g (-x+) (g Ωf)(x)=(-x+)-4 (g Ωf)(x)=-x-3 a=-, b=-3 a=-, b=-3

23 34 (( f Ωg )Ωf )(a) =( f Ωg )( f(a)) =( f Ωg )(a+) =(gωf ) (a+)= (gωf )()=a+ g(f())=a+ g(3)=a+ f()=3 -;3!; 3+4=a+ a= 35 y=f(x) y=f (x) y=f(x) y=x x -4x+6=x x -5x+6=0 (x-)(x-3)=0 x= x=3 (, ), (3, 3) " (3- ) +(3 -) =' ' 36 y=x y y=g(x) y=x y y f (d)=k e d c b a y=f(x) f(k)=d k=c f (d)=c (gωf )(d) =g(f (d)) =g(c)=b O a b c d e x f (b)=l f(l)=b l=a f (b)=a (f Ωg)(c)=f (g(c))=f (b)=a (gωf )(d)+(f Ωg)(c) =b+a a+b 37 x + x + x - x - x +x+ = x - x +x+ = (x-)(x +x+) = x- x x-6 - x-3 3-x x x-6 = + x-3 x-3 x+x-6 = x-3 x-6 = 3 x-3 (x-3) = 34 x-3 = 3x- x x -4 x -3x+ 3x- x-3 = - (x-)(x+) (x-)(x-) (3x-)(x-)-(x-3)(x+) = 35 (x-)(x+)(x-) x -6x+8 = (x-)(x+)(x-) (x-4)(x-) = (x-)(x+)(x-) x-4 = (x+)(x-) 38 x-4 x- (x+)(x-) x -5x+6 x +5x+4 x-3 _ x -6 x -4 x-4 (x-)(x-3) (x+)(x+4) x-4 = 35 _ 35 _ 5 (x+4)(x-4) (x+)(x-) x-3 x+ = 5 5 x+ x+ x+ 3

24 39 a bx+c 5+ 5 x- x +x+ a(x +x+)+(bx+c)(x-) = 53 (x-)(x +x+) (a+b)x +(a-b+c)x+a-c = 53 x - x+4 (a+b)x +(a-b+c)x+a-c 53= 53 x - x - x a+b=0, a-b+c=, a-c=4 a=, b=-, c=- abc=8 8 x -=(x-)(x +x+) (x-)(x +x+) x+4=a(x +x+)+(bx+c)(x-) x+4=(a+b)x +(a-b+c)x+a-c x a+b=0, a-b+c=, a-c=4 a=, b=-, c= x+ x+ x+3 x+4 ={ - }+{ - } x+ x+3 x+ x+4 = + (x+)(x+3) (x+)(x+4) (x+)(x+4)+(x+)(x+3) = 3 (x+)(x+)(x+3)(x+4) (x +0x+) = 3 (x+)(x+)(x+3)(x+4) x- x x+ x-3 ={ - }+{ - } x- x+ x-3 x 3 3 = + (x-)(x+) x(x-3) 3x(x-3)+3(x-)(x+) = x(x-)(x+)(x-3) 6(x -x-) = x(x-)(x+)(x-3) x+3 x+7 x+ x x+4 x+8 x+ x+6 (x+4)- (x+8)- = x+4 x+8 (x+)- (x+6)- =-5-5 x+ x+6 ={- }+{- } x+4 x+8 =-{- }-{- } x+ x+6 = x+4 x+8 x+ x+6 ={ - }+{ - } x+ x+4 x+6 x+8 = + (x+)(x+4) (x+6)(x+8) (x+6)(x+8)+(x+)(x+4) = (x+)(x+4)(x+6)(x+8) 4(x +0x+8) = (x+)(x+4)(x+6)(x+8) x -x-3 x +x- + x+ x- (x -x-)- (x +x-)+ = + x+ x- (x+)(x-)- (x-)(x+)+ = x+ x- ={x-- }+{x++ } x+ x- =x+ - x- x+ (x+)-(x-) =x+ (x-)(x+) =x+ x - 4

25 x- x+ < x - x-3 x +x -x-3 -x- -x- x -x-3 x+ =x-- x+ 4 x +x- x- =x++ x =;!; {-;3!;}+;!; {;3!;-;5!;}+;!; {;5!;-;7!;} =+;!; {;7!;-;9!;}+;!; {;9!;-; ;} =;!; {-; ;}=; ; x +x x +4x+3 x +9x x +6x = + x(x+) (x+)(x+3) 3 6 =+ - (x+3)(x+6) x(x+6) ={;[!;- }+{ - } x+ x+ x+3 =+{ - }-{;[!;- } x+3 x+6 x+6 =0 4 (a+b)+(a-b) + 3 a-b a+b (a-b)(a+b) = 33 (a+b)-(a-b) - 3 a-b a+b (a-b)(a+b) a = = a b b x+ x- < x + x- x -x x- x- x- ; ; 0 x-y x+y-(x-y) - 3 x+y x+y = 33 y y-(x+y) - x+y x+y -x+y x-y = = -x x x+ +;[@; x = 5 5 (x-3)(x+)-5 x-3-3 x+ x+ x+ x = 5 x -x-8 x+ x+ x+ = 3_ 3 x (x+)(x-4) x+ = x(x-4) 43 x + 3 ={x-;[!;}+= +=6 x {x+;[!;}={x-;[!;}+4= +4=8 x+;[!;= ' x - 3 ={x+;[!;}{x-;[!;} x =( ' ) = 4' x - 3 ={x-;[!;}3+3{x-;[!;} x = +3 =4 44 a x-y x+ b x x(x-4) 6 4' 4 x +x+=0 x+0 x x++ =0 x+ =- x x x + ={x+ } -=(-) -=- x x 5

26 x + ={x+ } -3{x+ } x x x =(-) -3 (-)= {x + } +{x + } x x =(-) + = x + =7 x {x+;[!;}-=7, {x+;[!;}=9 x+;[!;=3( x>0) x + ={x+;[!;}3-3{x+;[!;} x =3-3 3= y- x+;]!;= x=-;]!;= y y+ = =-y z= z z (-y) y z+ = + x (-y) (y-) y- z+ = =;!; ;!; (y-) 47 y x=3y ;3{;=;; =k(k+0) x=3k, y=k y y y - + x +y x-y x+y (k) k k = 5- + (3k) +(k) 3k-k 3k+k 4k k = - +;5@kK; 3k k =; 3;-+;5@; =-;6*5$; (x-y) (3k-k) = 5 x -y (3k) -(k) k = 9k -4k k = 5 =;5!; 5k -;6*5$; ;5!; 48 x+y x+y = =k(k+0) 5 7 x+y=5k, x+y=7k x=k, y=3k x -xy k -k 3k -k 5 = 55 = =- xy+y k 3k+(3k) k 6-6 x+y x+y = 5 7 4x+7y=5x+0y 3y=9x y=3x xy+0 y=3x x -xy x -x 3x -x 5 = 55 = =- xy+y x 3x+(3x) x 6 49 x+y y+z z+x = = =k(k+0) 3 4 x+y=k yy y+z=3k yy z+x=4k yy ++ (x+y+z)=9k x+y+z=;(; k yy - z=;%;k - x=;#; k - y=;!;k 6

27 x:y:z=;#; k:;!; k:;%;k x:y:z=3::5 x:y:z=3::5 x=3k, y=k, z=5k(k+0) xy-yz+zx 3k k-k 5k+5k 3k 5= 5 x +y +z (3k) +k +(5k) 3k = 3 35k 5 y=- + x+ x=- y= x y y=0, x=0 x=0, y=0 y - O x =;3!5#; 3::5 ;3!5#; {x x+-} {y y+} 50 x-3y+z=0 yy [ 6x+y-z=0 yy _+ 0x-5y=0 y=x y=x x-6x+z=0 z=4x 5x-y+z 5x-x+ 4x = 5 x+y+z x+ x+4x x = =: 9 : : 9 : 9x 5 a+b+3c+0, (b+3c)+(3c+a)+(a+b) k= a+b+3c (a+b+3c) k= = a+b+3c a+b+3c=0, b+3c=-a 3c+a=-b a+b=-3c -a -b -3c k= = = =- a b 3c, k= k=- - -x+ -(x+3)+7 7 y= 33 = 333 = 3 - x+3 x+3 x+3 x=-3, y=- x y y=0, x=;!; x=0, y=;3!; {x x+-3} {y y+-} 4x -4x y= 3= 3 3-x x-3 -(x-3)-6 y= 34 x-3 6 y=-3- x-3 x=;#;, y=- x y y=0, x=0 x=0, y=0 [x x+;#;] {y y+-} -3 y - O 3 y O - 3 x x 7

28 53 ;A;(x+)-;A;+3 ax+3 y= 3= 3455 x+ x+ -;A;+3 y= +;A; x+ x=-;!; y=;a; ;3!;(3x+b)-;3!;b- x- y= 3= x+b 3x+b ;3!;b+ y=- +;3!; 3x+b x=-;3b; y=;3!; -;!;=-;3B;;A;=;3!; b=;#; ab= 54 y=-;[#; x 3, y - 3 -x+3 y=- -= 35 x-3 x-3 a=-, b=3, c=3 abc= ax- a(x-)+a- a- y= 3 = = +a x- x- x- x=y=a (a) b=a=5 a+b= x+4 3(x+)- y= 3 = 5=- +3 x+ x+ x+ x=- y=3(-3) y=-x+k(-3) 3=+k k= 57 x=-, y=4 k y= 3+4 (k+0) yy x+ (3, ) k = 3+4 k=-5 3+ k= (x+) y= 3+4= 3 x+ x+ 4x-7 y= 3 x+ a=, b=4, c=-7 a+b+c= x=3, y= k y= 33 + (k+0) yy x-3 (, 0) k 0= 3+ k= -3 a+b+k=-3++= 59 x+3 (x+)- y= 33= 33 x+ x+ y=- 3+ x+ y=-;[!; x -y 8

29 y=;#;, x=0 y=3, x=-3 [y y ;#; yæ3], {x -3 x<- -<x 0} 60 x y= 3= 3+ x- x- x æ3, xæ3 x -3 x=3, ()=;#; x=-3, ()=;4#; yy ;#;+;4#;=;4(; ;4(; 6 3x+k 3(x+)+k-6 k-6 y= 33 = 33 = x+ x+ x+ k>6 k-6>0 y k 0 x a 3 x=0 5 O a x 5=;K; k=0 - x=a 4 3a+0 4= 33 a= a+ a+k=+0= 6 f (x)=( fωf)(x)=f( f(x)) -3 x+ x + x- x- f (x)= = 5 =x x+ - x- x y 3 y O O x x f (x)=f (x)=f fl (x)=y=f «(x)=x f 0 (x)=f _50+ (x)=f(x) 0+ f 0 (0)=f(0)= 355 =;; 9 ;; ;; 9 ;; 0-0+ f (0)= 355 =;; 9 ;; 0- f (0)=( fωf)(0) f (0)=f( f(0))=f{;; 9 ;;} 5+ 9 f (0)= 5 =0 5-9 f (0)=( fωf )(0) f (0)=f( f (0))=f(0) f (0)=;; 9 ;; f«(0);; 9 ;;, 0 0=_50+ f 0 (0)=f(0)=;; 9 ;; 63 (f ) =f f (x) f(x) -x+3 y= 33 x x- y(x-)=-x+3, (y+)x=y+3 y+3 x= 33 y+ x y x+3 y= 33 x+ x+3 f(x)= 33 x+ a=, b=3, c= a=, b=3, c= 9

30 ax+b f(x)= 33 x+c -cx+b f (x)= 3343 x-a -cx+b -x = 333 x-a x- c=, b=3, a= 64 (f Ωg) (5)=(g Ωf)(5)=g ( f(5)) 5+7 f(5)= 33 =4 5- g ( f(5))=g (4) g (4)=k g(k)=4 k- 33 =4 3k+ k-=k+4 k=-; ; -; ; 65 (f ΩfΩf )(3)=f (3) f Ωf=I f (3)=k f(k)=3 k =3, k+5=3k+6 k+ k= y= + x- y=;[#; 3 y= + x- x - O x y - mx-y-m+=0 m(x-)-y+=0 yy m (, ) y y=m(x-)+ m>0 m>0 67 x+3 y= 3335 y=x- x- x x =x- x- (x-) =x+3 x -4x-=0 (x+)(x-6)=0 x=- x=6 (--3)(65) " {6-(-)} +{5- (-3)} =8' 8' 68 ()æ0, ()+0 'ƒ+x +xæ0 xæ- 'ƒ-x -xæ0 x - x 'ƒx+3 x+3æ0 xæ-3 'ƒ-x -xæ0 x ;!; -3 x ;!; 'ƒx+ x+æ0 xæ- 'ƒ5-x 5-x>0 x<5 - x<5 - x -3 x ;!; - x<5 30

31 69 -'<0 " (-') = -' =-(-') =-+' a< a-<0 " (a-) = a- =-(a-) =-a+ x>5 4-x<0, 5-x<0 "( 4-x )Ω - 5-x = 4-x - 5-x "( 4-x )Ω - 5-x =-(4-x)+(5-x) "( 4-x )Ω - 5-x = 70 -+' -a+ x-y=a +6-4a=(a-) +>0 x+y=a +6+4a=(a+) +>0 " (x-çy)ω -" (x+çy)ω = x-y - x+y " (x-çy)ω -" (x+çy)ω =x-y-(x+y)=-y " (x-çy)ω -" (x+çy)ω =-8a -8a 7 'ƒa-3'ƒ-a=-" (a-3)(-a) a-3<0, -a<0 a-3=0 -a=0 a 3 " (a+) +" (a-4) = a+ + a-4 =(a+)-(a-4) =6 6 7 'a =-Æ;bA; (a+0) 'b a>0, b<0 " (a-b) -" b + a = a-b - b + a =(a-b)-(-b)+a =3a 3a 73 '5 '3 + 3 '3+ '5-'3 '5('3 -) '3('5+'3) = ('3 +)('3 -) ('5-'3)('5+'3) ' 5 -'5 ' 5 +3 = ' 5 -'5+' 5 +3 = 3-'5+' 5 = -'-'3 3 +'+'3 {(-'3)-'}{(-'3)+'} = 3 {(+')+'3}{(+')-'3} (-'3) - = 3 (+') -3 -'3 '-'6 = 5= 3 ' 'ƒx+-'ßx 'ƒx++'ßx 'ƒx++'ßx 'ƒx+-'ßx ('ƒx+-'ßx) +('ƒx++'ßx) = 33 ('ƒx++'ßx)('ƒx+-'ßx) x+-'x +å'x+x+x++'x +å'x+x = 33 ('ƒx+) -('ßx) =4x+ 74 <'< <3-'< 3-' =. a= b=(3-' )-=-' -b= a-b -(-') -(-') -b= -(-') '- '+ -b= 5-+' ('-)('+) -b=('+)-+' -b='- '- 3

32 75 '5+ '5- x+y= 3+ 3 '5- '5+ ('5+) +('5-) x+y= 3 ('5-)('5+) x+y=;; ;;=8 '5+ '5- xy= 5 _ 5 = '5- '5+ x +3xy+y =(x+y) -xy+3xy x +3xy+y =(x+y) +xy x +3xy+y =8 + x +3xy+y = x+y='6, x-y=', xy=4 'ßx+'ßy ('ßx+'ßy) = 'ßx-'ßy ('x-'y)('x+'y) x+y+'ßxßy = x-y '6+'4 = 33 ' ='3+' '3+' 77 +x -x +' æ -æ +x= >0, -x +x ' +ßx ' -ßx -' = - -x= >0, ' -ßx ' +ßx ('ƒ+x) -('ƒ-x) = 'ƒ-x'ƒ+x +x-(-x) = 5 " -x x = " -x ' x= 4 ' 4 5 ' æ -{} ' = = ;: ;; ' ; 78 x=+'3 x-='3 x -4x+=0 yy x -x -x+5 x -4x+ x+3 x -4x+ < x - 4x - x + 5 x -4x +x 35 x -3x -x+5 x -3x -x+3 35 x -3x -x+ x -x -x+5 =(x -4x+)(x+3)+ = () x -x -x+5 = 55 =4 4 (x-) ('3) 79 (4x+4y)-(xy+8)'=8 x, y 4x+4y=8, xy+8=0 x+y=, xy=-8 x +y =(x+y) -xy x +y = - (-8)= y=-'ƒx- y=-'x x y O 3 x y=- 'ƒ x- {x xæ}, { y y } y=-'ƒ-x+-=-'ƒ-(x-)- y=-'ƒ-x x y 3

33 , y - {x x }, { y y -} y='ƒ3x--=æ 3{x-;3@;}- y=' 3x x ;3@;, y - O - [x xæ;3@;], { y yæ-} y='ƒ6-x+='ƒ-(x-3)+ x y='ƒ-x y= ' 6-x+ y x 3y '6+ O {x x 3}, { y yæ} y 3 y O - -3 y=- '-x+- x y= ' 3x-- 3 x y-cæ0 yæc {y yæc} c=- y='ƒax+b- (, ) ='ƒa+b- yy ='ƒ-a-, ='ƒ-a 4=-a a=-4 a=-4 b=8 a+b+c=-4+8+(-)= y='ƒ-x+ x, y - y=" -(x-)+- x x-, y='ƒ-x+3- y y+ y y='ƒ-(-x)+3- x -x y='ƒx+3- a=, b=3, c=- a+b+c= 8 y='ƒ-x++b y-b='ƒ-x+ -x+æ0 x y-bæ0 yæb {x x }, {y yæb} a=b=3 a+b=4 4 8 ax+bæ0 axæ-b {x x } a<0 x -;ab; -;ab;= b=-a yy 84 y='ƒ-3xƒ+a+=æ- 3{ x - ;3A;}+ y y='ƒ-3x 'ƒ 5+a+ 'ß 3+a+ x ;3A;, y -5 - O a x 3 x=-5,y='ƒ5+a+ x=-,y='ƒ3+a+ -5 x - 'ƒ5+a+ 'ƒ3+a+ 33

34 3 'ƒ3+a+=3, 'ƒ3+a= 3+a=4 a= ()='ƒ5+a+='ƒ5++= y=' ax(a>0) x 3, y - y=" a(x -3)- yy (5, 0) 0=' a-, ' a=, a=4 a= a= y=" (x -3)-='ƒx-6- a=, b=-6, c=- a+b+c= y='ƒ4x-8=" 4(x-) y y=' 4x y=x+k x y= 'ƒ ƒ4x-8, O x y=x+k y k y=x+k(, 0), 0=+k k=- y='ƒ4x-8 y=x+k, 'ƒ4x-8=x+k x +(k-)x+k +8=0 D D 3=(k-) -(k +8)=0 4-4k-4=0 k=- - k<- k=- k<- k>- - k<- k=- k<- 87 k>- ƒ y='ƒ6-x='ƒ-(x-3) y y='ƒ-x y=-x+k x 3 y= 'ƒ 6-x, O y=-x+k - y k y=-x+k(30), 0=-3+k k=3 3 x y='ƒ6-x y=-x+k, 'ƒ6-x=-x+k x -(k-)x+k -6=0 D D 3=(k-) -(k -6)=0 4 -k+7=0 k=;&; 3 k<;&; 88 3 k<;&; y=4-'ƒx+6 {y y 4} {x x 4} y=4-'ƒx+6 y-4=-'ƒx+6 y -8y+6=x+6 x=;!;y -4y+5 34

35 x y y=;!;x -4x+5 (x 4) a=;!;, b=-4, c=5, d=4 a+b+c+d=: : : : 89 y='ƒ-xƒ+a+ y-='ƒ-xƒ+a (y-) =-x+a x=-(y-) +a x y y=-(x-) +a g(x)=-(x-) +a g()=3 -(-) +a=3 a=4 g()=-(-) +4=4 4 f(x) g(x) g()=3 f(3)= f(3)='ƒ-3 +a+= 'ƒ-3 +a=, -3+a= a=4 f(x)='ƒ-x +4+ g()=k f(k)= f(k)='ƒ-k +4+= 'ƒ-k +4=0, -k+4=0 k=4 g()=4 90 y=-'ƒ-x y=-'ƒ-x y=x -'ƒ-x=x -x=x x +x-=0, (x-)(x+)=0 y O y=x x y=-'ƒ -x yy x= x=- x 0 x=- (-, -) a=-, b=- a+b= (fωf Ωf )(3)=f (3) fωf =I f (3)=k f(k)=3 'ƒ5k-=3 5k-=9 k= f (3)= (fωf Ωf )(3)= 9 (g Ωf ) ()=( f Ωg)() =f (g()) =f (4) g()='ƒ4+5+=4 f (4)=k f(k)=4 'ƒk+3=4 k+3=6 k=3 f (4)=3 (g Ωf) ()= (fω(gωf) Ωf)()=( fωf Ωg Ωf)() =( g Ωf)() =g (f()) +5 =g (3) f()= 3=3 + g (3)=kg(k)=3 'ƒk-=3 k-=9 k=: : (fω(gωf) Ωf)()=: : : : 35

36 III. 94 d d=0-6=-6 6, 0, -6, -, a, 5, a, 7 +d +d +d a +d=5 a +3d=7 a =9, d=6 a =5+6= d=;4!;-;4#;=-;!; yy yy a=0, d=7-0=-3 a«=0+(-)_(-3)=-3+3 a«=3 a«= a«=3-5 a =3_-5=- a«=3(+)-5=3- a«-a«=3--(3-5)=3 a«=-7+9 a =-7_+9= a«=-7(+)+9=-7+ a«-a«=-7+-(-7+9)=-7 -, 3, -7 ;4#;, ;4!;, -;4!;, -;4#;, -;4%; -;!; -;!; -;!; -;!; d=;!;-;3!;=;6!; a, ;3!;, ;!;, a +;6!; +;6!; +;6!; a +;6!;=;3!; a =;3!;-;6!;=;6!; 97 a, d a =3 a+d=3 yy a =3 a+6d=3 yy, a=, d= a«=+(-) =-, 99 -=99 =00 a«=-, 00 a =;!;+;6!;=;3@; -, -8 9, -;4#;, -;4%; ;6!;, ;3@; 95 3, 6, 9, y a=3, d=6-3=3 a«=3+(-)_3=3 0, 7, 4,, y 98 a, d a +a =(a+5d)+(a+4d)=6 a+9d=6 yy a +a =(a+7d)+(a+5d)=70 a+d=35 yy, a=, d=3 a =a+30d=+30 3=9 9 36

37 a, d (a+d)+(a+5d)=0, a+6d=0 a+3d=0 yy 3 a+d= yy, a=3, d=- 3, (+) -8+{(+)-}_=30 =36 =8 8 0 a, d a =70 a+6d=70 yy a º=6 a+9d=6 yy, a=88, d=-3 a«=88+(-) (-3)=9-3<0 >:ª3 :? a=-40, d=-37-(-40)=3 a«=-40+(-) 3=3-43>0 >: 3 :? x, x -, x+3 x - x x+3 (x -)=x+(x+3) x -3x-5=0, (x-5)(x+)=0 x=;%; x=- ;%; - 04 a, b, c a+c b= yy -c, b, 4a -c+4a 4a-c b= b= yy 4 a+c 4a-c = 4 (4a-c)=4(a+c) 4a=6c a=;#;c yy b=;4%;c ;#;c+;4%;c ;; 4 ;;c a+b = 5 = 5 =;; 4 ;; ;; 4 ;; c c c 05 a-d, a, a+d (a-d)+a+(a+d)=5 3a=5 a=5 yy (a-d) +a +(a+d) =83 yy d =4 d= 3, 5, 7 3, 5, (50-0) S«= =0 = 37

38 -0 a =50+0d=-0 d=-6 a º=50+9 (-6)= a, d 5(a+37) S = =05 a=5 a =5+4d=37 d= a d S º=0 0(a+9d) =0 a+9d=4 yy S º=440 0(a+9d) =440 a+9d=44 yy a=3, d= 30( 3+9 ) S º= 5 = , 07,, y, 97 0, 5 97=0+(-)_5 = , 97, 0 0(0+97) = S () S () =+4+y+00, 50, 00 50(+00) S () = =550 3 S (3) S (3) =3+6+y+99 3, 33, 99 33(3+99) S (3) = = S (6) S (6) =6++y+96 6, 6, 96 6(6+96) S (6) = =86 3 S S=S () +S (3) -S (6) = = , 5, + 00 (+)(-5+5) =00 = d=5 d=;@9); =8, d=;@9); -9, 4 S«{ (-9)+(-) 4} S«= S«= -3 38

39 S«={-: 4 :} -:ª;8^; : : 4 :, S«= =-0 8, -0 a«a«=-9+(-) 4=4-33 a«=4-33< { (-9)+7 4} S = =-0 3 S«= -3 S«=(-) -3(-) = -7+5 S«0 a«=s«-s«=4-5 (æ) a º= S«= -+3 a =S = - +3= a º=S º-Sª a º=(0-0+3)-(9-9+3)=7 a +a º=+7=9 9 5 a =8, a =-54 r a -54 r= = 3=-3 a 8 a =a r =8 a = a =, a = (-3)=-6 a =', a =- r a r= =- a ' a = {- }=-;!; ' ' a =-;!; {- }= ' ' 6, -6 -;!;, a, ra =-8, a =64 a r =-8 yy a rfl =64 yy r =-8 r=- r=- a = a«= (-)«=(-)«a =(-)fi =(-) =6 7 a«=(-)«, a =6, (-4)«=-04=(-4)fi -=5 = a = = a = = 6 a = 3=;!;, ;!; a 9 a, r a +a =6 a+ar=6 a(+r)=6 a +a =8 ar +ar =8 ar (+r)=8 r =3 a +a =ar +arfi a +a =ar (+r) ' yy yy 39

40 a +a =r a(+r) a +a =3 6= a r a -a =a-ar =a(-r )=56 yy a +a +a =a+ar+ar =a(+r+r ) =4 yy a(-r ) a(-r)(+r+r ) 56 = 5 = 3 a(+r+r ) a(+r+r ) 4 -r=4 r=-3 a(-3+9)=4 a= a =ar = (-3) =6 6 r =rfi, rfi =43 r=3 a +a =r +r = =80 80 a 6 a 3=6 a= a«= 3«3«>0000 3«> =87, 3 =656 -æ8 æ a, ar, ar a+ar+ar =3 yy a ar+ar ar +a ar =- yy a ar ar =-k yy a(+r+r )=3 yy a r(+r+r )=- yy ar=-7 k=-(ar) =-(-7) = , a, b a =b yy a, b, 30 b=a+30 yy a =a+30, a -a-30=0 (a+5)(a-6)=0 a=-5 a=6 a, b a=6, b=8 b-a= 5 f(x)=x +x+a x+, x-, x- f(-), f(), f() { f()} =f(-)f() (a+3) =(a-)(a+8) a +6a+9=a +7a-8 a=7 f(x)=x +x+7 x+ f(-)=7 7 6, -3 S«{-(-3)«} S«= 5=;4!; {-(-3)«} -(-3) ' ' 0 S º ' {(') -} '(3-) S º= 3 = '- '- S º=3(+') 40

41 3, 96 3 «=96 «=3=fi -=5 =6 6 S 3(fl -) S = =89 - S«3«S«=;#;+;(;+: :+y+ 3 ;#;, 3 ;#;(3«-) S«= =;4#;(3«-) 3- ;4!; {-(-3)«} 3(+') 89 ;4#;(3«-) 96 S rl-a 96-3 S= = =89 r- - 7 a, r, S«a(-rfl ) S = 3=4 yy -r a(-r ) S = 3= yy -r a(-rfl )(+rfl ) = -r +rfl =3 rfl = a(-r ) S = -r a(-rfl )(+rfl +r ) S = -r a(-rfl ) S = (+rfl +r ) -r S =4(+rfl +r ) () S =4(++4) S =8 8 8 a, r a +a =0 ar+ar =0 ar(+r )=0 a +a =40 ar +arfi =40 ar (+r )=40 r =4 r= ( r>0) 0a=0 a= a«= «= «(«-) S«= =«- - 9 yy yy a«=«, S«=«- x, (x+) S«x [{(x+) }«-] S«= (x+) - x {(x+) «-} S«= x +x (x+) «- S«= ( x>0) x+ 30 æ a«=s«-s«=(3«± -3)-(3«-3) = 3«yy = a =S =3-3=6 = a«= 3«(x+) «- x+ a«= 3«4

42 3 S«= 3«+k æ a«=s«-s«=( 3«+k)-( 3«+k) =4 3«yy = a =S = 3+k=6+k yy {a«} = 6+k=4 k=- - S«= 3«+k +k=0 k=- 3 a r a =ar=4 yy a =ar =3 yy r =8 r= r= a=4 a= S«(«-) S«= 5 =«± - - S«>000 «± ->000 «± >00 =5, =04 +æ0 æ S S=_(+0.05)+_(+0.05) +y +_(+0.05) _(+0.05), _(+0.05){(+0.05) -} S= (+0.05)- _.05(.6-) S= =.6() y 9 0 S S=0+0(+0.)+0(+0.) +y +0(+0.) 0, {(+0.) -} S= (+0.)- 0(3.-) S= 5=75() a 0 a(+r)+a(+r) +y+a(+r) 0 00 a(+0.06){(+0.06) -} =0fl (+0.06)- 0fl _0.06 a=.06(.06 -) a= = (.8-) y 9 8 y _(+0.07) 0(+0.) 0(+0.) y 0 0(+0.) 0 0 4

43 =000_3.87=3870 yy a 3 a 3 y a+a(+0.07)+a(+0.07) +y+a(+0.07) a(.07 -) a(3.87-) = 5= =4a =4a a= = a a (0000) (+0.05) =0000_.05 =0000_.6=5000() yy a a0 a+a(+0.05)+y+a(+0.05) a{(+0.05) -} a(.6-) = 55 = (+0.05) =3a a=5000 a= =65() 3 y y 90 a a a a(+0.07) a(+0.07) a(+0.07) 65 y a(+0.07) = 3 + ()= 3 k= k+ a=-3, d= a«=-3+(-) =-5-5=9 = ()= (k-5) 39 3 i=,, 3, y, 8 ()=+3+3 +y+3 3 j=,, 3, y, j(j+) () = + + +y (+) 40 0 (3a -) - (a +) k= = {(3a -) -(a +) } = (8a -0a -3) =8 a -0 a - 3 = = (a +b ) = (a +a b +b ) k= 0 k= 0 k= 0 k= 0 0 k= k= k= (a +b ) = (a +b )+ a b k= k= 0 k= 7 k= k= 43

44 60=40+ a b a b = = 3 k= 3-3 =3 3-3 =3( + +y+ ) 3 - ( -) 3 =3 =6( -) - 6( -) 43 (k +)- (k -) k= = (k +)-[ (k -)-( -)] k= = (k +)- (k -)+( -) k= = {(k +)-(k -)}+( -) k= k= = + - k= = +- +-=9 +-0=0 (+)(-0)=0 = (3+4k)=3+ (3+4k) k=0 (+) (3+4k)= (3+4k)= +5+3 (k+3 )= k+3 k= k= k= - k= (+) («-) = = ++3 «-3 k= k= 0 k= 0 k= k= k= k= k = k - k (+)(4+) (+)(+) = =;6!;(+)(8+--) (+)(7+) = «-3 (k -k+)+ (i +i-) k= = (k -k+)+ (k +k-) = {(k -k+)+(k +k-)} = k = k 0 = 55= (k-3)=4 (k-3) k=5 0 k= 0 k= 0 k= k=+ 4(k-3)=4[ (k-3)- (k-3)] k= k= (+5)(+6) 4(k-3)=4[ -3(+5) 4 5 4(k-3)=-{ -3 4}] 4(k-3)= +0+8 ABC= 0 8= k= a«=(+) 0 ()= k(k+) k= 0 k= (+)(7+) k= +5 k=5 0 i= +5 0 k= 4 44

45 ()= k + k 0 0 ()= + 6 ()= («-) a«=++ +y+«= =«- - S«S«= ( -) S«= - («-) S«= - - S«=«± -- «± S«S«= a = +3 a«=s«-s«=( +3)-{(-) +3(-)} =+ (æ) a =S =4 a«=+ (æ) 5 k a = k(6k+) k= k a = (6k +k) k a =6 + = k= k= k= S«S«= a =3«- k= k= 5 k= 5 k= 0 k= a«=s«-s«=(3«-)-(3«-) =3«-3«= 3«(æ) 0 k= a =S = a«= 3«(æ) a «3 «5 a «= 5=3 3 «a 0 5= 3=30 a 30 0 k= 5 (k+l)= k+ l 0 (k+l)= +0l 0 k= (k+l)=0l+55 ()= (0l+55) ()=0 l ()= ()=00 j j(j+) i= i= k { i} j= = k j= =;!;{ j + j} k(k+)(k+) k(k+) =;!;[ 5 + ] 6 k(k+)(k+) = 5 6 () = j i= j(j+) k j= 4 k= k= k(k+)(k+) 5 6 =;6!;{ k +3 k + k} 4 5 =;6!;[{ } ] 6 =;6!;( ) 4 k= 0 k= k j= 0 l= 4 k= 0 0 k= l= = l= 4 k= 45

46 5 m { i} = i= m (+) m = 3=;!; (+) = = =;!;{ + } m(m+)(m+) m(m+) =;!;[ 35+ 3] 6 m(m+)(m+) = 3=56 6 m(m+)(m+)=336=6 7 8 m= m = m = a«= 5= 5 (+) - 4(+) a«=;4!;{;!;- 3} + S«= a =;4!; {;k!;- 3} k= k= k+ S«=;4!;[{-;!;}+{;!;-;3!;}+y S«=+{;!;- 3}] + S«=;4!; {- 3}= + 4(+) a«= 5 = ++3+y+ (+) a«= 3 ={;!;- 3} (+) + S«= a = {;k!;- 3} k= k= k+ S«=[{-;!;}+{;!;-;3!;}+y S«=+{;!;- 3}] + S«= {- 3}= (+) k= 0 =:ª4º: { 4-4} k= 4k+ 4k+5 =:ª4º:[{;5!;-;9!;}+{;9!;-; 3;}+y+{;4 ;-;4 5;}] =:ª4º: {;5!;-;4 5;}= () = 9 = [ 5- ] k= k(k+) (k+)(k+) = [{ 5-5}+{ 5-5}+y = [+{ 5-5}] = { 5-5}= a«= 'ƒ++'ƒ+ 'ƒ+-'ƒ+ a«= ('ƒ++'ƒ+ )('ƒ+-'ƒ+ ) a«='ƒ+-'ƒ+ {a«}6 6 a = ('ƒk+-'ƒk+ ) k= 9 k= a =('3-' )+('4-'3 )+y a =+('6å4-'6å3) a =-'+'6å4=8-' (4k+)(4k+5) k(k+)(k+) 6 k= a«= 5 'ƒ++' ('ƒ+-' ) a«= 5 ('ƒ++' )('ƒ+-' ) ' 46

47 a«='ƒ+-' {a«}5 5 a = ('ƒk+-' k) k= a =('4-')+('6-'4)+y+(' 3-' 30) a =' 3-' a =3' 3' 58 a«=+(-) =- 3= k= ' a +'ßa =;!; ('ƒk+-'ƒk-) =;!;{('3-)+('5-'3) =;!;{+y+(' 5-' 3)} =;!;(' 5-)= k= = 3= (' k+-'k) f(k) k= 'k+' k +å k= =('-)+('3-')+y =+(' 0å0-'ß9å9) =-+' 00=9 9 S«= a = + k= 5 k= a«=s«-s«=( +)-{(-) +(-)} =+(æ) a =S =3 a«=+(æ) 5= k= a a k= k= 3 'ƒk++'ƒk- 5 (k+)(k+3) =;!; { 5-5} k= k+ k+3 k= =;!;[{;3!;-;5!;}+{;5!;-;7!;}+y =;!;[+{ 5-5}] + +3 =;!; {;3!;- 5} +3 = 3(+3) 6 S=+ ;!;+3 {;!;} +y+30 {;!;} - > ;!;S=+ ;!;+ {;!;} +y+9 {;!;} +30 {;!;} - > ;!;S=+ ;!;+3 {;!;} +y+30 {;!;} -30 {;!;} -{;!;} - > ;!;S= 3-30 {;!;} =-fi {;!;} -;!; S=4-fl {;!;} =4-{;!;} > S= y+ -> S= y+ + -S=0-(+ +y+ + ) ( -) -S=0- =- + - S= -= {;!;} y (), (, ), (,, 3), (,, 3, 4), y (+) ++3+y+= k= 5 k= 3(+3) = , 4 47

48 (+) ++3+y+= k= 5 k= k(k+) 5 k= 50 S º S º= 6 S º=+(7 4) 6 k(k+) S º= 5+(++3+y+4) k= S º= { k + k}+ k k= k= k= S º= { } S º= (496+36)+05 S º=86+05= =34 ;6 4;34 («-) ++4+y+«= =«- - 6 fl -=63, 7 -= k- k «_ = 8 34 ; 8; 64 ()+() 3 {;!;}, {;!;, ;@;}, {;3!;, ;@;, ;#;}, y + ;8%; 63 3 {;!;}, {;4!;, ;4#;}, {;8!;, ;8#;, ;8%;, ;8&;}, y ;!;, ;4!;, ;8!;, y 3 ««=64=fl =6 ;6 4;6, 3, 5, 7, y ;6 4;6 3,, 3, 4, y,,, 4, 8, y, « =3 5 ++y+= k= 334=66 k= 66+5=7 ;8%;7 (+) ++3+y+= k= 3 k= =9, = k {; 4;, ; 3;, ; ;, y, ;6(;, y} 48

49 00 ;6(; 65 7 ;6(; 3 {(, )}, {(, ), (, )}, {(, 3), (, ), (3, )}, 4 {(, 4), (, 3), (3, ), (4, )}, y + (0, 6) y+4= y+4=300 (0, 6) 300+0=30, ,, 3, 4, 5, y 3, 3, 5, 7, 9, y 4, 4, 7, 0, 3, y {a«} a«=+(-) 9 = aª=9 9-8= a«+3=a«a«-a«=-3 {a«} 50-3 a«=50+(-) (-3)=-3+53 a =-3k+53=4 k= a«+a«a«=, a«=a«+a«{a«} -5, a -a =-3-(-5)= a«=-5+(-) =-7 0 a = (k-7)= k-7 0 k= 0 a = -40= a«=a«a«{a«} a a a º r 5= 5= 55 =rfi a a a a a a º =5 rfi +rfi +rfi =5 a a a 3rfi =5 rfi =5 a º 55 =r =(rfi ) =5 =5 5 a º 70 a«=a«+,, 3, y, - a =a +_ a =a +_ a =a +_3 +> a«=a«+(-) a«=a + {++3+y+(-)} (-) a«=5+ a«= a = (k -k+5) k= 0 k= k= 0 k= 49

50 0 4 0 a = a = a«=a«++3,, 3, y, - a =a + +3 a =a + +3 a =a > a«=a«+(-) +3 a«=a +{++3+y+(-)}+3(-) (-) a«=3+ +3(-) a«= + 0 a = (k +k) k= 0 0 a = + 6 a = a«=a«+«+,, 3, y, - a =a + + a =a + + a =a + + +> a«=a«+«+ a«=a +( + +y+«)+- a«=+ +- («-) a«= a«=«+- a«=« k= - k= a«= a«,, 3, y, - + a =;!;a a =;3@;a a =;4#;a - _a«= 5a«3 - a«= y 5a 3 4 a«= = a º=;3 0;=; 5; ; 5; 74 a«={- 5} a«a«={- }{+ } a«- + a«= 5 5a«, 3, 4, y, a =;!; ;#;a a =;3@; ;3$;a a =;4#; ;4%;a - + _a«= 5 5a« a«= y 5 5 a + + a«= 5 = 5 a =;@0!; k=0 0 50

51 75 a«-a«+=0 a«=;!;a«- a«-a=;!;(a«-a) a«=;!;a«+;!;a ;!;a=- a=- a«+=;!;(a«+) {a«+} a +=+=3 ;!; a«+=3 {;!;} «a«=3 {;!;} «- S«= a = [3 {;!;} -] -{;!;} «S«=3 3- -;!; S«=6 {- 5 }- «a«=3 {;!;} «- S«=6 {- 5 }- «76 k= k= 3a«-a«=3 a«=;3@;a«+ a«-a=;3@;(a«-a) a«=;3@;a«+;3!;a ;3!;a= a=3 a«-3=;3@;(a«-3) {a«-3} a -3=4-3= ;3@; a«-3= {;3@;} «a«=3+{;3@;} «a -a º=[3+{;3@;} ]-[3+{;3@;} ] a -a º={;3@;} -{;3@;} a -a º={;3@;} [-{;3@;} ]={;3@;} 640 ;9%;= 3 N= a«=3a«-a«a«-a«=(a«-a«) {a«-a«} a -a =3a -a =a a«-a«=a «,, 3, y, - a -a «=a a -a «=a a -a «=a +> a«-a«=a «a«-a «=a (++ +y+«) «- a«-a «=a - a«-a «=a («-) a«=a («-)+a a =85 a ( -)+a =85 a =;3!; a«=;3@; («-)+;3!; a =;3@; ( -)+;3!;= a«-a«-a«=0 a«-a«=-;3@;(a«-a«) {a«-a«} a -a =-= -;3@; a«-a«= {-;3@;} «,, 3, y, - 5

52 a -a «= a -a «=-;3@; a -a «={-;3@;} +a«-a«={-;3@;} «a«-a «=+{-;3@;}+{-;3@;} +y+{-;3@;} «-{-;3@;} «a«-a «= 3 -{-;3@;} a«-a «=;5#; [-{-;3@;} «] a«=;5*;-;5#; {-;3@;} «p=;5*;, q=-;5#; p-q=;; 5 ;; ;; 5 ;; 79 a«a«= 55 a«+3 a«+3 3 = 55 =+ a«a«a«3=b«b = 3=;!; a«a b«=3b«+ b«+;!;=3{b«+;!;} [b«+;!;] b +;!;=;!;+;!;= 3 3«- b«+;!;=3«b«=3«-;!;= 3 a«= 3 a«= 3 3«- 3«- 80 S«=a«+(=,, 3, y) S«=a«++ a«=s«-s«(=,, 3, y) a«=a«++-(a«+)=a«-a«+ a«=a«- a«-=(a«-) {a«-} a -=--=- a«-=(-) «=-«a«=-«+ a ºº= y+ =;6!;(+)(+) yy =, ()= =, ()=;6!; 3= ()=() = =k, y+k =;6!;k(k+)(k+) (k+) y+k +(k+) =;6!;k(k+)(k+)+(k+) =;6!;(k+){k(k+)+6(k+)} =;6!;(k+)(k +7k+6) =;6!;(k+)(k+)(k+3) =;6!;(k+){(k+)+} {(k+)+} =k+, y = 44 yy (-)(+) + 5

53 =, ()= 4=;3!;, ()=;3!; 3 ()=() = =k, y k + 4 = 44 (k-)(k+) k+ 4 (k+)(k+3) y (k-)(k+) (k+)(k+3) k = k+ (k+)(k+3) k(k+3)+ = 435 (k+)(k+3) (k+)(k+) = 435 (k+)(k+3) k+ k+ = 4= 44 k+3 (k+)+ =k+, 8 «> yy =5, ()=fi =3, ()=5 =5 ()>() =5 =k(kæ5), >k >k, ± >k kæ5 k -k-=(k-) ->0 k >k+ ± >k =k +k >k +k+=(k+) =k+, y+ 3<-;!; yy 3 =, ()=+ 3=;4%;, ()=-;!;=;#; ;4%;<;#; =k(kæ), y+ 3<-;k!; 3 k yy (k+) y k (k+) <-;k!;+ yy (k+) k k+ - yy k+ -() {- }-[-;k!;+ 3 ] k+ (k+) = 3 >0 k(k+) -;k!;+ 3 <- (k+) k y+ 3 3 (k+) <- k+ =k+, 53

54 83-4 xx =-4 x= i 6 ' 6=4 7 xx =7 x -7=0, (x-3)(x +3x+9)=0-3 3'3i x=3 x= 55 9 xx =9 x -9=0, (x -3)(x +3)=0 x= '3 x= '3i 9 '3-6 xx =-6 x +6=0, (x +4i)(x -4i)=0 x fl'6å4=fl" fl = = fi'ƒ-3=fi" (-)fi =- 'ƒ-0.008= " (-0.) = IV. " ' 6=" ' 6="ç " 6 ='6 "ç 'å6_"ç 'å6= "ç 'å6_ "ç'å6 = "ç " _ "ç" 4 = '_ '4 = 'ƒ_4 = '8= " = fi"ç3 ( ')fl - "ç'6å4=fi" (fi ) " fl -fl" fl =fi" ( )fi " 4 - = 4- =- =- æ æ +4 =æ æ +( ) 3 fi +4fi fi +( )fi æ æ =æ æ + 5 fi + æ æ =æ æ (+fi ) 3 fi (+fi ) æ æ ="ç'ß = " = æ 7 +9 =æ (3 ) +(3 ) (3 ) +(3 ) æ =æ æ =æ 3 (3 +) 4 3 (+3 ) æ =" 3 =3 =8 fiæ 'x _ æ 'x _æ fi'x = fi"ç 'x _ "ç'x _ "fiç'x 'x fi'x 'x fi"ç'x "çfi'x "ç 'x = fi'x _ fl'x _ 'x 'x fi'x fl'x = ' ;4!; _3 -;!; -;4#; =( ) ;4!; _(fi ) -;!; -;4#; 8 ;4!; _3 -;!; -;4#; = ;4#; _ -;%; -;4#; 8 ;4!; _3 -;!; -;4#; = ;4#;-;%;-{-;4#;} 8 ;4!; _3 -;!; -;4#; = =;!; 9 -;#; _6 ;4!; " 8 =(3 ) -;#; _( ) ;4!; {(3 ) } ;!; 9 -;#; _6 ;4!; " 8 =3 _ (3 ) -;#; 9 -;#; _6 ;4!; " 8 =3 _ 3 fl 9 -;#; _6 ;4!; " 8 =3 ±fl _ 9 -;#; _6 ;4!; " 8 =3 _ =54 54

55 [{; 6;} -;3!; ] ;#; _{: 6 :} ;!; =[{ } -;3!; ] ;#; 3 _{ } ;!; (3 ;!; +)(3 ;!; -)(8 ;3!; +)(8 ;3!; -) ={(3 ;!; ) -}{(8 ;3!; ) -} ={ } -;!; 3 _ 3 ;!; =(3-)(8 ;3@; -) =( ) -;!; _3_ -;!; ={( ) ;3@; -} 87 "ça 'a_ 'a=a ;3@; a ;4!; _a ; ; "ça 'a_ 'a=a ;3@;-;4!;+; ; "ça 'a_ 'a=a ;!; = ;#; _3_ -;!; = ;#;-;!; _3 =_3 =6 ;!; 54 6 k=;!; ;!; 88 4 æ 4'4_ =(4_4 ;!; _4 4 ;4!; ) ;3!; '4 =(4 +;!;+-;4!; ) ;3!; =(4 ;4(; ) ;3!; =4 ;4#; =( ) ;4#; = ;#; k=;#; ;#; 89 (A-B)(A +AB+B )=A -B (a ;3!; -b ;3!; )(a ;3@; +a ;3!; b ;3!; +b ;3@; ) =(a ;3!; -b ;3!; ){(a ;3!; ) +a ;3!; b ;3!; +(b ;3!; ) } =(a ;3!; ) -(b ;3!; ) =a-b (A+B)(A-B)=A -B =( -)=6 90 x ;!; +x -;!; = x++x =4 x+x = x +3x x +3x x +x =8 x +x +3(x+x )=8 x +x +3 =8 a-b 6 yy x +x = x+x =(x ;!; +x -;!; ) -= -= x +x =(x+x ) -3(x+x ) x +x = -3_= 9 a ;!; +a -;!; =3 a++a =9 a+a =7 a ++a =49 a +a =47 a ;#; +3a a -;!; +3a ;!; a +a -;#; =7 a ;#; +a -;#; +3(a ;!; +a ;!; )=7 a ;#; +a ;#; +3 3=7 a ;#; +a ;#; =8, yy yy yy yy a ;#; +a -;#; + = =;5@; ;5@; a +a

56 9 x=4 ;3!; + ;3!; x =(4 ;3!; + ;3!; ) x =4+3 4 ;3@; ;3!; +3 4 ;3!; ;3@; + x =6+3 4 ;3!; ;3!; (4 ;3!; + ;3!; ) x =6+6x 4 ;3!; ;3!; =( ) ;3!; ;3!; = ;3@;+;3!; = x -6x= x =5, 4=5 x =;5!; x, x x -x = x(x -x ) = x -x x+x x(x+x ) x + {;5!;} -5 = 455 =-;^5@; -;^5@; ;5!; =, 3 = (3-3 ) 45= 45= (3 +3 ) 3-3 (3 ) -(3 ) = 45 = = =;6&; ;6&; + 95 a +a 43=, a a -a a (a +a ) a =, 4 = a (a -a ) a - a +=(a -) a =3 a ='3 '3 a +a 43= a -a a +a =(a -a ), a +a =a -a a =3a a a =3 a =' =3 53=3 ;[!; yy (9_53) =3 9_53=3 ;]#; yy 9_53 =3 ;]#; 3 ;[!; 53 9=3 ;]#; ;[!;, 3 =3 ;]#; ;[!; ;]#;-;[!;= 97 A, B, C A=(0 ;!; ) ;3!; =0 ;6!;, B=5 ;!;, C=(8 ;!; ) ;3!; =8 ;6!; 6,, 6 6 A=0 ;6!; B=5 ;!; =5 ;6#; =(5 ) ;6!; =5 ;6!; C=8 ;6!; 0 ;6!; <8 ;6!; <5 ;6!; A<C<B A<C<B 98 log x- (-x +8x-7) x->0, x-+ x>, x+3 yy -x +8x-7>0 x -8x+7<0, (x-)(x-7)<0 <x<7 yy <x<3 3<x<7 x 4, 5, =5 5 56

57 99 logå(a -3a-0) a>0, a+ yy a -3a-0>0 (a+)(a-5)>0 a<- a>5 yy, a>5 a>5 300 log 0.5=x 8 =0.5=;4!; ( ) =, = 3x=- x=-;3@; log =x (0.) =0.00=(0.) x=3 logæ 8=-;3$; x -;3$; =8=3 x=(3 ) -;4#; =3 =; 7; log x=- x={ } =(') = ' log {log (log x)}=0 log (log x)=4 = log x=3 =3 x= =8 -;3@; 3 ; 7; 8 30 ' logå 7=- a =7=3 a>0 a=3 -;#; log '3 b=3 b=('3) =(3 ;!; ) =3 ;#; ab=3 -;#; _3 ;#; =3 = 30 x=log ('3+') 3 ='3+' 3-3 =3-4 ='3+'- 3 '3+' 3-3 ='3+'-('3-')=' ' 303 ;!; log ;4ª9;-log ; 4; =log {4ª9;} ;!; -log ; 4; =log ;7#;-log ; 4; {;4ª9;} ;!; =[{;7#;} ] ;!; =;7#; =log {;7#; ; 4;} =log {;7#;_: 3 :}=log = ;!; log 3+3 log '-log '6 =log 3 ;!; +log (') -log '6 =log '3+log '-log '6 '3_' =log 5=log = '6 log º ;3%;-log º ;4&;+ log º 3+;!; log º 49 =log º {;3%;} - log º ;4&;+log º 3 +log º 49 ;!; =log º [{;3%;} ;4&;_3 _49 ;!; ] =log º {: 9 :_;7$;_9_7} =log º 00=log º 0 = log º 0= 3 log '+log ' 0-;!; log 8 =log ( ') +log ' 0-log 8 ;!; =log +log ' 0-log ' 8 ;!; ='8=' _' 0 =log 53 =log '5=log 5 ;!; ' =;!; log 5=;!; ;!; 57

58 304 (log 3+log 9)(logª +log 6) =(log 3+log 3 )(log 3 +log 3 ) ={log 3+;3@; log 3}{;!; log +;3$; log } =;3%; log 3 ;; 6 ;; log =;%8%; log 3 log 3 =;%8%; log 4-3 log =log 4 -log =log 6-log 8 =log ;; 8 ;;=log ()=5 log = log 3 log 5 log 6 log 8 log º 3 log º 5 log º 6 log º = _ log º log º 3 log º 5 log º 6 3 log º = 3=3 ;%8%; 3 log º 305 log 3 log x=log 3 log 3 log x=log 3 log 3 ;!; log x=;!; log 3 log x= x= a b = a logå a b =logå, logå a +logå b =0 +3 logå b=0 logå b=-;3@; logå a b =logå a +logå b =3+ logå b logå a b =3+_{-;3@;}=;3%; (log 3+ log 5)log ' 5 8 ;!; =(log 3+ log 5)log 5 ={log 3+_;!; log 5}_6 log =(log 3+log 5)_6 log 6 =log 5_ 5 log 5 =6 306 log º 5=log º ;; º;; log º 5=log º 0-log º log º 5=-a log º 0.7=log º ; 0 0; ;3%; 6 log º 0.7=log º 7-log º 00 log º 0.7=log º ( _3 )-log º 0 log º 0.7=log º +log º 3 -log º 0 log º 0.7=3 log º + log º 3- log º 0.7=3a+b- log º ; 5;=log º 3_0 log º ; 5;=log º -log º (3_0) log º ; 5;=log º -(log º 3+log º 0) log º ; 5;=log º -log º 3- log º ; 5;=a-b- log º ' 30=;!; log º 30 log º ' 30=;!; log º (3_0) log º ' 30=;!;(log º 3+log º 0) log º ' 30=;!;(log º 3+) log º ' 30=;!;(b+) -a a-b- 3a+b- ;!;(b+) 58

59 307 3 ;[!; =a, 3 =b, 3Ω =c ;[!;=log a, y=log b, z=log c log bc log b+log c log 'a bc = 55= 5 log 'a log a ;!; log b+ log c log 'a bc = 5 ;!; log a y+z log 'a bc = 5=x(y+z) x(y+z) ; [; =6 x=log 6 ;[!;= 3=log 3 log 6 43 =6 y=log 6 ;]!;= 35=log 43 log 6 ;[!;+;]!;=log fi +log 3fi ;[!;+;]!;=log 6 (fi _3fi ) ;[!;+;]!;=log 6 6fi ;[!;+;]!;=;3%; ;3%; 309 x -9x+3=0 a, b a+b=9, ab=3 a+b a +b =;å!;+;!;= =;3(;=3 ab log (a +b )=log 3= 30 x -5x+3=0log º a, log º b ()=log º a+log º b=5 ()=log º a log º b=3 log a b+log b a log º b log º a = + log º a log º b (log º a) +(log º b) = 4 log º a log º b (log º a+log º b) - log º a log º b = 4 log º a log º b 5 -_3 = =;; 3ª;; ;; 3ª;; log º {- } k= k = log º 00 (k-)(k+) = log º 3 k= k =log º +log º +log º =+y+log º =log º { _y_ } =log º =log º 0-log º =log º 0-log º (_0 ) =log º 0-(log º +log º 0 ) =log º 0-log º - log º 0-log º k= 00 k= = 3= (' k+-'k) f(k) k= 'k+' k +å k= =('-)+('3-') =+y+(' 0å0-'ß9å9) =-+' 00=9 99 log { 5}=log 9=log 3 = k= f(k) 33 k - 3 k 5 <00<5 log 5 <log 00<log 5 <log 00<3 59

60 a= b=log 00-=log 00-log 5 00 b=log =log a +4 ;b!; =4 +4 log 4 =6+4 log 5 4 a +4 ;b!; =6+5= 34 log 53=log(5.3_0 ) log 53=log 5.3+log 0 log 53= log 53=.785, log 5.3=log(5.3_0) log 5.3=log 5.3+log 0 log 5.3= log 5.3=.785, log 0.053=log(5.3_0 ) log 0.053=log 5.3+log 0 log 0.053= , log =0 log 3.74= = log = log 374=+0.579= log = = log x=.369 log.34 x.34log x x 3 x=34 log x= =-+( ) = log.34 x.34 log x- x 0 x=0.34 log x=-.6308= log x=(--)+( ) log x= log.34 x.34 log x -3 x 0 x= KEY Poit 37 log 5 =30 log 5=30 log ;; º;; log 5 =30(-log )=30(-0.300) log 5 =0.97 log log =-0 log =-0_0.300 log =-6.00= log =(-6-)+(-0.00) log =

61 log log {;4!;} ºº =log 4 log {;4!;} ºº =-00 log 4 log {;4!;} ºº =-00_ log log {;4!;} ºº = log {;4!;} ºº =-60.0= log {;4!;} ºº =(-60-)+(-0.0) log {;4!;} ºº = log {;4!;} ºº fi 63 log 8fi 6 6 log 8fi < log 8<63 ;5^0@; log 8<;5^0#; 5 5_;5^0@; 5 log 8<5_;5^0#; 8.6 log 8 fi <8.9 log 8 fi 8 8 fi log (log x)=5 log x=fi =3 log x3 x log 5 =0 log 5=0 log ;; º;; log 5 =0(-log ) log 5 =0_(-0.300) log 5 =3.98 log a=4 log log 9= log 3=_0.477=0.954, log 0= log 9<0.98<log 0 3+log 9<3.98<3+log 0 log(9_0 )<log 5 <log(0_0 ) 9_0 <5 <0_0 5 =9._0 5 9 b=9 a+b=4+9=3 3 3 log A=+a(,0 a<) a x +5x+k=0 +a=-;%; yy a=;k; yy 0 a< +a=-;%;=--;!; +a=(--)+{-;!;} +a=-3+;!; =-3, a=;!; -3_;!;=;K; k=-3-3 6

62 3 log x log ;[!; log x -log ;[!;= log x+log x log x -log ;[!;=3 log x=() log x log x<3 6 3 log x<9 3 log x 3 log x=6 3 log x=7 3 log x=8 log x= log x=;3&; log x=;3*; x=0 =00 x=0 ;3&; =00 ' 0 x=0 ;3*; =00 ' 00 00, 00 ' 0, 00 ' log x log 'x log x+log 'x=log x+;!; log x log x+log 'x=;#; log x=() log x 4 4<log x<5 log x=4 log 'x= ;#; 0 6<;#; log x<: : ;#; log x ;#; log x=7 log x=: 3 : log 'x=;4!; log x=;4!;_;; 3 ;; log 'x=;6&;=+;6!; log 'x ;6!; ;6!; log x a log x=4+a(0 a<) log 'x=;!; log x log 'x=;!;(4+a) log 'x=+;ƒ; log 'x;ƒ; a+;ƒ;= a=;3@; log x=4+;3@;=;; 3 ;; log 'x=;4!; log x ;6!; 34 T=Tº+(T -Tº)0-0.0t Tº=0, T =0, T=5 5=0+(0-0)0-0.0t 00_0-0.0t =5, 0-0.0t =; 0; -0.0t=log ; 0;=-log 0-0.0t=-(+log ) -(+log ) +0.3 t= 3 = 4= m 50lg 50=0+80_ =; 8º0;, 3 56 =6, 3 56 =6 ;5N6; log 3=log 6 56(log +log 3) = 33 log 3 56( ) = =

63 0, {,,,,, },, 0 A={4, 8,, y}b={,, 4, 8} 7 A 03 {6, 8} {0, 0, 03, y} [;!;, ;3!;, ;4#;, y] {7, 4,, y} 04 ({,, })-()+({})-({0}) =3-0+-= ({a, b, c})=({e, f, g})=3 A={0}, (A)= ({3, 5, 7})-({3, 7})=3-= ({, })= ({0})=({})= 06 I. A={,, 3} B _+=3, _+=5, _3+=7 A={,, 3}, B={3, 5, 7} C 3-=, 3-=, 3-3=0 5-=4, 5-=3, 5-3= 7-=6, 7-=5, 7-3=4 C={0,,, 3, 4, 5, 6} (A)=3, (B)=3, (C)=7 (A)+(B)+(C)= x x -4 x 5, -5 x- 4 5 x- - x- x- x- x- =-, =-, =0, x- x- =, = x-3, -,, 3, 5 A={-3, -,, 3, 5} A A={,, 4, 8, 6}B={,, 4, 8} B,A {},AA A {} A {8},B {,, 4, 8, 6},A 09 A,, {0}, {0, } 0 A {0,, } A 63

64 0 A,,, {3, 4} 4 A -=5 A{a, b, c, d} a, b A, a,b,{a, b} (A)=4 {},A {b},a {{a, b}},a 3 -=3 -=7 -=5 fi -=3 fi -=3 4 A,B B,A A=B A={, 3, 5, 5} A=B a-=3, b-=5 a-=5, b-=3 a=5, b=7 a=7, b=5 a+b= 5 X {a, b, c, d, e} a, b, c X fi -= -=3 3 {a, b, c}, {a, b, c, d}, {a, b, c, e} 3 6 A={, 3, 5, 5} A 3 = =8 A 8-=7 a<x Xa b X Xb Xa b A fi = =8 b A XA, A X a b A A=B -<A -<B a -3a=- a -3a+=0, (a-)(a-)=0 a= a= yy 3<B 3<A a +a=3 a +a-3=0, (a-)(a+3)=0 a= a=-3 yy, a 8, 3, 4 6 «=6= -4=4 =8, 4, 6, 8, 0 - fi =04-3=99 64

65 A b f b f {a, c, d, e} fl - =64-6= b fl =fi =3 f fl =fi =3 b f fl = =6 b f 3+3-6=48 9 A,X,B XB, 3 (X)=3 B 3 X {, 3, 5}, {, 3, 8}, {, 3, 9} 3 0 A={x- -4<x 3} -4<x 3-6<x- A a -6<a B={x+a - x<7} - x<7 -+a x+a<7+a B b -+a b<7+a A, B A,B B A -+a -6 7+a -+a -6, 7+a> -6<a -5 KEY Poit A,B, A,B A B <A <B -a += -a+8= -a +=, a = a= a=, A={, 5}, B={, 3, 7} a=-, A={, 3}, B={, 3, 9} -a+8=, a=7 A={, }, B={-47,, 3}, a=- - A;B=A A,B A;=A'=A (A;),(A') 3 A'B=B A,B A={8, 6, 4, y} A,B 8,, 4, =5 5 4 b B A a x A'B={3, 4, 5, 8} a +4=4 a +4=8 a +4=4, a=0 b B A B A A={3, 4, 5}, B={0, 3, 8} A'B={0, 3, 4, 5, 8} a x B A b a b a b<a b a b a x b a x 65

66 a +4=8, a= a= A={3, 5, 8}, B={4, 5, 8} A'B={3, 4, 5, 8} a=- A={3, 5, 8}, B={-4,, 8} A'B={-4,, 3, 5, 8} a= 5 A;B={, 4}A'B={,, 3, 4, 5, 7} (A;B),X X,(A'B) (A;B),X,(A'B) {, 4},X,{,, 3, 4, 5, 7} X {,, 3, 4, 5, 7}, 4 X {, 3, 4, 5} 6 A;B={} <A a -4a+5=, a -4a+3=0 (a-)(a-3)=0 a= a=3 a=, A={, }, B={-,, } A;B={, } a=3, A={, }, B={, 3, 5} A;B={} a=3 A'B={,, 3, 5} {,, 3, 5} 7 A X X,A {, }'X={,, 3} X 3 4 X = =4 4 8 (A;X)=B;X=X X,B XB A B X, 3 X fl = =8 B X, 3 X fl = =8 B X, 3 X fl = =8,, X 8_3=4 4 9 AÇ,BÇ B,A U A A-B+ B 30 A;BÇ =A-B= A,B A B -<B a-=- -a=- a-=-, a= A={-, 0}, B={-, 0, } A,B -a=-, a= 66

67 A={-, 3}, B={-, 0, }, a 33 B-A={, 5, 8} (B-A)'X=X (B-A),X 3 X, 5, 8 A;X= X, 3, 7 A'(B;C) A;(B'C) U A U A XU, 5, 8, 3, 7 B C B C X B;(A'C) U A AÇ ;(B'C) U A = =6 A;B={a, b, d} A Ba, b, d B C BÇ ;(A'C) U A B C A fi = =4, {c}, {e}, {c, e} 6 4,, {c}, {e}, {c, e} 3 B C A;B={}A-B={} A={, } A, x +(a-)x+b=0 x=, x= +(a-)+b=0 a+b=0 yy 4+(a-)+b=0 a+b=- yy a=-b= B={x x +(-3+a)x+b=0} ={x x -5x+4=0} ={x (x-)(x-4)=0} ={, 4} B +4= U A B A-B A-BA;B U U U A B A B A B A;BÇ A;BÇ AÇ ;B U U U A B A B A B (A'B)Ç U A ; = AÇ ;B ; = A;B (A'B)Ç,A;B B ; = A;B U A B 67

68 35 A;X=X X,A (A-B)'X=X (A-B),X (A-B),X,A A-B={x x 3} {x x 3},X,{x x 5} X={x p x q} p= 3 q 5 q 3, 5 3+5= AÇ 'BÇ =(A;B)Ç ={, 4, 5, 7} A;BÇ =A-B={, 4} B-A={7} B={, 3, 6, 7} {, 3, 6, 7} 37 (A-B)Ç ;BÇ =(A;BÇ )Ç ;BÇ (A-B)Ç ;BÇ =(AÇ 'B);BÇ (A-B)Ç ;BÇ =(AÇ ;BÇ )'(B;BÇ ) (A-B)Ç ;BÇ =(AÇ ;BÇ )' (A-B)Ç ;BÇ =AÇ ;BÇ AÇ ;BÇ =AÇ AÇ,BÇ B,A 38 (p) 3 q 5 7 A-B U A A;(AÇ 'B)=(A;AÇ )'(A;B) ='(A;B) =A;B (A'B);(AÇ ;BÇ )=(A'B);(A'B)Ç = A 4 B 3 6 B 7 5 x (A-B);(A-C)=(A;BÇ );(A;C Ç) =A;(BÇ ;C Ç) =A;(B'C)Ç =A-(B'C) (A-B)'(A-C)=(A;BÇ )'(A;C Ç) =A;(BÇ 'C Ç) =A;(B;C)Ç =A-(B;C) {(A;B)'(A-B)};B ={(A;B)'(A;BÇ )};B ={A;(B'BÇ )};B =A;B+A 39 {(A;B)'(A;BÇ )}'{(AÇ 'B);(AÇ 'BÇ )} ={A;(B'BÇ )}'{AÇ '(B;BÇ )} =(A; U)'(AÇ ') =A'AÇ =U 40 A U=(A;U)'(A'U)Ç A U=A'UÇ A U=A' A U=A A B=(A;B)'(A'B)Ç A B=(B;A)'(B'A)Ç A U=B A A =(A;)'(A')Ç A ='AÇ A =AÇ AÇ BÇ =(AÇ ;BÇ )'(AÇ 'BÇ )Ç AÇ BÇ =(A'B)Ç '(A;B) AÇ BÇ =A B 68

69 A AÇ =(A;AÇ )'(A'AÇ )Ç A AÇ ='UÇ A AÇ = 4 A B=(A;BÇ)'(AÇ ;B) A B=(A-B)'(B-A) ~ A B C A B C A B C A B C A B C (A B)C=A(BC)=(A C)B 4 A,B A'B=B (A'B)=(B)=9 (A'B)=(A)+(B)-(A;B) (A)=(A'B)-(B)+(A;B) =6-+5= (A;B)=(A)-(A-B) =-8=4 (A'B)=(A-B)+(A;B)+(B-A) =8+4+6=8 ((A'B)Ç )=(U)-(A'B) =5-8=7 U A B = 44 (A;B)=(A)+(B)-(A'B) =7+0-=5 (B;C)=(B)+(C)-(B'C) =0+6-6=0 (C;A)=(C)+(A)-(C'A) =6+7-= (B;C)=0 B;C= A;B;C= (A;B;C)=0 (A'B'C) =(A)+(B)+(C)-(A;B) =-(B;C)-(C;A)+(A;B;C) = = Ua Ab B (U)=48, (A)=3, (B)=30 a b AÇ ;BÇ (AÇ ;BÇ )=((A'B)Ç )=5 (A'B)=(U)-((A'B)Ç ) =48-5=43 a (A-B) (A-B)=(A'B)-(B) =43-30 =3 3 69

70 46 A, B, C x (A'B'C) =(A)+(B)+(C)-(A;B) =-(B;C)-(C;A)+(A;B;C) 70-4= x x= 47 U A B (U)=40, (A)=5, (B)=8 A;B (A;B) A'B=U (A'B)=(A)+(B)-(A;B) 40=5+8-(A;B) (A;B)=3 m=3 (A;B) B,A (A;B)=(B)=8 M=8 M+m=8+3= KEY Poit (A;B) (A;B) (A) (A;B) (B) A'B=U U, A, B, C A, B, C (U)=50, (A)=3, (B)=8, (C)=, (A;B;C)=7 ((A'B'C)Ç )=(U)-(A'B'C) 4=50-(A'B'C) (A'B'C)=46 (A'B'C) =(A)+(B)+(C)-(A;B) =-(B;C)-(C;A)+(A;B;C) (A;B)+(B;C)+(C;A) =(A)+(B)+(C)+(A;B;C) =-(A'B'C) = =33 (A;B)+(B;C)+(C;A) -3_(A;B;C) =33-3_7= A, B, C U A A, B, C B C A;B;C (A;B)+(B;C)+(C;A) (A;B;C) (A;B)+(B;C)+(C;A) -3_(A;B;C) 49 AB=(A-B)'(B-A) ={3, 4}'={3, 4} (AB)C={3, 4}{, 3, 5} ={4}'{, 5}={, 4, 5} +4+5=0 AB=(A'B)-(A;B) ={, 3, 6, 7, 8, 9} 70

71 A={, 3, 4, 6, 7} 3 B={,, 4, 8, 9} B = , 5 A, B (A)=33, (B)=0 A;B (A;B)=6 35 (A'B) (A'B)=(A)+(B)-(A;B) (A'B)= (A'B)= A ;A =A (A '(A ;A )) =(A 'A ) =(A )+(A )-(A ;A ) =(A )+(A )-(A º) = = A ;A =A (A ;A )'A =A 'A =A =A l=6 A º,A A 'A º =A Aª,A Aª 'A =A A B (A 'A º );(Aª 'A ) =A ;A =A =Aμ m=7 (A 'A );(A 'A ) =(A 'A );(A 'A ) =A '(A ;A ) =A 'A =A =A«=36 l+m+=6+7+36= A A ={, 4, 6, y}, A ={3, 6, 9, y}, A ={4, 8,, y}, y A,A A,A l k,kl Aμ,(A ;A ) Aμ,A, Aμ,A m 4 6 m 4 6 m4 6 (A 'A ),A«A,A«, A,A«8 8 4 (m )+() =+4= ABBCCA A A A B C B C B C AB BC CA 7