oÿÅ ³oÿÅ - انتشارات خیلی سبز · ÐÕÖ ³ m ¸ â c À ¸ c É " Ò Å Á ½...
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Transcript of oÿÅ ³oÿÅ - انتشارات خیلی سبز · ÐÕÖ ³ m ¸ â c À ¸ c É " Ò Å Á ½...
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y xf x f x( ) ( )
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{( , ) , ( , ) , ( , )}1 2 2 5 3 10
y x
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y
x x f x f x1 2 1 2
<< ÛÛ >>( ) ( ) f f
y y x fx
x x f x f x1 2 1 2
< Þ £( ) ( ) x
[ , ]2 5 [ , ]1 4 g
y x y x= [ ] y x= [ ]
yx xx
=³<
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x x f x f x1 2 1 2
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( , )-¥ 0( , )-¥ 1
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p( , )0 + ¥( , )1 + ¥
( , )0 p
a f a f a( ) ( )2 1 3 1++ >> -- f
- - -[ , ]2 1 ( , )- -2 1 ( , )2 + ¥ ( , )-¥ - 2f f
f a f a a a af( ) ( ) ( , )2 1 3 1 2 1 3 1 2 2+ > - ¾ ®¾¾¾¾ + < - Þ > Þ + ¥SwH ²»qº
a f a a a== -- -- ++ ++{( , ),( , ),( , )}1 1 1 2 3 2 2 2
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32 ( , )- -3
3
2 ( , )- 2
31 ( , )-2 1
y x
2 3 1 2 2+ < - < +a a aþ²H
J
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2 3 1 2 33
2+ < - Þ < - Þ < -a a a a
a a a- < + Þ - <1 2 2 3 - < < -3
3
2a
f x c( ) ==
a<< a>> f x ax b( ) == ++
y ax bx c== ++ ++
yx
==
( , )++¥¥ ( , )--¥¥
y x== cos y x== sin
y xa== log y ax==
0 1<< <<aa>>
y a x== --( )12
y a x== -- ++ ++( )2 7 1 a
a > 7
2 a < 3 3 7
2< <a
a a a
a a a
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f g++ g f
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x x
( )| |
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ííïï
îîïï
1
2 1 2
2
( , )-1 1 ( , )1 3 ( , ]1 2 ( , )0 2
( , )0 2 x =1
x = 2 ( , ]12
( , )1 3
( , )-1 1
y f x== ++| ( ) |3 1 y f x== --( )2 ( , )-¥ - 5 ( , )1 2
( , )-1 1 ( , )- -2 1
f x( )- 2 f x( ) f x( )
f x( )+1
x
( , )-1 1
y x x= -23 2 y x x= + [ ] y x x= -[ ] y x x= [ ]
f ( )12
f ( )0
yx
x [ ]x
f x x( ) | |== --24
( , )0 2 ( , )-2 0 ( , )-1 1 ( , )- -3 1
y x= -24 y x= 2y x= -| |2
4
( , )2 + ¥ ( , )-2 0
y x= -| |24
( , )2 + ¥ ( , )-2 0
d b [ , ]c d [ , ]a b
127
712
412
112
f xx x
xx x
( ) ==-- -- << --
-- ££ <<-- ³³
ìì
ííïï
îîïï
2 3 4
3 4 2
3 2 2
[ , )- +¥4 ( , ]-¥ 2
f xx x
x x( ) ==
++ ££
-- >>
ììííïï
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3
2
1 0
0
f x x x( ) | |= f x x x( ) | |= + f x x xx x
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2 0 f x x( ) | |=
f x x x
x x( ) =
- £
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2
2
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1 1
f x x x( ) | |= - -2 1 f x x( ) | |= - -2 1 f x xx x( ) | |= +
y f x== --( )1 y f x== ( )
( , )- -3 1
[ , ]- -4 3
( , )-1 1 [ , ]1 2
y x= 1 y x= - -2 y x= - +log2 2 y x= -2 2
a ( , ]--¥¥ a f x x x( ) | |== 2
-1y x x== ++ | | y x x== [ ]
[ , ]3
2
7
3 [ , ]0
1
2 [ , ]1
2
3
2 [ , ]0 1
k f xx
k xx
( ) ==-- >>
==++ <<
ìì
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1
1 1
a f x x xx a x
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ììííïï
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21 0
3 0
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b a-- ( , )a b f x x x( ) | | ( )== -- 1
12
13
b a-- ( , )a b f x x x( ) | |== --2 2 x >> 0
12
13
y x x== -- ++ ++| | | |1 1
( , )-¥ 3
2 ( , ]-¥ 1 [ , )- +¥1
2 [ , )- +¥2
g x x x( ) | | | |== -- -- --2 1
m f m== --{( , ),( , ),( , ),( , )}1 1 3 6 2 2 10 202
b a-- f x x x== ++ -- -- ++{( , ),( , ),( , )}1 2 7 2 10 0 4
2 x a bÎÎ[ , ]
f x m x( ) ( )== ++3 1
4m
m g x ax( ) == f x a x( ) ( )== --2
3
[ , ]--1 2 f x x x( ) == -- ++3 6 22
{ : | | }x x -- <<1 2 f x x x( ) == -- --2
2 3
m [ , )1 ++¥¥ f x m x x( ) ( )== -- ++13
2
m ³ 2 m £ -2 0 2< £m - £ <2 0m
f ( )2 f x a x ax( ) ( )== -- ++ ++2 2 32
x x f x f x1 2 1 2<< ÛÛ >>( ) ( ) x2 x1 f x x( ) | |== 1
( , )0 1 ( , )-1 1 ( , )-2 0 ( , )- -3 1
a ( , )--¥¥ a f x xx( ) == ++
--2 1
3
- 1
3
( , )-- ++ ¥¥2
y xx
= +-
2 1
1 y x
x= +
-1
2 y x
x= -
+2 3
1 y x
x= -
+1
3
f x x x x( ) | | ( )== ++ 1
f g++ g f
f g++ g f
f g-- g f
f g´́ g f
a f a f a( ) ( )3 2 1-- >> ++ f
-2 -1 y xf x== ( ) f ( )3 0== f
[ , )3 + ¥ ( , ]-¥ 3 - ( , )0 3 [ , ]0 3
y x x== --( )f(x)2 f ( )2 0== f
g x x x( ) == -- --2 10
2 f x x x( ) | | | |== -- ++ --2 3
(
m [ , ]--1 2 f x x m x( ) ( )== -- ++ ++22 1 1
- < <3
2
3
2m 1
2
3
2£ £m - < <1
1
2m - £ £1
1
2m
y x x== ++ --21| |
[ , )1
2+ ¥ [ , )- + ¥1
2 [ , )0 + ¥ [ , )- + ¥1
y f x f x== ++ -- --( ) ( )2 1 2 f
( , ]-¥ 2
3 [ , )2
3+ ¥ ( , ]-¥ 1
3 [ , )1
3+ ¥
( y f== -- --( ) f( | x | )2 1 f
( , ] [ , )-¥ - + ¥1 3 [ , ]-1 3 ( , ] [ , )-¥ - + ¥3 1
f x( ) f x
x x x
x x
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++ -- ££ ££ ++
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2
2
6 5 3
4
5
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f
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f x xx x( ) | | | ( ) |== --1 1
( , )-¥ 0 ( , )1 + ¥ ( , )0 + ¥ ( , )0 1
f x a a x x( ) ( ) | |== -- ++ --3 2 12 a
( , )- -2 1 ( , )1 2 ( , )23 17
2
+ ( , )3 17
21
-
m f x xx( ) | (x m) |== -- 2
a ( , ]--¥¥ a f x x( ) == --
--1
2
f ( )1 ( , )--¥¥ 3 ( , )3 ++ ¥¥ f x x ax a( ) == ++ --
-- ++2 1
1
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2 52
-
g f x x xg x x
( )( )
== ££ -->> --
ììííïï
îîïï
21
1
y x x= + | | y x= - -| | x y x= - 2 y x= | |
g f
g f f g´ fg f g- f g+g f y y x
xf x
1 1 2 0 3
4 1 3 2 0
--( )
1 0 2, , x
y y f g+
fg
2 gf
fg 2f g-
2 1 2 4 2 0 2 2 1 2 2 3 0 1 6 0 5 2f g- = - + - - = -{( , ( ) ),( , ( ) ) ,( , ( ) )} {( , ) ,( , ) ,( , 66)} fg = ´ ´ - - ´ = -{( , ),( , ( ) ,( , )} {( , ) ,( , ) ,( , )}1 4 2 0 2 1 2 3 0 1 8 0 2 2 0
gf
= --
={( , ) ,( , ) ,( , )} {( , ) ,( , )}12
40
1
22
0
31
2
42 0
g( )0 gf
x =0
fg
2 2 2 2
14
202
12
3
01 8 0 4=
-- = -{( , ),( , ) ,( , ( ) )} {( , ) ,( , )}
g( )2 fg
2x = 2
( )( ) ( ) ( ) ,f g x f x g x D D Df g f g+ = + =+ ( )( ) ( ) ( ) ,f g x f x g x D D Df g f g- = - =-
( )( ) ( ) ( ) ,fg x f x g x D D Dfg f g= ´ = fgx f x
g xD D D x g xfg
f g( ) ( )( )
, { | ( ) }= = - =0
D Dg f= +¥ = -[ , ) , { }0 1 g x x( ) = f x xx
( ) = +-
11
D D D D Df g f g fg f g+ -= = = = +¥ -[ , ) { }0 1
D D D x gfg
f g= - = = +¥ -{ | } ( , ) { }0 0 1 g( )0 0= fg
g == -- --{( , ),( , ),( , ),( , )}1 2 2 3 0 4 1 1 f == --{( , ),( , ),( , ),( , )}1 2 3 4 1 0 2 1
gf
fg( )2 3= g f- ( , )1 2- f g+ ( , )-1 3
gf
fg g f- f g+ -1 g f
f g
f g f g f g
+ = - + + - +¯ ¯ ¯ ¯ ¯ ¯
{( , ) , ( , ( )) , ( , )}1 2 1 1 0 2 2 1 3 f g+ ( , )-1 3
g f
g f g f g f
- = - - - - -¯ ¯ ¯ ¯ ¯ ¯
{( , ) , ( , ) , ( , )}1 1 2 1 2 0 2 3 1 g f- ( , )1 2-
fg f g= ´ = - ´ ´ - ´{( , ) , ( , ( )) , ( , )}1 2 1 1 0 2 2 1 3 . ( ) ( )f g´ 2
gf
= - -{( , ) , ( , ) , ( , )}11
21
2
023
1
gf
gf( )1 -2
0
gf
= -{( , ) , ( , )}11
22 3
( , )a b fÎ y
( , )a b c+ f g+ ( , )a c gÎ
y f g== --1 1 g x xx( ) == ++
--33 f x x
x( ) == --
--
2
2
1
4
g f g ±2 f
-3 ±1 ±2 1 1f g
- g x = -3 f x = ±1
2 3f g-- g x x( ) == --1 2 f == --{( , ),( , ),( , ),( , )}1 1 1 2 0 4 2 0
D Df g = -{ , , }1 1 0 [ , ]-1 1 g -1 f x f g f g= Þ - = - = - =1 2 3 2 1 3 1 2 2 3 0 4( ) ( ) ( ) ( ) 2 3f g-
x f g f g= Þ - = - = - =0 2 3 2 0 3 0 2 4 3 1 5( ) ( ) ( ) ( ) x f g f g= - Þ - = - - - = - =1 2 3 2 1 3 1 2 1 3 0 2( ) ( ) ( ) ( )
2 3f g-
f g++ g f
[ , ]-2 1
xfg xf g x
- -
+
2 1 0 1
0 1 2 2
4 3 3 3
4 4 5 5
(x)( )( )
f ( )-1
f g-- g xx xx x
( ) ==++ >>
-- ££ììííîî
2 1 1
3 1f x
x xx x
( ) ==-- ³³
<<ììííîî
1 2
2 2
g f
x f x x g x x f g x x x f g x x³ Þ = - = + Þ - = - - + Þ - = - -2 1 2 1 1 2 1 2( ) , ( ) ( )( ) ( ) ( )( ) 1 2 2 2 1 1< < Þ = = + Þ - = -x f x x g x x f g x( ) , ( ) ( )( ) x f x x g x x f g x x£ Þ = = - Þ - = +1 2 3 3( ) , ( ) ( )( )
f g-
f g+ g f
g x x( ) = -1 f x x( ) = +1
21
fg f g- f g+
( )( )f g x x x x+ = + + - =1
21 1
3
2
( )( ) ( )f g x x x x- = + - - = - +1
21 1
1
22
fg x x x x x( ) ( )( )= + - = + -1
21 1
1
2
1
21
2
fg
f g- a a- ¢ ¢ +a a g x a x b( ) = ¢ + ¢ f x ax b( ) = +
y x x== ++ [ ]
y = 4 2/ y = 3 1/ y = 2 1/ y = 2y x= y x= [ ] y x x= + [ ]
y x= [ ] y x= [ , ) ,[ , ) ,[ , ) ,[ , ) ,...- - -2 1 1 0 0 1 1 2
y = 3 1/ [ , )2 2 1k k +
f g++ f ( )( )fg 0 g( )--3 f f g++
-9-18
y f x= ( ) y x= - +2 22 ( , )±1 0 ( , )0 2 y f x g x= +( ) ( )
y x= - +1
31
g x f g x f x x x x x( ) ( )( ) ( ) ( )= + - = - + - - + = - + +2 21
31 2
1
31
2 2
Þ - = - - + - + = -g( ) ( ) ( )3 2 31
33 1 18
2 fg f g( ) ( ) ( )0 0 0 1 1 1= = ´ = -18 -18
1
( )( )f g++ 2 g == --{( , ),( , ),( , ),( , )}1 2 0 3 2 4 3 0 f == --{( , ),( , ),( , )}2 5 3 4 0 2
f g++ 2 g == {( , ),( , )}2 3 5 1 f == {( , ),( , )}1 3 2 5
{( , ) , ( , )}1 4 2 11 {( , ) , ( , )}1 4 2 7 {( , )}2 7 {( , )}2 11
( )( )2 4f g-- -- g x x( ) == --1 2 f x x( ) == ++ 1 -9 -32fg g == {( , ),( , ),( , )}1 5 2 6 3 0 f == {( , ),( , ),( , )}1 2 2 3 3 4
gf f
g--
{ , }- 3
20 { , }3
21-
{ , , }0 13
2 { , , }- 3
20 1
fgx f x
g x
x
x( ) ( )
( )= =
+
-
1
21
1
f f++ --1 f == -- --{( , ),( , ),( , ),( , )}1 0 1 2 0 1 2 1
( , )2 0 ( , )0 0 ( , )-1 1 ( , )1 2
x f== ( )1 ( )( )f gf x++
3 g == {( , ),( , ),( , )}1 4 2 3 4 1 f == {( , ),( , ),( , )}1 2 2 4 3 4
12
5 127
712
57
f ( )3 ( )( )fg x x== ++2 1 g == --{( , ),( , ),( , )}1 3 2 5 3 4
( h x g x( ) ( )==
--1
8f g-- == --{( , ),( , )}1 4 3 1 f == {( , ),( , ),( , )}1 4 2 3 3 4
( , )11
8- ( , )3
1
5 ( , )3
1
5- ( , )1
1
8
( )( )f g++ --2 1 g xx xx x
( ) ==³³ --
-- << --ììííîî
22 2 f x
x xx x
( ) ==++ >>-- ££
ììííîî
1 0
1 0
-6 -4
( ). (x)g f g-- g == -- --{( , ),( , ),( , )}2 1 1 2 0 1 f x x( ) == --1 2
f ( )1 ( )( )f g x x-- == ++ 2 ( )( )f g x x++ == --2 1 g f
-2 -4 ( g( )2 ( )( )f g x x-- == ++2 1 ( )( )f g x x++ == ++4 1
2 g f
( ( )( )fg x g x x( ) == --1 f x x( ) == ++1
Æ ( , ]-¥ 1 [ , )1 + ¥ fg g x x( ) == --4 2 f x x( ) log==
( , )-2 2 ( , )-2 0 ( , )0 2 [ , ]-2 2
( ( )( )( )( )f g xf g x
++--
g f
( )( )f g x2 2++ g x x( ) == --1 f x x( ) == --2 2
( am n++ {( ,a)}1 f g++ g x x m( ) == -- 3 f x n x( ) == -- 3
12
f g f gf g
3 2 2 3++++
g x x( ) == ++1 f x x( ) == --1
y x x= - ³( ) ,1 02 y x x= - ³1 0, y x x= - Î( ) ,1
2 y x x= - Î1 ,
( )fg g x x x( ) == -- --21 f x x x( ) == ++ --2
1
g f
( )( )f g x++ g f
h x x x( ) = +24
h x x x( ) = + +23
h x x( ) = 5h x x( ) = 3
( )( )f g++ 1
2g f
3 5/
2 5/
f g++ 2 g f
( )( )f g x++ == 0
fg g f
fg g f
( , )0 1
( , )2 3
( , )-2 0
( , )-3 5
fg g f
( y fg x== ( )( ) g f
( , ) [ , ) { }-1 0 1 2 3
( , ) { }- -1 3 0
( , )1 2
( , )-1 0
( x yf x g x
==--1
( ) ( )g f
( , ) ( , )-2 2 2 7
[ , ) [ , ]- -23
22 7
( , ) { , }- - -2 73
22
[ , ) ( , )- -23
22 7
fg == --{( , ),( , )}4 5 2
3
5g n n== -- -- -- ++{( , ),( , ),( , ),( , )}4 1 2 1 2 5 3 2 f n== -- ++ --{( , ),( ,m),( , ),( , )}1 3 4 2 1 3 1
2
( n m++
-7 g
gf f
y x= -2 6
y x=
y x= -
y x= +1
fg g f
fg f g+ f g- g ffg
f x( ) f f x gofg f x( ( )) g
f f g gofg
gof fog
gofgof
f g ( , )b c ( , )a b fÎ f gofg
( , )( , )
( , )a b fb c g
a c gofÎÎ
üýþ
Þ Î a b cf g
gof
¾ ®¾ ¾ ®¾
g f f
xg
-- ---- --
2 1 0 1 2
3 1 1 0 2
gof f == -- --{( , ),( , ),( , ),( , )}1 1 2 0 1 3 0 2
1 1
2 0
1 3
0 2
f
f
f
f
¾ ®¾ -
¾ ®¾
- ¾ ®¾
¾ ®¾
1 1 1
2 0 1
1 3
0 2 2
f g
f g
f g
f g
¾ ®¾ - ¾ ®¾
¾ ®¾ ¾ ®¾ -
- ¾ ®¾ ¾ ®¾
¾ ®¾ ¾ ®¾ -
f ggof
g gof = - -{( , ),( , ) ,( , )}1 1 2 1 0 2
gof
Dgof ={ , , }1 2 0 gof Rgof = - -{ , , }1 1 2 gof fog = - -{( , ),( , ) ,( , )}1 1 0 3 1 2 fog fof ={( , ),( , ) ,( , )}1 3 2 2 0 0 fof
hof gofhog( ) ( )
( )2 1
3
++ h g x x( ) == --2 1 f == -- --{( , ),( , ),( , ),( , )}1 1 2 1 3 2 1 0
53
52
73
32
gof g f gf
fg x x( ) ( ( )) ( ) (( , )
( )( )
1 1 1 21 1
1 1
2 1= ¾ ®¾¾¾ = - ¾ ®¾¾¾¾ -- Î=-
= -11 1 3) - = -
hof h f hff( ) ( ( )) ( )( , )( )2 2 1
1
2
2 1
2 1= ¾ ®¾¾¾ = =Î
=
. h( )1 1
2= ( , )0 1 ( , )2 0 h
hog h g hg x xg( ) ( ( )) ( )( )( )3 3 5
3
2
2 1
3 5= ¾ ®¾¾¾¾ = = -= -
=
. h( )5 15
2
3
2= - = - y x= -1
2
x y2
1+ = x h
- +
-=
-
-=
31
2
3
2
5
2
3
2
5
3
fog goff g x--
++( ) g f
xf x
-- ---- --
1 2 0 1 2 3
2 0 1 3 3 2( )
-1
- 2
3
- 4
3
gof fog
1 1 2
2 3 2
2 3 2
1 0 1
0 2
g f
g f
g f
g f
g f
¾ ®¾ - ¾ ®¾
¾ ®¾ ¾ ®¾ -
- ¾ ®¾ ¾ ®¾ -
- ¾ ®¾ ¾ ®¾
¾ ®¾ ¾ ®¾¾ 3
fog
- ¾ ®¾ ¾ ®¾
- ¾ ®¾ ¾ ®¾
¾ ®¾ ¾ ®¾ -
¾ ®¾ - ¾ ®¾
¾ ®¾ ¾ ®¾
1 2 3
2 0 2
0 1 1
1 3
2 3
f g
f g
f g
f g
f g
3 2 3f g
gof
¾ ®¾ - ¾ ®¾
fog gof- fog gof Dfog = - -{ , , , , }1 2 1 2 0 Dgof = - -{ , , , }1 2 0 3 = - -{ , , }1 2 0 fog gof- = - - - - - - - = - - - -{( , ),( , ) ,( , ( )} {( , ),( , ) ,(1 1 3 2 2 2 0 3 1 1 2 2 4 0,, )}4 f g+ = - + - + - + - + + = -( ,( ) ( )),( , ),( , ),( , ),( , )} {( ,1 3 1 2 3 3 10 2 2 3 0 02 1 1 44 2 6 12 2 3 0 3),( , ),( , ),( , ),( , )}- - f g+
fog goff g
-+
= - - - -{( , ),( , ) ,( , )}12
22
4
304
3
- - + = -14
3
4
31
goff x x( ) = -2
1 f x( ) x g x( ) gof x( )
gof x g f x xx gx
( ) ( ( )) ( )= ¾ ®¾¾¾¾¾ = - +-
, ÁI] ¾M nj
.´ÃÀjï¶ nHo¤ 21
23 1 11 3 2
2= -x g x x( ) = +3 1
fog x f g x xx fx
( ) ( ( ))= ¾ ®¾¾¾¾¾¾ = + - =+
, ÁI] ¾M nj
.́ ÃÀjï¶ nHo¤ 3 1
2
3 1 1 33 1 1 3x x+ - =
fof x f f x xx fx
( ) ( ( )) ( )= ¾ ®¾¾¾¾¾ = - - =-
, ÁI] ¾M nj
.́ ÃÀjï¶ nHo¤ 21
2 21 1 xx x4 2
2-
gog x g g x xx gx
( ) ( ( ))= ¾ ®¾¾¾¾¾¾ = + ++
, ÁI] ¾M nj
.́ ÃÀjï¶ nHo¤ 3 1
3 3 1 1
gof x( ) g x x( ) == ++21 f x x( ) sin==
2 2- cos x 2 2+ cos x 12- cos x cos2 1x +
g f x x( ( )) sin= +21 sin x x g
g f x x x( ( )) cos cos= - + = -1 1 22 2
12- cos x sin2 x
fog x( ) g x x( ) == -- 1 f x x( ) == ++32
fg x(x) x(x)
(fog) (x) (x )= += -
ìíï
îïÞ = - +
332
1
1 2 fog y x= - +( )1 2
3
keH» ÏI£TºH
¯IM ¾M
2¾ ®¾¾¾¾keH» â½pHkºH ¾M ÏI£TºH
IÀ n¼d¶ ÁITwHn nj
( )+¾ ®¾¾¾¾¾¾¾1x
fog x( ) == 1 g x x( ) == --2 1 f x x x( ) == --23
52
32
fog x( )
fog x f g x f x x fx( ) ( ( )) ( ) ( )= = - ¾ ®¾¾¾¾¾-2 12 1
IÀ ÁI] ¾M nj
´ÃÀjï¶ nHo¤ ¾¾ = - - - = - + - + = - +( ) ( )2 1 3 2 1 4 4 1 6 3 4 10 4
2 2 2x x x x x x x
4 10 4 12x x- + = fog x( ) =1
4 10 3 010
4
5
22 5
2 0x x S ba- + = ¾ ®¾¾ = - = + = =>D /
g fog g x xx( ) == ++
--1
3 1f x x
x( ) == --++
2 1
3
fog
fog x f g xxxxx
xxxx
( ) ( ( ))( )
( )= =
+-
-
+-
+=
+-
-
+-
+
21
3 11
1
3 13
2 2
3 11
1
3 13
==
+ - +-
+ + --
= - +-
2 2 3 1
3 1
1 9 3
3 1
3
10 2
x xx
x xx
xx
- +-
= +-
¾ ®¾¾¾¾ - + -
- + -
xx
xx
x x
x x
3
10 2
1
3 13 3 1
3 10 32
¸Ãõw» ¸ÃÎoö ( )( )� ���� ��� � ��� ���= - + Þ - + =
+ -
( )( )10 2 1 13 2 1 0
10 8 2
2
2
x x x x
x x
g x( )
g fog
b fog x( ) == 7 g x x( ) == ++2 1 f x x x b( ) == ++ ++2
7 25/ 10 25/ 8 25/ 9 25/
fog x f g x x x b x x x b x x b( ) ( ( )) ( ) ( )= = + + + + = + + + + + = + + +2 1 2 1 4 4 1 2 1 4 6 22 2 2 == 7
Þ + + - = ¾ ®¾¾¾¾ =4 6 5 0 02x x b ¾zÄn ¦Ä ô£Î D
6 4 4 5 0 36 16 5 536
16
9
45
9
4
29
47 25
2 - - = Þ = - Þ - = = Þ = + = =( ) ( ) ( ) /b b b b
y x x== ++ ++24 5
f x x x
g x x
( )
( )
== ++
== ++
ììííïï
îîïï
24
5
f x xg x x( )( )
== ++== ++
ììííïï
îîïï
21
2
f x xg x x x( )( ) ( )( )
== ++== -- ++
ììííïï
îîïï
17
2 6
g f x x x( ( )) ( )= + +2
4 5 gof f g x x x x( ( )) ( )= + + = + + +2 1 4 4 1
2 2 fog
f g x x x x x x x( ( )) ( ) ( )= - + + = + - + = + +2 6 17 4 12 17 4 52 2 fog
fg f
gof fog g f
f fog g g fog f
g fog x x x( ) == -- ++2 6 52 f x x( ) == --2 1
x x23 6- + x x2
3 3- + x x23 1- + x x2
3 2- +
f x x f g x g x( ) ( ( )) ( )= - Þ = -2 1 2 1 g x( ) x f x( ) 2 1 2 6 5 2 2 6 6 3 3
2 2 2g x x x g x x x g x x x( ) ( ) ( )- = - + Þ = - + Þ = - + fog
f fog x x x( ) == -- ++4 6 52 g x x( ) == --2 1
x x23- + x x2 - x x2 2- + x x2
1- +
f x x x( )2 1 4 6 52- = - + g f
2 11
2x t x t- = Þ = + t 2 1x - f
f t t t( ) ( ) ( )= + - + +41
26
1
25
2 t
f t t t t t t( ) = + + - - + = - +2 22 1 3 3 5 3
f x( ) g x x( ) == --2 1 fog x xx( ) == --
++1
2
xx
-+4
2 x
x-+1
5 x
x+-1
5 2 2
2 1
xx
-+
f ( )1 0= x f x xx( )2 1
1
2- = -
+
f ( )- = -11
2x x =1
2 11
2x t x t- = Þ = +
f x xx f t
t
ttt
tt( ) ( )2 11
2
1
21
1
22
1 2
1 4
1- = -+
¾ ®¾¾¾ =
+ -
+ += + -
+ += - KveoM
tt + 5
g x( ) f g x x x( ( )) == --25 f x x x( ) == -- --2
6
4 - x 3 - x 2 - x 1- x
f g x g x g x x x( ( )) ( ) ( )= - - = -2 26 5
Þ - = - +g g x x2 25 6 :g g x( )
Íμ]
Joò
g g x x g x g xg x
2 21 5 6 0 3 2 0- - - + = Þ + - - - = Þ
=� ��� ���( ) ( ( ))( ( ))
( ) xxg x x
-= -
ìíî
2
3( )
+
¾ ®¾ - + = - + Þ - = -1
4 2 2 2 21
45
25
4
1
2
5
2g g x x g x( ) ( )
nm]¾ ®¾¾ - = ± - Þ- = - Þ = -
- = - -g x x
g x x g x x
g x x( ) ( )
( ) ( )
( ) (
1
2
5
2
1
2
5
22
1
2
5
2)) ( )= - Þ = -
ì
íï
îï 5
23x g x x
x - 2 3 - x g x( )
g g g20 0 6 0 0 2 3( ) ( ) ( )- - = Þ = - IÄ x =0 f g x g g x x( ( ))= - - = -2 2
6 5
g( )0
f x( ) f x f x gof
g f g f x( ) g
R Df g
f x xx( ) = +
-2
3g x x( ) = -1 g f gof
x = 4x = 3
f x = 3 f
x = 2x = -4 g -4 f x = 2 f
D x D f x Dgof f g= Î Î{ | ( ) } f x Dg( )Î x DfÎ
gof
D x D f x D x x x xgof f g= Î Î = Î - ¹ = ¹ = -{ | ( ) } { | } { | } { }2 1 23
2
3
2
Df =
D x xg = ¹{ | }2
f x x( ) = -2 1
g x xx( ) =
-3
2
D x D f x D x x x xgof f g= Î Î = Î + ³ = ³ = +¥{ | ( ) } { | } { | } [ , )2 1 3 1 1Df =
D x xg = ³{ | }3 f x x( ) = +2 1
g x x( ) = - 3
D x D f x D x xx x xgof f g= Î Î = ¹
-¹- = ¹ ¹ = -{ | ( ) } { | } { | } { , }11
1 11
211
2
D x xf = - = ¹{ } { | }1 1
D x xg= - - = ¹-{ } { | }1 1
f x xx( ) =
-1g x x
x( ) =+2
1
x = 1
2
xx -
= -1 1
D x x x xgof = ³ - + £ = ³ - £ = -{ | } { | } [ , ]2 2 3 2 7 2 7
D x xf = ³ -{ | }2D x xg = £{ | }3
f x x( ) = + 2 g x x( ) = -3
fog g x x( ) == ++2 1 f x x x( ) == -- --2 2
[ , ]0 1 [ , ]- 1
20 [ , ]- 1
21 [ , ]-2 1
g x x:2 1 01
2+ ³ Þ ³ - g f
f x x x x x x x: ( )( )2 0 2 0 2 1 0 2 12 2- - ³ Þ + - £ Þ + - £ ¾ ®¾¾¾ - £ £¾zÄn »j ¸ÃM
D x D g x D x x xfog g f= Î Î = ³ - - £ + £ ={ | ( ) } { | } {1
22 2 1 1
SwH ¿ÄkM ¸ÄH� �� �� ³³ - + £1
22 1 1| }x :fog
D x xfog = ³ - £ = -{ | } [ , ]1
20
1
20 x £0 2 1 1x + £
- 1
2
2 3 3 0- - < 3 x =1x =1
fof f x x( ) == --2 [ , ]-2 0 [ , ]0 2 [ , ]-2 2 ( , ]-¥ 2
D x D f x D x xfof f f= Î Î = £ - £{ | ( ) } { | }2 2 2 x £ 2 f D x xfof = £ ³ - = -{ | } [ , ]2 2 2 2 x ³ -2 2 4- £x 2 2- £x
gof g x xx( ) == --
--4
2f x x( ) == 2
( , ]-¥ 0 ( , ]-¥ 1 ( , ]-¥ 2 ( , ]2 4
f x( ) gof x ¹ 2 x £ 4 g 2 4 2
x x£ Þ £ f x( ) ¹ 2 f x( ) £ 4 2 2 1
x x¹ Þ ¹ ( , ) ( , )-¥ 1 1 2 ( , ] { }-¥ -2 1 f
x
( , ]-¥ 2 [ , ]1
32 [ , ]-1 14 [ , ]0 5
g( )x x x= -52 g f x( ) f x gof
5 0 5 02x x x x- ³ Þ - ³( )
xx x
0 5
5 0 0( )- - + -
0 3 1 5 1 3 61
32
1 3£ - £ ¾ ®¾ £ £ ¾ ®¾ £ £+ ¸x x x [ , ]0 5
[ , ]1
32 3 1x -
gof g f
83
73
103
D x D f x D x f xgof f g= Î Î = £ <{ | ( ) } { | ( ) }2 4 x < 4 g x £ 2 f
f x( ) < 4 y x= -3 12
( ) ( , )0 3 ( , )2 0 f
3 3
24 1
3
2
2
3- < Þ - < Þ > -x x x
22
3
8
3- - =( ) ( , ]- 2
32 - < £2
32x gof
gof fog( ) ( )0 0++ g x x( ) == f x x( ) cos==
2 fog g == {( , ),( , ),( , ),( , )}5 7 3 5 7 9 9 11 f == {( , ),( , ),( , ),( , )}7 8 5 3 9 8 11 4
gof
fog goffof( ) ( )
( )-- ++1 0
1g f
-1
59
- 5
9
( )( ) ( )( )fog fog-- ++ ==2 1 5 g == -- -- --{( , ),(c, ),( , )}2 1 3 31
3f a b== --{( , ),( , ),( , )}1 2 1 2
( a b c++ ++
. fog( )4 35= g( )4 7= f ( )7 5=
fog x gof x( ) ( )= f g¹ g f. fog g( ) ( )5 2= g x x( ) = -2 1 f x x( ) =
. fog x x( ) = - 2 g x x( ) = -24 f x x( ) = -2
4
............... ...............
x53 4 1
2x x- + 3 4 12x x- + x5 x5
3 4 12x x- + 3 4 1
2x x- + x5
fog g f
{( , ) , ( , )}- -1 3 2 1
{( , ) , ( , )}3 3 2 1-
{( , ) , ( , )}1 2 2 3- -
{( , ) , ( , ) , ( , )}1 2 2 3 4 1- -
( fog( )2 g x xx( ) ==
--1 f x x( ) [ ]==
-1 -2 -3 -4
( fog gof( ) ( )1 2 1 2-- -- -- g x x x( ) == ++ ++22 1 f x x( ) | |==
4 2 4 2 1( )- 4 1 2( )-
( )( )fog f-- --2 2 g xxx x
x x( ) ==
++--
³³
++ <<
ììííïï
îîïï
1
12
2 22
f xx xx x
( ) ==++ ³³ --
-- << --ììííîî
3 1 1
4 1
-17 -55
( g f x( ( )) g x xx( ) == ++
--2 22 f x x
x( ) == --++
2 1
1
2x x x +1 x -1( , )pp pp2
3
2fog x( ) g x x( ) tan== f x x
x( ) ==
++1 2
-cosx -sin x cosx sin x
( )( )gof pp4 g x x x( ) == --1 2 f x x( ) sin==
2 22 1
2
fog( )pp3 g x x( ) cos== 2 2 f x
xx
( )1 2 12
== --
3
2 12
a b c++ ++ fog x x x( ) == ++ ++2 12 g x x bx c( ) == ++ ++2 f x x a( ) == ++2 2
-3 -1
fofof ( )1 f xx x
x x( ) ==
-- ³³
<<
ììííïï
îîïï
3 0
02
-2 fof x( ) f x x x A== -- ÎÎ{( , ), }2 1 A == { , , , , }1 2 3 4 5
(cos )x ¹¹ 0 fof x(sin )2 f xx xx x
( ) ==-- ³³++ <<
ììííîî
1 0
1 0
cos2 x -sin2 x cos2 x sin2 x
( )( )fof x f x x x( ) | |== --
x x+ | | -x xy fof x( ) f x f x( ) ( )++ == ++2 3 2
-8 -4 f ( )1 fof x x( ) == ++4 15 f x( )
-27 -10
a fog x gof x( ) ( )-- == 6 g x x( ) == --2 f x x a( ) == ++3 -1 -2
( ( )( )gof x == --5 g x x( ) == --1 2 f x x x( ) == ++ --3 12
-1 - 4
3 43
( ( )( ) ( )( )gof x fog x== g x x( ) == ++ 4 f x xx( ) == --
++2 1
2
-1 -7 -1 -7( fog f g x x( ) == ++ 2 f x x( ) ( )== --2 3
2
32
12
-1fof x( ) == 0 f ( )0 1== f ( )2 0== f
-2 -1 x g f x( ( )) == --2 g x x x( ) == ++ --2 2 f x x x( ) [ ] [ ]== ++ --
Æ �-
( )( )gof x == 27 g x x( ) == 3 f x x x( ) [ ] [ ]== ++ --2
x fog g x x( ) == -- 12
f x x xx x
( ) == -- ³³++ <<
ììííïï
îîïï
21 0
2 1 0
-1 12
x gof g x x( ) == -- ++1
22 f x x x( ) == ++2
3
( ( , )-1 4 ( , )-2 1 ( , )-3 2 ( , )-4 1
( x fog g x x( ) ( )== --1
23 f x x x( ) == ++ --2 2
( , )1 5 ( , )-2 1 ( , )-1 5 ( , )-5 1
y == 3 gof g x x( ) == ++4 1 f x(x) x== ++2
4 5/ x fog -- 1
46 x f g x x x( ) == --
4 9, 144, 1
49, 1
94,
N d d d( ) == -- ++20 80 5002
d t t( ) == ++4 3
(
(
f x( ) x ³³ 0 f x x( )21 2-- ==
1 2+ x 2 2- x 1 2+ x 2 1+ x
( f x( ) ( )( )fog x x x== ++ ++8 6 52 g x x( ) == ++2 1
2 32x x+ + 2 4
2x x- + 2 2 32x x- + 2 3 1
2x x+ +
( f x( )1-- f x x x( )-- == -- ++3 4 52
x21+ x x2
4 5+ + x2 3+ x x24 5- +
g x( ) ( )( )gof x x== 2 f x xx( ) ==
--2 x
x+1 x
x -1 xx-1 x
x +1g x( ) f x x( ) ( )== ++ 1 2 fog x x( ) == 4
( )x +1 4 x41+ x2 1+ x2
1-
g x( ) ( )( )fog x xx== ++ 2 f x x
x( ) ==--1
g x xx( ) = -
+22 2 g x x
x( ) = ++
22 2 g x x
x( ) = +-2
2 2 g x xx( ) = -
+2
2 2( ( )( )f g x++ f x(g(x)) x== ++ --2 2 f x x x( ) == -- --2 2
x x2 2+ x x2 2- x2 1+ x2 1-
f ( )3 fog x xx( ) ==
--3 g x x( ) == --2 1
-2 -4g( )--2 f g x x x( ( )) == ++4 6
2 f x x( ) == ++2 42
-1 ( a ( )( )gof a == 15 g x f x( ) ( )== ++ --2 2 3 f == {( , ),( , ),( , ),( , )}5 2 3 4 1 8 6 9
( , )4 1 ÎÎgof ( , )4 2 ÎÎ fog g a b== {( , ),( , ),( , ),( , )}1 2 3 1 3 1 f =={( , ),( , ),( , ),( , )}2 1 3 2 4 5 1 7
( , )a b
( , )5 4 ( , )4 3 ( , )4 5 ( , )3 4
( a g f a( ( )) == 5 g == {( , ),( , ),( , ),( , )}1 2 5 4 6 5 2 3 f x x x( ) == ++
a g f a( ( )) == 3 g == -- -- -- -- --{( , ),( , ),( , ),( , )}2 1 1 4 2 3 4 3 f xx x
x x( ) ==
³³
-- -- <<
ììííïï
îîïï
00
-1 -4
72
119
g( )3 f x x( ) == --2 1
92
112
( f ( )5 g x x( ) == ++3 4 g f
g gof == {( , ),( , ),( , )}1 1 4 7 8 9 f == {( , ),( , ),( , ),( , )}1 5 2 3 4 7 8 9
{( , ),( , ),( , ),( , )}5 1 2 3 7 7 9 9 {( , ),( , ),( , ),( , )}2 3 8 9 11 4 7 {( , ),( , ),( , ),( , )}0 9 7 4 3 1 5 5 {( , ),( , ),( , ),( , )}5 1 3 5 7 7 90
( gof g x x( ) == --21 f x x( ) == -- 2
gof x x x( ) = - +24 3 Dgof = - ±{ }1 gof x x( ) ( )= - -1 2
2 Dgof =
gof x x( ) ( )= - -1 22 Dgof = - ±{ }1 gof x x x( ) = - +2
4 3 Dgof =
( D Dfog gof g x x( ) == --2 12 f x x( ) == -- 1
( , )1 +¥ [ , )1 +¥ ( , )0 +¥ [ , )0 +¥
( gof g x x( ) ==--6
5f x x( ) == --3 2
( , ] { }-¥ - -3
211 ( , ] { }-¥ -3
25 - -{ }11 -{ }5
( fog g x x( ) == 3 f x x( ) ==--2
1
fog g x x x( ) ,== ++ ££ ££2
1 0 2 f x x x( ) ,== ££ ££20 1
( , ]0 2 [ , ]0 2 { }1 { }0
( fog g x x( ) ==--6
3 5f x x( ) == --3 2
gof x( ) g x x( ) == --1 2 f x x( ) == --4 12
( , )-1 1 ( , )- 1
2
1
2 [ , ]-1 1 [ , ]- 2
22
2fog x( ) g x
x( ) ==
--1
22f x x( ) ==
++11
( gof g x x( ) == f x x( ) sin==
( , )- -2p p ( , )p p2 2 ( , )03
2
p ( , )- p p2 2
( fof x( ) f x x( ) == --1
( fof f x x x( ) == --
[ , )1 + ¥ [ , ]0 1 [ , )0 + ¥ { , }0 1
( gof g xx x
( ) ==--1
42
f x x x( ) | |== ++
( , )0 +¥ -{ }0 -{ , }0 8 ( , ) ( , )0 8 8 +¥
( fog g x x( ) ( )== 1
4f x x
x x( ) ==
-- ++ ++2 2
( , )-1 1
2 ( , )-2 0 ( , )1
2+ ¥ ( , )- + ¥1
2
( fog g x x x( ) log ( )== ++22 2 f x x( ) == --3
[ , ) ( , ]- -4 2 0 2 [ , ] ( , ]- -4 1 1 2 [ , ]-2 0 [ , ]-4 2
f x( )2 1++ Df == --[ , ]2 6 f x( )
f x x x( ) == ++ --62 y f x== --( )1 2
[ , ]-1 3
2 [ , ]-2 3 [ , ]-3 2 [ , ]-5 5
y f x== --3 3 4( ) [ , ]1 4 y f x== --( )2
[ , ]5 8 [ , ]5
3
8
3 [ , ]2
3
5
3 [ , ]2 5
b a-- fof f x x( ) == -- ++1 1 [ , ]a b
¥ fog g x x( ) == -- 1 f x x( ) == ++2
1
[ , )- +¥1 [ , )1 +¥ + fog x( ) g x x( ) == --4 2 f x x( ) == --4 2
[ , ]0 4 ( , ]-¥ 4 ( , ]-¥ 0 [ , )4 +¥
( )( )fog x g f
[ , )3 + ¥
[ , )4 + ¥
[ , ]-2 3
[ , )0 + ¥
gof g f
( [ , ]-1 5
( , ]0 4
( , ]1 3
( , )0 4
y f x== --( )3 f x( )
( )f g of++ g x x( ) == f x x( ) == --1 2
[ , ] { }- -1 1 0 [ , ]-1 0 [ , ]0 1 [ , ]-1 1
fog g x x x( )1 1== ++ f x x x( ) == --2 2
gof g x x x( ) == -- 2 f x xx
( ) == ++--1
1
2
2
R - -{ , }1 1 ( , )-1 1 { }0 [ , )0 1
( fog g x x x( ) tan ; | |== << pp2 f x x
x( ) == --1 2
[ , ) ( , ]-1 0 0 1 [ , ) ( , ]- p p40 0
4 [ , )p p4 2
[ , ]- p p4 4
f x( ) f xx x( )-- == --12 1
[ , )1 +¥ [ , )-1 1 [ , )0 1 [ , )-1 0
gof x( ) f x( ) g x x( ) == --3 f x x( ) log== --2 1
[ , )9 +¥ ( , ]-¥ 3 ( , ]1 9 ( , )1 +¥
y gog x== ( ) y g x== ( )
[ , ]0 3
[ , ]3 4
[ , ]3
43
[ , ]03
4
fog x( ) g x( ) f x x( ) == --21
h x f x( ) ( )==--1 g x x f x( ) ( )== -- x f
(
fog f x x x( ) == -- -- 2
g x x( ) = -1 g x x( ) sin= +3 g xx
( ) = 1
2 g x x x( ) = + +2
1
B A B AB
xy
5 7 2 3
2 1 4 0
--
f = -{( , ) ,( , ) ,( , )}1 1 2 4 0 2
{( , ) ,( , ) ,( , )}1 2 5 1 3 2-
a f a a a== ++{( , ),( , ),( , ),( , )}2 3 3 1 1 42
-2 -2
y y x x a a a a a1 2 1 2
2 22 2 0 1 2= Þ = + = Þ + - = ¾ ®¾¾¾¾¾¾ = -: ,SwH oÿÅ KÄHoò Íμ]
( , )1 4 ( , )1 1 a = 1a = -2 {( , ) , ( , ) , ( , ) , ( , )}2 3 2 3 2 1 1 4- a = -2
f == --{( , ),( , ),( , ),( , ),( , ),( , )}2 1 4 2 1 1 5 1 7 3 0 2
f
ff = -
¯
( , ) ( , ) ( , )
( , ) ( , )( , )
2 1 4 2 7 3
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x
y x==
x y ax bcx d
= ++
y ax b== ++ y x==
y xa== log y ax==
y x== tan y x== cos y x== sin
y x x= -2 2
D4
0 3 1={ , , } D3
1 2= [ , ] D2
1= +¥( , ) D1
2 5= [ , )
[ , ]a b ( , ]a b [ , )a b ( , )a b x a bs Ï ( , )
y x== --| |2[ , )-3 1 -{ }2 { , , }1 2 3 ( , )0 + ¥
f f( ) ( )1 3 1= = ( , )0 + ¥
[ , )-3 1 -{ }2 { , , }1 2 3
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0
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ìíï
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2 0
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f b a- =1 ( ) f a b( ) = f f f f( ) , ( ) ( ) , ( )- = = Þ = =- -1 2 3 0 2 1 0 3
1 1
y x=
y xy x y x x y= Þ =3 3
x y y x+ = Þ + =2 5 2 5
f = -{( , ) ,( , ) ,( , )}1 2 1 3 0 2 f -1 f
f - = -12 1 3 1 2 0{( , ),( , ) ,( , )}
-1 f -1
f -1
y x= -| |1 : | |x y= -1 y x =1
y
f f f
x y y x f f
f b-1 ( ) f a b( ) = f b-1 ( ) b f
f --12( )
f == --{( , ),( , ),( , ),( , )}2 1 1 4 0 2 4 1
f x x x( ) == ++3
f x x( ) == --4
1
f -1 f - =12 0( ) ( , )2 0 f -1 ( , )0 2 f
f - = -11 2 4 1 2 0 1 4{( , ) , ( , ) , ( , ) , ( , )}
f -1 . f - =12 3( ) f -1 f
x f -12( ) f x x( ) = +1
31 ( , )-3 0 ( , )0 1 f
13
1 21
31 3x x x+ = Þ = Þ = x 1
31x + f x( )
x y= +1
31 y x y x= +1
31 f -1 f - =1
2 3( )
f x( ) x f x x x( ) = +3 f -12( )
f - =12 1( ) x = 1 x x3
2+ =
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3
1
2
0
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1
. f - =12
3
2( )
x f x b( ) = f b-1 ( )
f -1 ( , )b a f ( , )a b
f --1 f x x x( ) == ++2
( , )1 3 ( , )0 1 ( , )11
4 ( , )1
41
f f ( , )a b f -1 ( , )b a
( , ) ( , ) ( )1
41 1
1
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1
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t¼§÷¶ nj¾ ®¾¾¾ ¾ ®¾¾ = +f
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1
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f ( )2 ( , )1 1-- f --1 f x x bx( ) == ++3
-4 -2( , )-1 1 f ( , )1 1- f -1
f x x bx b b( ) ( ) ( )( , )= + ¾ ®¾¾ = - + - Þ = --3 1 1 31 1 1 2
f ( ) ( )2 2 2 2 8 4 43= - = - = f x x x( ) = -3
2
f -1 f f -1 f f f -1
f -1 f
f-- A
x y B
¢¢C C
y f-- x D
E
f x x( ) == --1
f x x( ) = -1
f xx x
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-- <<
³³
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0
f
y x=
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y f g x== ---- --( )( )1 1
-1
f - = -11 2 1 3 2 0 0 1{( , ) , ( , ) , ( , ) , ( , )} g- = - - -1
1 1 0 2 3 1 4 3{( , ) , ( , ) , ( , ) , ( , )} g-1 f -1 -1 x f g- --1 1
0 2 1 1 0 1 1 21 1 1 1g g f f- - - -
¾ ®¾¾ - ¾ ®¾¾ ¾ ®¾¾ - ¾ ®¾¾, , ,
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g f a--1 2( ( ))
-1 -2 -3 -4( , )2 3 ( , )a a2
3- f a a a a a2 2
2 2 0 1 2- = Þ - - = Þ = - IÄ f = {( , ) , ( , ) , ( , ) , ( , )}2 3 2 2 2 3 1 4 a = 2
f = -{( , ) , ( , ) , ( , ) , ( , )}2 3 1 2 2 3 1 4 : a = -1 g-1
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f --1
y x f f -1
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=-2
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21y x= -3
1y x x= -2 2x >1
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y=
-2
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Þ = +2 1y x
Þ = +y x 1
2
Þ - =yx
12
Þ = +yx
12
Þ = +log2
1x yÞ = -y xlog
21
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Þ = +y x 13
Þ - + = +y y x22 1 1
Þ - = +( )y x1 12
nm]¾ ®¾¾ - = +y x1 1Þ = + +y x1 1
f x x- = +1 1
2( )f x
x- = +1
12( )f x x- = -1
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1( )f x x- = + +1 1 1( )
. y >1 x >1
f x x x x( ) = + +3 23 3
x - -1 13 x - +1 1
3 x + +1 13 x + -1 1
3
f ( )1 7= f x =1 x = 7 f -1
7( ) f -1
7 1 1 2 1 13 + - = - =
y x f -1 f f -1 f f -1 f
D Rf f- =1 R D
f f- =1
f f -1
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2 4( )
f f f -1
x ³ -3 [ , )- + ¥3 f
f x x x- = - + ³ -12 4 3( ) ,
f x x xx
( ) == --££
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1
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x 2 4
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xx
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y ax b x ay b y f x x ba
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f -1 f f f -1
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f x x x- = + - £ £12 1 3 3( ) , f x x x- = + - £ £1 1
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f x x x- = + - £ £1 1
21 2( ) , f x x x- = + - £ £1 1
21 2( ) ,
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y x x y x y f x x= - Þ = + Þ =
+Þ = +-
2 1 2 11
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1
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3
a b
b a
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f x x( ) == -- --2 1
f x x x x- = + + ³1 22 3 2( ) , f x x x x- = + - ³ -1 2
2 1 1( ) ,f x x x x- = + + ³ -1 2
2 3 1( ) , f x x x x- = + - ³1 22 1 2( ) ,
y x= - -2 1
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y xx
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x ³³ 2 f x x x( ) ( )== -- 2x
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1( xx
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y x x x x x x y y y= - = - ³ ¾ ®¾¾ = - ³( ) , ,2 2 2 2 22 2·»nH»
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. f - =13 3( ) f ( )3 3= : x = 3
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y ax bcx d
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f x ax bcx d
f x dx bcx a
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-1 x
y xx
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3 2 y x
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f f -1 f a d+ =0
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3 d a
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( , )11
2y x = 1
2
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a d k k k+ = + + = Þ = -2 0 1
f b a-- ==( ) f a b( )==
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y x== f-- f
y x= y x=
f a b( )== ( , )a b f-- f
f ( , )b a ( , )a b f a b-- ==( )
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( , )3 3 ( ( , )4 4 ( ( , )5 5 ( ( , )6 6 ( ( , )2 8 y x x= -2
3 x y= f y x= f -1 f
y f x x xy x
x x x x x x= = -=
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2 2 2 833 4
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f a bf a b
( )( )( )
= + Þ= + = -
- = - - =ìíî
32 8 2 1
1 2
f ( )2 1= - f ( )- =1 2
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210 32 10 22
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f f of x-- ( ) f f-- x f-- x fof x-- ( ) :
x DfÎÎ x
f f x x x Rf( ( )) , ( )-- == ÎÎ f f x x x Df-- == ÎÎ( ( )) , ( )
y x== fof-- f of--
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f of-1 fof -1
y fof x x= =-1 ( ) x R xfÎ Þ < £1 3
y f of x x= =-1 ( ) x D xfÎ Þ - < £2 4
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[ , ]1 5 f f of-1
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f g gof x x( )== fog x x( )== f of x x-- ==( ) fof x x-- ==( )
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2 2 2 0¾õ£º nnHj¼μº nj
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3( ) ( 1
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y f x x f y= - ¾ ®¾¾ = -( ) ( )3 1 3 1·»nH» y x
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f f x
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4 2 2 4( ) ; (
f x x x- = - + < <14 2 0 4( ) ; ( f x x x- = - - < <1
4 2 0 4( ) ; (y x x= -| |2
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f x x( ) ( )= -2 21
( , )0 1 ( , )-¥ -1( , )1 + ¥ ( , )-1 0
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y x y x y x x yxx
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y x x= + [ ] ( y xx
= ++
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2( y x= log ( y x= 2 (
( y x x= + +3
1 ( y x x= -3 23 ( y x= [ ] ( y x x= -4 2
2 ((
f f -1 y x= f f -1
f -1 f f b a- =1( ) f a b( ) =
( 2 3 31f f-- -- ++( ) ( ) f == -- -- --{( , ),( , ),( , ),( , )}1 2 3 1 3 4 4 3
( ( ( (( f f++ --1 f == -- --{( , ),( , ),( , ),( , )}1 0 1 2 0 1 2 1
( , )2 0 ( ( , )0 0 ( ( , )-1 1 ( ( , )1 2 (a b c d++ ++ ++ f a c d b-- == -- ++ --1
1 1 2{( , ),( , )} f a b== ++{( , ),( , )}2 1 3
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f -- --15( ) f x
x xx x
( ) ==++ ³³
++ <<ììííîî
4 3 3
1 3
( -6 ( -2 ( -4 (b f -- ==1
21 4( ) f -- ==16 1( ) f x ax b( ) == ++
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( , )/2 2 5 ( ( , )- -2 1 (
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1
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13
( 12
( ( (
( ( , )a a++ 2 f x xx( ) == --
--4
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( a f g a-- ==12 6( ( )) g x x
x( ) ==-- 1 f == {( , ),( , ),( , ),( , ),( , )}2 5 6 3 3 7 4 1 1 9
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f ( )2 ( , )1 2 f x x ax b( ) == -- ++ ++3
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x x( ) ;== ++ >>2
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[ , )2 + ¥ ( ( , )0 + ¥ ( [ , )0 + ¥ ( (
f x x( ) == -- ++3 1
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83
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3
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y x= +2
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y x= -3
2
3
2(
f x x( ) == ++ --1 2
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f [ , )2 ++¥¥ f x x x( ) == --24
f xx x
x x( ) ==
³³
-- <<
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02
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f x x x- = - - £ £1 4
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f x x x- = - £ £1 4
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( f --1 f --1 f
x +12
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y +12
( y -12
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( ( -1( -2 (
a f x ax( ) == ++ 1 -1 ( -1
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y x==--11
f xx
- = +1 1 1( ) ( f xx
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11( ) ( f x
x- =
+1 1
1( ) (
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--4
2
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++2 3
-1( ( - 3
2( 3
2(
f x x( ) == -- 1f x x- = +1 2
1( ) ( ) ( f x x- = -1 21( ) ( ) ( f x x- = +1 2
1( ) ( f x x- = -1 21( ) (
( , ]--¥¥ -- 1 f x x( ) == 2
f x x- = - -1 ( ) ( f x x- = -1 ( ) ( f x x- = -1 ( ) ( f x x- =1 ( ) (( f x x( ) == ++3
f x x x- = - ³1 23 0( ) , ( f x x x- = - ³ -1 2
3 3( ) , ( f x x x- = + ³1 2
3 0( ) , ( f x x x- = + ³ -1 23 3( ) , (
( y x== -- --2 1
f x x x x- = - + - £1 24 5 2( ) ; ( f x x x x- = - + £1 2
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4 5 1( ) ; ( f x x x x- = - + ³1 24 5 1( ) ; (
x ³³ 1 y x x== --2 2f x x- = - -1 1 1( ) ( f x x- = + -1 1 1( ) ( f x x- = - +1 1 1( ) ( f x x- = + +1 1 1( ) (
( f x x x( ) == -- ++24 5 x ³³ 2
f x x- = + -12 2( ) x ³ -1 ( f x x- = - +1
1 2( ) x ³ 1( f x x- = + -1
2 2( ) x ³ 1( f x x- = - -1 1 1( ) x ³ 2 (
( f xx x
x x( ) ==
³³
-- -- <<
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îîïï
00
f x x x- = -1( ) | | ( f x x x- =1( ) | | ( f x x- =1 2( ) ( f x x- = -1 2( ) (f x x x( ) | |== ++ ++ 2
y x x= + ³2
20, ( y x x= - ³2
20, ( y x x= + ³ -2
2 2, ( y x x= - ³ -22 2, (
( y x x== --| |2
f x x- = - - <1 1 1 1( ) , x ( f x x x- = - + <11 1 0( ) , (
f x x x- = - - < <11 1 0 1( ) , ( f x x x- = + - < <1
1 1 0 1( ) , (( f x( ) | x | | x |== -- -- ++2 6 1
f x x x- = + >1 1
32 3( ) , ( f x x x- = + >1
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f x x x- = - - < <1 1
21 4 8( ) , ( f x x x- = + > -1
7 4( ) , (
y x x== --3 | |
y x x= -2
4
| | ( y x x= + | |4 ( y x x= +3
8
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x( )
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¹¹
==
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00 0
f x x x- = Î -10( ) | | , x { } ( f x x x x( ) | | ,= Î (
f x x x x- = Î1( ) | | , ( f x x x x- = Î -10( ) | | , { } (
y xx==
++1 | |
f x xx x- = - >1 1 1( ) | || | , | | ( f x x
x x- =-
<11 1( ) | | , | | (
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x x- =-
>11 1( ) | | , | | (
y x x x== ++ ++ ++3 23 3 2
y x= - - +1 13 ( y x= - + -1 1
3 ( y x= - +1 13 ( y x= - -1 1
3 (
( , )--1 0 y x x== -- ++4 22 1
y x= - +1 ( y x= -1 ( y x= - -1 ( y x= +1 (y x== --++
10 21
f x x- = + -11 2( ) log( ) ( f x x- = + +1
1 2( ) log( ) ( f x x- = + -1
2 1( ) log( ) ( f x x- = + +12 1( ) log( ) (
f x x( ) == ++ 2 -1( ( -1(
y x== f --1 f x a x a x a x x( ) ( ) ( ) ( )== ++ ++ ++ ++ ++ ++1 2 4 34 3 2
( ( ( (fof --1 f == {( , ),( , )}1 2 2 3
{( , ),( , )}1 1 3 3 ( {( , ),( , )}2 1 3 2 ( {( , ),( , )}1 1 2 2 ( {( , ),( , )}2 2 3 3 (( g fog x x( ) == g f == {( , ),( , ),( , )}1 4 2 3 3 5
g ={( , ),( , ) ,( , )}4 1 3 2 5 3 ( g ={( , ),( , ) ,( , )}4 1 2 2 3 3 ( g ={( , ),( , ) ,( , )}4 1 3 2 5 5 ( g ={( , ),( , ) ,( , )}4 4 3 3 5 5 (
g( )0 f x x( ) == --2 1
( 12
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( ( ( (
fof --1 [ , ]2 5 f x x( ) == -- 1
( ( ( (
( y f of x== ++ --1 1 ( ) f x x( ) == --1 ( , ]-¥ 1 ( ( , ]-¥ -1 ( [ , ]-1 1 ( [ , ]0 1 (
( gof --1 g == --{( , ),( , ),( , ),( , )}2 3 1 4 4 1 3 0 f == --{( , ),( , ),( , ),( , )}1 2 2 5 0 3 4 1
gof - = -15 3 1 1{( , ),( , )} ( gof - = -1
2 0 1 4{( , ),( , )} ( gof - =12 4 3 5{( , ),( , )} ( gof - =1
0 0 1 3{( , ),( , )} (
( g of-- --1 1 g == -- --{( , ),( , ),( , ),( , )}1 3 5 21
20 4 6 f == -- --{( , ),( , ),( , ),( , )}0 1 2
1
23 2 1 5
{( , ),( , ) ,( , )}1
25 5 2 1
1
2- ( {( , ),( , ) ,( , )}- -1
1
25 2 2 3 (
{( , ),( , ) ,( , )}- -11
2
1
25 2 1 ( {( , ),( , ) ,( , )}2 1
1
25 1 3- - - (
( ( ) ( ) ( )( )fog f og-- -- --++1 1 15 1 g x x( ) == 3 f x x( ) == --1
83
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1 21gof g a== -- --{( , ),( , )}2 1 1 6 2 f a== ++{( , ),( , )}2 1 3 7
-5 ( -1( ( (( ( )( )fog--1
4 g x xx( ) == ++
--5 2
2 1f x(x) x== ++2
( ( ( (( a ( )( )g of a-- -- ==1 1
8 g x x( ) == ++5 9 f == {( , ),( , ),( , ),( , ),( , )}5 2 7 3 1 4 3 6 9 1
( ( ( (( )( )fogof --1
3 g == -- -- --{( , ),( , ),( , ),( , )}0 2 2 4 3 2 4 2 f == -- --{( , ),( , ),( , ),( , )}2 3 1 2 4 1 3 0
( ( ( -1( -2 (( f og-- --1 1 g x x( ) == --2 5 f x x( ) == ++ 4
y x= + 2 ( y x= +12
( y x= - 3
2( y x= -2
3(
g of-- --1 1 x >> 0 g x x( ) == 2 f x x( ) == ++1 x2
1+ ( x21- ( x +1( x -1 (
fog x( ) x ³³ 0 g x x-- ==1 2( ) f x x-- == ++1 1( )
x x+ -1 2 ( x x+ +1 2 ( x21- ( x -1 (
( ) ( )fog x-- -- ==1 10 g x x x( ) == -- ++2 8 1
2 f x x( ) == ++ 2 -8 ( ( -10 ( (
a g a-- ==1 1( ) f -- ==15 3( ) g x f x( ) ( )== ++ ++2 1 1 f
( ( (( ( , )--1 1 f x x a( ) | |== ++3 a
( ( ( (
p x xx
( ) =+21
( h x xx
( ) = +21 ( g x x x( ) = - ( f x x x( ) = + (
f g x( ) f xg x x
x x( )
( )==
³³
-- <<
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îîïï
1
02
x + 7 ( x x24 3- + ( x - 3 ( | |x - 2 (
k g x k x x( ) ( )== -- 1 f x x x( ) == ++ ++21
12
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2( ( (
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3 1
21
1f x- - +( ) ( 13
1
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1
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2 42
x x
( ( ( ( (
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3
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g
12
3
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1g x- -( ) ( g x- -1 3
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4
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22
8 2 ( 6 2 ( 5 2 ( 4 2 (
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x( ) ==
++21
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3
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24
2
x2 1+ ( 2x
( 2x (
f xx
x( ) == --++
2 1
2 1
y xxx= -
+log 2
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= +-
log2
1
1( y x
x= +
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2
1
1(
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y x= 5
1
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f x--1 (sin ) f x xx
( ) ==++21
sin| cos |
xx
( | cos |sinxx
( cot x ( tan x (( )( / )fof --1
4 5 f x x x( ) [ ]== ++
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