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A {0, 1, 2} B {a, z} A B {(0, a),(0, z),(1, a),(1, z),(2, a),(2, z)} .. n(A) 3 , n(B) 2 n(A B) 3 2 6 ** A B (cartesian) A B 2. A B B A ( A B B A ) n(A B) n(B A)

A {0, 1, 2} B {a, z} B A {(a, 0),(a, 1),(a, 2),(z, 0),(z, 1),(z, 2)} A B B A .. n(A B) n(B A) 6 ** A B B A A B ( ) 3. (r) ()

C {(1, 0),(3, 5),(3, a),(a, 5),(0, 1)} D { 0, 1,(3, a),(a, 5)} E { } F { } C () E () .. D ( 0 1 ) F ( )

5 314 4. r {(x, y) A B | } r (x, y) A B ( x A y B )

A {0, 1, 2} B {3, 4,5,6, 7, 8, 9} r {(x, y) A B y 2x 1}| r {(1, 3),(2,5)} r {(y, x) B A y 2x 1}| r {(3, 1),(5, 2)} r {(x, y) A B x y }| r {(2, 4)} r {(x, y) A B y 5x }| r ** A B x y .. (x, y) R R 5. ( x, y) r {(1, 2),(2, 5),(3, 8),(7, 8)} 6. (D) (R) A { 3, 2, 0, 1, 2} B {3, 4,5,6, 7, 8, 9} r {(x, y) A B y 3x 2 }| .. r {(1, 5),(2, 8)} rD {1, 2} rR {5, 8} 2r {(x, y) A B y x }| .. r {( 3, 9),( 2, 4),(2, 4)} rD { 3, 2, 2} rR {4, 9} r {(x, y) A B y 5x }| .. r rD rR

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317

9. (f) x y 1

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a , ( a ) (h, k) h b/2a k h c y 2f (x) x 4x 4 2g(x) (x 3) 1 f ( a ) h b/2a ( 4)/2( 1) 2 x 2 y 8 (h, k) ( 2, 8) fD R fR ( , 8] g ( a ) (h, k) (3, 1) gD R gR [1, )

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a , ( a ) (h, k) f (x) 2 x 1 g(x) x 1 f ( a ) (h, k) (1, 0) fD R fR [0, ) g ( a ) (h, k) (0, 1) gD R gR ( , 1]

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323

1. r A B rD A rR B A B ( f : A B ) fD A fR B A B ( ontof : A B ) fD A fR B A B ( 1 1f : A B ) fD A fR B y x A B ( 1 1ontof : A B ) fD A fR B y x A B A B 1-1 A B 1-1 A B

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y

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- A 3 32 512 A A 7 7 7 343

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A {0, 1, 2, 3} , B { 2, 1, 0, 1, 2} - A B f (x) x< ( < ) 3 4 5 5 300 - A B x f (x) 2 2 5 5 100 3. .. 1 2x , x [a, b] f [a, b] 2 1x x 2 1f (x ) f (x ) f [a, b] 2 1x x 2 1f (x ) f (x ) ** 2 .6 4. y f (x)

5 326 f (x) 2x 3 f (3x 1) f ( ) 2( ) 3 .. f (3x 1) 2(3x 1) 3 6x 5 f (3x 1) 6x 5 f (x)

A 3x 1 A 1x 3 f (3x 1) 6x 5 A 1f (A) 6( ) 5 2A 33 A x .. f (x) 2x 3 f (3x 1) 6x 5 f (2) 2 3x 1 .. x 1 x 1 f (3x 1) 6x 5 f (2) 6(1) 5 1 f (x) 2x 3 f (3x 1) f (x) f (3x 1) 2(3x 1) 3 6x 5 x f (x) f (x) 3f (x) 2x 3 x 2 f (x) 3f (3x 1) 6( ) 5 3 f (x) 42 5. r 1r 1r {(y, x) | (x, y) r } 1r r ( x y) r {(3, 7),(4, 8),(5, 9),(6, 7)} 1r {(7, 3),(8, 4),(9,5),(7,6)} r {(x, y) y 2x 3 }| 1r {(x, y) x 2y 3 }| y 1 312 2r {(x, y) y x }| 1r {(x, y) y 0.5 x 1.5 }| 1r {(y, x) y 2x 3 }| ( x y )

327

6. 1 rrD R 1 rrR D ( 1 ffD R 1 ffR D )

r {(3, 7),(4, 8),(5, 9),(6, 7)} 1rD 1rR 1 rrD R 1rD {7, 8, 9} 1 rrR D 1rR {3, 4,5,6}

r {(x, y) y x 2 }| 1rR x 2> 1 rrR D 1rR [2, ) 7. () ( 1f ) f f f {(3, 7),(4, 8),(5, 9),(6, 7)} ( 1-1 (3,7) (6,7)) 1f {(7, 3),(8, 4),(9,5),(7,6)} ( (7,3) (7,6)) 1f ( ) f ( )

f (x) 2x 3 1f (x) f (x) 2x 3 1f (2x 3) x A 2x 3 A 3x 2 1f (2x 3) x .. 1 A 3f (A) 0.5 A 1.52 1f (x) 0.5 x 1.5 x y .. f (x) y 2x 3 x 2y 3 y 0.5 x 1.5 .. 1f (x) 0.5 x 1.5

5 328 f (x) 2x 3 1f (5) f (x) 2x 3 1f (2x 3) x 2x 3 5 x 4 x 4 1f (2x 3) x .. 1f (5) 4

f (x 1) 4x 3 1f (x) f (x 1) 4x 3 1f (4x 3) x 1 A 4x 3 A 3x 4 1f (4x 3) x 1 .. 1 A 3f (A) 1 0.25 A 0.254 1f (x) 0.25 x 0.25

f (x 1) 4x 3 1f (5) f (x 1) 4x 3 1f (4x 3) x 1 4x 3 5 x 2 x 2 1f (4x 3) x 1 .. 1f (5) 1 1 1(f ) f 8. 1r ( 1f ) r ( f) y x ( x y) r 1r

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y

x

y

x r

r-1

(-3,-1)

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329

r 1r r 1r 1r r 1 4 21 9(3)4 4 1r r 3 4 274 9. () g(f (x)) (g f)(x) g f fR gD f g f {(0, 3),(1, 4),(2,6)} g {(3, 7),(4, 8),(5, 9),(6, 7)} g f g f fR gD ( 3, 4, 6) g(f (0)) g(3) 7 , g(f (1)) g(4) 8 , g(f (2)) g(6) 7 (g f)(0) 7 , (g f)(1) 8 , (g f)(2) 7 g f {(0, 7),(1, 8),(2, 7)}

0 1 2

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5 330 f (x) 2x 3 g(x) 3x 4 (g f)(x) (g f)(x) g(f (x)) g(2x 3) 3(2x 3) 4 6x 5 (g f)(x) 6x 5 g(x) 3x 4 f (x) (g f)(x) g(f (x)) 3(f (x)) 4 (g f)(x) 6x 5 3(f (x)) 4 6x 5 .. f (x) 2x 3 (g f)(x) 6x 5 g(x) 3x 4 f (2) (g f)(2) g(f (2)) 3(f (2)) 4 (g f)(2) 6(2) 5 7 3(f (2)) 4 7 .. f (2) 1 (g f)(x) 6x 5 f (x) 2x 3 g(x) (g f)(x) g(f (x)) g(2x 3) (g f)(x) 6x 5 g(2x 3) 6x 5 ( A 2x 3 ) .. g(x) 3x 4 (g f)(x) 6x 5 f (x) 2x 3 g(1) (g f)(x) g(f (x)) 6x 5 g(1) f (x) 1 .. 2x 3 1 x 2 x 2 (g f)(x) g(f (x)) 6x 5 (g f)(2) g(1) 6(2) 5 7 10. 1 1 1(f g) g f

331

3(g f)(x) x 7 f (3x 1) x 2 1(f g) ( 2) 1 1 1(f g) ( 2) (g f )( 2) 1 1(g (f ( 2)) f (3x 1) x 2 1f (x 2) 3x 1 x 0 1f ( 2) 3(0) 1 1 1 1 1(g (f ( 2)) (g (1)) 3(g f)(x) g(f (x)) x 7 .. 1 3g (x 7) f (x) x 2 1g (1) f (2) f (2) f (3x 1) x 2 .. x 13 513 3f (2) 2 .. 1 53(f g) ( 2) 11. f gR D gof fD D f gR D gof fD D (g f)(x) (1) g f(x) ( f(x) x ) .. (2) gofD g(f(x)) f(x) x () (3) gofR f f(x) g(f(x)) g(f(x)) () 21g(x) 1 x 2f (x) 4 x gofD gofR 21(g f)(x) 1 f(x) ; .. 21 f(x) 0 1 f (x) 1 x 21 4 x 1 20 4 x 1 < 4 1 23 x 4 < x gofD [ 2, 3) ( 3, 2]

5 332 ; f(x) 2x x 0 >R 24 x 4 < .. 20 4 x 2< < f(x) [0,2] f g 20 f (x) 2 0 f (x) 4< < < < 23 1 f (x) 1 < < .. 20 1 f (x) 1< < .. 211 1 f (x) < .. gofR [1, ) ** g f

2g(x) x 6 2f (x) 3 x gofD

g f 22 2 2(g f)(x) 3 x 6 3 x 6 9 x 29 x 0 > x [ 3, 3] ! (g f)(2) f (2) .. g f 2 g f x 12. (f g)(x) f (x) g(x) , , ,

1 xf (x) x 2 (f g)(x) x 2 (f g)(2) (fg)(2)

f(2) xf ( ) xx 2 x 2x 2 x 4 .. f(2) 4 g(2) (f g)(x) x 2 x 2 f(g(2)) 4 .. 1f (4) g(2) () 4g(2) 24 2 .. (f g)(2) f(2) g(2) 4 2 6 (fg)(2) f(2) g(2) 4 2 8

333

13. f g f g f gD D D ( , , , ) g(x) 0

x 3f (x) x 1 2

2x 3x 2g(x) x 9 fgD

fD {1} R gD {3, 3} R fg f gD D D {1, 3, 3} R ** fg .. 2 2x 3x 2x 3 x 2(fg)(x) x 1 x 3x 9 { 3} R ! f gD f gD fgD .. f / gD g(x) 0 x 1, 2 f / gD {1, 3, 3, 1, 2} R

.. 5

1. [ / 2522] x y x y a a 0

1. 2. 3. 4. 5. 2. [ / 2533] 2f {(x, y) | y 4 x } R R g {(x, y) | y x 2 } R R h {(x, y) | y x 2 0 R R x 0 }< 1. (f g) h 2. (f g) h 3. (f h) g 4. (f g) h

1. 1 2. 2

y

x O a

-a

a

-a

y

x O a

-a

a

-a

y

x O a

-a

a

-a

y

x O a

-a

a

-a

y

x O a

a

-a

-a

5 336 3. [ / 2521] 2 21r {(x, y) | y 2 x } R R 1 2r r 1. 2. 3. 4. 5. 1. 4. 4. [ / 2537] R 2 2A {(x, y) | x y 16 }

337

6. [ / 2528] 3 y x x y 3 1. {(x, y) | x 0 >R R , 0 y 3< < y x 3 }> 2. {(x, y) | y 0 >R R , 0 x 3< < x y 3 }< 3. {(x, y) | 3 y 0 <

5 338

-4 -2 3 3

-4 -2 3 -2 3

10. [ / 2527] :( 2A {(x, y) | y x 3 } R R B {(x, y) | 2y 3(x 1) 4x } R R 1. ( 1, 2) A ' B 2. 3 3( , )2 4 A ' B 3. 3 3( , )2 4 A ' B 4. ( 1, 2) A ' B 11. [ / ..2541] S { x | x x 5 }< 3 2 24x x 4x af (x) x bx 4 a S, b S (a, b) S S f (1) 0 1. 15 2. 18 3. 20 4. 22 12. [ / 2540] :( A {0, 1, 2, 3} P(A) A r A P(A) r {(a, B) | a 2, a B > a 1 B } r 13. [ / ..2543] S 2x 8x 20< A { x S | x } B { x S | x } (A B) (B A) 1. 11 2. 15 3. 21 4. 23 14. [ / 2521] ; 1r {(x, y) | y 3 x } R R , 2

2r {(x, y) | y }1 x 3 R R

A B 1r 2r A B 1. 2. 3. 4. 5. 1. 4.

339

15. [ / 2528] ; 2r {(x, y) | x y 6y 10 } R R 1. 1rD R 1rR { y | y 0 } > 2. 1rD { x | x 0 } > 1rR R 3. 1rD R 1rR { x | x 1} > 4. 1rD { y | y 1} > 1rR R 16. [ / 2528] 2f(x) x 1 1. fD { x | x 1} fR { x | 2 x 0 } < 2. fD { x | x 1 x 1} fR { x | 2 x 0 } < < 3. fD { x | x 1} fR { x | x 2 < x 0 } 4. fD { x | x 1 x 1} fR { x | x 2 < x 0 } 17. [ / ..2544] 3 2 2 2r {(x, y) | 2x 3xy x y 0 } R R 1r 1. 1 1( , ]3 2 2.

1 1[ , )2 3

3. 1 1( , ) ( , )3 3 4. ( , ) 18. [ / ..2543] 2r {(x, y) | y 9 x } 21s {(x, y) | y }x 9 . 1r sD R . 1r sR D (0, ) 1. . . 2. . . 3. . . 4. . . 19. [ / 2529] 2 21r {(x, y) | x y 1} R R 2 2 1r {(x, y) | y 1}x 1 R R A 1r B 2r A B 1. [0, 1] { 1} 2. (0, 1] { 1} 3. (0, 1] 4. { 1} 10. 3 11. 3 12. 12 13. 2 14. 3 15. 3 16. 4 17. 1 18. 2 19. 2

5 340 20. [ / 2541] 24r {(x, y) | y 2 }(x 1) 4 R R r 1. ( , 2) [3, ) 2. ( , 2) (3, ) 3. ( , 2] [3, ) 4. ( , 2] (3, ) 21. [ / 2535] R 2 2r {(x, y) | 9x 4y 18x 16y 11 0 } R R r rD R 1. [ 1, 3] 2. [ 5, 1] 3. [ 1, 1] 4. [ 5, 3] 22. [ / ..2542] r 221 xr {(x, y) | y }1 x 1. 1r rD [ 1, 1], D [ 1, 1] 2. 1r rD [ 1, 1], D [0, 1] 3. 1r rD [0, 1], D [ 1, 1] 4. 1r rD [0, 1], D [0, 1] 23. [ / ..2546] 2x 4r {(x, y) | y }x 2 . r4 R . 1rR [0, 4) (4, ) 1. . . 2. . . 3. . . 4. . . 24. [ / 2532] 2 x , x [ 2, 3]f(x) x 5 , x (3, 8)

x 2 , x ( 2, 0]g(x) 4 x , x (0, 4]

A f B 1g A B' 1. ( 2, 0) [2,6] 2. [ 2, 0] (2,6) 3. [2, 6] 4. ( 2, 0)

341

25. [ / 2524] 2f {(x, y) | y x 2x 2 3 x 2 } < (1) fR { y | 3 y 6 } < < (2) fR { y | 1 y 6 } < (3) h f hD { x | 1 x 1} < h(x) f(x) 1h 1 1 1. (1) 2. (2) 3. (3) 4. (1) (3) 2 5. (2) (3) 2 26. [ / 2538] 2r {(x, y) | y x < y 2x }> 1r 1. [0, 2] 2. [0, 4] 3. ( , 0] [2, ) 4. ( , 0] [4, ) 27. [ / 2537] :( * 21r {(x, y) | x y 2 0 } < 22r {(x, y) | ln y x 0 } > 1 2(r r ) 1. [1, 2] 2. ( , 0] 3. 1( , 1] [ , 1]2 4.

1( , ] [1, 2]2 28. [ / ..2547] :( r {(x, y) | x y > 2 2y x 2x 3 } . rD [1, ) . rR ( , ) 1. . . 2. . . 3. . . 4. . . 29. [ / ..2546] :( r {(x, y) | 0 x, 0 y 5 < < < 2 2x y 2x 6y 8 } < . rD [0, 3] . 0 c (3, c) r c 5 1. . . 2. . . 3. . . 4. . . 20. 1 21. 3 22. 2 23. 3 24. 1 25. 4 26. 3 27. 4 28. 2 29. 4

5 342 30. [ / ..2544] 21yr {(x, y) | }x 1 . rD ( , 1) (1, ) . 1 1 xr {(x, y) | y }x 1. . . 2. . . 3. . . 4. . . 31. [ / 2523] ; A {1, 2, 3, 4} r A A 1. 1r {(x, y) A A | y x } 2. 22r {(x, y) A A | y x } 3. 3r {(1, 1),(2, 4),(4, 1)} 4. 4r {(1, 1),(2, 4),(3, 3),(4, 1)} 5. 5r {(1, 2),(2, 3),(3, 4),(4, 1)} 32. [ / 2532] ; * D {2,5,6, 7, 8} D 1. {(x, y) | y sin( (x 5))}6

2. {(x, y) | y x 2 } 3. 2{(x, y) | y x 4x } 4. {(x, y) | y x 4 } 33. [ / 2537] R I 2A { x | x 2 8 } I 1B { x | 1 0 }x R A B B 1. {( 3, 1),( 2, 2),( 1, 3),(1, 4),(2,5)} 2. {( 3, 0),( 2, 1),(1, 1),(2, 2),(3, 3)} 3. {( 3, 1),(0, 2),(1, 1),(2, 3),(3, 4)} 4. {( 3, 1),( 2, 4),(1, 5),(2, 2),(3, 1)}

343

34. [ / 2523] ; A B f {(x, y) A B | y x 2 } 1. f x y 2. f 1 1 x 3. f A B A B 4. 2. 3. f 5. 35. [ / 2527] 1. {(x, y) A A | y x } , A {1, 2, 3} 2. 2{(x, y) | x y 1} R R 3. {(x, y) | y x 2 } R R 4. {(x, y) B B | y x 2 } , B { 2, 1, 0, 1, 2} 36. [ / 2522] ; f 1 1 A B 1. A B 2. A B 3. B A 4. A B 5. A B 37. [ / 2526] ; A {1, 2} 1. A A 4 2. A A 4 3. A A 1 4. A A 38. [ / 2539] ; A 10 A A A 1. 1002 2. 10002 3. 2100 4. 21000 30. 2 31. 3, 4 32. 4 33. 4 34. 2 35. 3 36. 1 37. 2 38. 2

5 344 39. [ / 2520] ; A {1, 2, 3} , B {2, 3, 4} 1 1 A B 1. {(1, 3),(2, 4),(3, 3)} 2. {(2, 2),(3, 3),(4, 1)} 3. {(1, 1),(2, 2),(3, 3)} 4. {(1, 2),(3, 3),(2, 3)} 5. 1. 4. 40. [ / 2540] ; A 8 B 6 A B 3 (B A) (A B) 1. 3 2. 5 3. 10 4. 20 41. [ / ..2542] A {1, 2, 3} B {3, 4} S { f : A B A B | f } S 1. 120 2. 240 3. 360 4. 480 42. [ / ..2541] A {1, 2, 3} B {a, b, c, d} { f : A B | f 1 1} 1. 40 2. 34 3. 30 4. 24 43. [ / 2539] A {1, 2, 3, 4,5,6, 7} B {a, b} A B 1. 14 2. 63 3. 126 4. 252 44. [ / ..2543] A {1, 2, 3, 4,5} B {a, b} S { f | f : A B } S 1. 22 2. 25 3. 27 4. 30 45. [ / 2540] A {1, 2, 3, 4,5} S f f : A A 1 1 f(1) 3 S 1. 40 2. 48 3. 56 4. 72

345

46. [ / 2536] A { 2, 1, 0, 1, 2} f : A A f(x) 0 x 0 f(x) 0 x 0 1. 160 2. 80 3. 64 4. 16 47. [ / ..2544] A {1, 2, 3, 4} S { f : A A | f (x) x 1 < x A } S 48. [ / 2531] A {1, 2, 3, 4,5} f : A A x A , f(x) x f(x) 3 1. 24 2. 29 3. 72 4. 120 49. [ / ..2544] A, B F A {1, 2, 3, 4,5,6} B {{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}} F {f : B A | f (x) x x B } F 1. 24 2. 60 3. 100 4. 120 50. [ / 2535 / ..2546] :( A {1, 2} B {1, 2, 3, ..., 10} 1 1{ f | f : A B x A 1 f (x) x } 1. 16 2. 17 3. 18 4. 19 51. [ / ..2548] :( A {1, 2, 3, 4} B {1, 2, 3, 4,5} f A B f (1) 2 f (2) m m f 1. 75 2. 150 3. 425 4. 500 52. [ / ..2541] :( A {1, 2, 3, 4,5,6} B {1, 2, 3} f : A B f (1) 1 f (2) 2 f (3) 3 1. 530 2. 612 3. 702 4. 814 39. 5 40. 41. 3 42. 1 43. 3 44. 4 45. 2 46. 2 47. 96 48. 3 49. 4 50. 2 51. 3 52. 3

5 346 53. [ / 2534] ; * a, b 2f(x) a sin x bx cos x x x R f(2) 3 f( 2) 1. 3 2. 1 3. 1 4. 5 54. [ / 2525] 1. r {(x, y) | x , y R R x 3y }2x 1 1r {(x, y) | x , y R R x 3y }1 2x 2. f(x) x 5 2x 25g(x) x 5 f g 3. 2f(x) x 4 , x 2 > 2g(x) x 2x 3 22f x 4( )(x) , x 2g x 2x 3 > 4. f(x) x 3 x 3 x 3g(x)

3 x x 3

> f g 55. [ / 2539] :( 22x x 0g(x) x x 0

>

x g( x x ) 1. x( x x ) 2. x( x x ) 3. 2 x( x x ) 4. 2 x( x x ) 56. [ / 2541] :( 2

2 , x 1f(x) (x 1) , 1 x 2

(x 1), x 2

f( x ) 4 0 1. ( 3,5) 2. ( 6, 1) 3. ( 5, 4) 4. (1, 6) 57. [ / 2533] :( f( x ) , x 3f(x) f(f(x 1)), 3 x 0

x 1 , x 0

h 5 f(3 h) f( h)f( 2)

347

58. [ / 2536] :( R f : R R

1 x, x 0f(1 x) 0 , x 0

1 x , x 0

2x y f(y x ) x y ( 2) f(3) 1. ( 4, 2] 2. ( 2, 2] 3. (2, 4] 4. (4,6) 59. [ / ..2545] 21f (x) 36 4x3 A { x | x [ 3, 3] f (x) {0, 1, 2, 3}} A 60. [ / ..2547] :( 1 x , x [0, 1]f (x) 1 x 1 , x (1, )

. 1f (x) f (x) x (1, ) . a 0> 2 1f (a) a 1. . . 2. . . 3. . . 4. . . 61. [ / 2520] 1. x 2F(x) x 4x 4 F(x) x 2. x y x y 0 x y x y < 3. r A B r A B 4. A B A B 7 5. f g g f 62. [ / 2528] * 1. A f : A B 1 1 B 2. f 1 1 1 1f f f f 3. 2g(x) x x 0> 1 1 4. xf(x) e 1 1 53. 4 54. 4 55. 4 56. 3 57. 1 58. 1 59. 5 60. 3 61. 1 62. 2

5 348 63. [ / 2540] ; I f {(x, y) | x 2y 12 x, y } I f f 1. {(8, 5),(4, 4)} 2. {(5, 8),(4, 4)} 3. {(2, 2),(4, 4)} 4. {(6, 3),(4, 4)} 64. [ / 2528] f(x) x 1 , g(x) x 1h(x) x 1. f h 2. h g 3. g f 4. h f 65. [ / 2526] R f g

R R u v R u v a R 1. g(u(f(a))) 2. u(g(v(a))) 3. f(u(a, a)) 4. g f u 66. [ / 2534] ; f g R R f(x) 1 x 1g(x) f(x) (g f)(x) 1. 1 x 2. 2 x 3. 11 x 4.

12 x

67. [ / 2535] ; f g 2f {(x, y) | x 2y 5 } R R g {(x, y) | 2x y 3 } R R g f 1. 2{(x, y) | x y 2 } R R 2. 2{(x, y) | x 4y 11} R R 3. 2{(x, y) | x 4x 2y 5 } R R 4. 2{(x, y) | 4x 12x 2y 4 0 } R R 68. [ / 2525] ; 2f(x) x 6 , 1g(x) x 3 1. 2

6(f g)(x) (x 3) 2. 21(g f)(x) x 3

3. fR R 4. gR {3} { x | x , x 3 } R R 5. gR {0} { x | x , x 0 } R R

349

69. [ / 2521] ; f(x) 3x , 2g(x) x 1 2x 2 x 0h(x)

2x 3 x 0 >

f (h g)(1) 1. 3 2. 5 3. 6 4. 10 70. [ / 2541] 1f(x) x 1 x 1 I g (f f)(f I) g(x) 1. 1 2.

2(x 1)(x 2)

3.

2(x 1) x(x 2) 4.

2(x 1) x(x 2)

71. [ / 2533] f g R R xf(x) 2 21 x 1g(x) x 20 x 1

<

n (g f)(n) 0 n 1. 1 2. 2 3. 3 4. 4 72. [ / 2524] 2f {(x, y) | y x 1 } 21g {(x, y) | y }1 x 1. 1 2

y(f g) {(x, y) | x }1 y

2. 2xf g {(x, y) | y }

1 x

3. 1(g f)(x) x 4. 1 2

21g {(x, y) | x 1 }y

5. g/ fD [1, ) 73. [ / 2523] (1) 3 3f(x) a x x 0 f f f f (x) x (2) 2xf(x) x g(x) x x R f g (3) f(x) x x R 1f 1f f 1. 2. 2 (1) (2) 3. 2 (1) (3) 4. 2 (2) (3) 5. (1) (3) 63. 1 64. 1 65. 3 66. 4 67. 1 68. 4 69. 1 70. 4 71. 3 72. 1 73. 3

5 350 74. [ / 2526] f g R c R , c 0 f(x) x c g(x) c x 1. f(x) g(x) x x R 2. f g 1 1 3. f gD R 4. f g 75. [ / 2526] f g x 3f(x) 2

x R g(x) x x R x 3 1 1[(f g)(x) (f g)(2)] /( x 2) 1. 2 2. 6 3. 1 4. 12

76. [ / 2520] A {1, 2, 3, 4} , B {1, 3, 4, 5} f {(1, 1),(2, 3),(3, 4),(4,5)} 1. 1f f A B 2. f f A A 3. 1f f A A 4. 1f f B A 5. 1. 4. 77. [ / ..2542] f (x) 4x 2g(x) x 1 x (f g)(x) (g f)(x) 78. [ / ..2545] :( 2

2 , x 1f (x) (x 1) , 1 x 2

x 1 , x 2

g(x) f (x) 2 k g(k) 5 (g f)(k) 1. 5 2. 6 3. 7 4. 8 79. [ / 2530] ; f {(x, y) | y 3x 2 } R R g {(x, y) | y 2x 7 } R R 1 1(g f )(2) 1. 176 2.

72 3.

16 4.

72

351

80. [ / 2540] 2f(x) x 2x 1 3 2g(x) x 3x 3x 9 1(f g )(7) 1. 2 2. 1 3. 1 4. 2 81. [ / ..2548] * 1f (x) det

1 11

1 x x 1

1. f 1 1 1 11f (x) det1 11

1 x

x 0 , x 1

2. f 1 1 1 1 1f (x) det 111 x

x 1 3. f 1 1 x 1det 0

1 11

1 x

4. f 1 1 21(f f)(x) det1 11

1 x x 1

82. [ / ..2544] xf (x) , x 11 x xg(x) , x 11 x 1. 1(f g) (x) x , x 1 2. 1 1(f g )(x) x , x 1 3. 1 x(f g)(x) , x 11 2x

4. 1 x(g f)(x) , x 11 2x

83. [ / 2539] :( 2r {(x, y) | x y 1} R R s {(x, y) | x y } R R 1. 1 1 1 1r s s r 2. 1 1 1r s r 3. 1 1 1r r r 4. 1 1 1s s s 84. [ / 2529] :( xf(x) 1 x 1f (x) 1. x1 x 2.

x1 x 3.

x1 x 4.

x1 x

74. 2 75. 1 76. 3 77. 0.2 78. 3 79. 1 80. 2 81. 2 82. 3 83. 3 84. 2

5 352 85. [ / 2529] :( r {(x, y) | y x x } R R r 1. 1 x , x 0r {(x, y) | y }x , x 0

>

R R

2. 1 x , x 0r {(x, y) | y }x, x 0

>R R

3. 1 x , x 0r {(x, y) | y }x , x 0

>R R

4. 1 x , x 0r {(x, y) | y }x , x 0

>R R

86. [ / ..2542] f (x) x 21A { x | f (x) [f (x)] 2 } R . 2x A [ x x 6 0 ] . 2x A [ x 2x 3 0 ] 1. . . 2. . . 3. . . 4. . . 87. [ / ..2546] :( 2f (x) (x 1) x 1< g(x) 1 x x 1< . 1 xf (x) 1 x 0< . 1 1 1 3(g f )( )4 4 1. . . 2. . . 3. . . 4. . . 88. [ / ..2545] :( f (x) x x 0> x , 0 x 1g(x) x 1 , 1 x

g g f 1. g(x) 1 x ( , 1) [1, )

2. 1g(x) f (x) x ( , 1) [1, ) 3.

2

2(x 1) , x 1g(x) (x 1) , x 1

>

4. 3g(x) x x ( , 1) [1, ) 90. [ / ..2547] :( * () 2xf (x) 1 x x ( 1, 1) .

21

1 1 4x , x 0f (x) 2x 0 , x 0

. f ( 1, 1) 1. . . 2. . . 3. . . 4. . . 91. [ / ..2546] :( * a 0 x3a(10 ) , x 1g(x) x 1 , x 1

>

gR ( 2.5, ) . 1g (a 1) log 2 . 1 3log(4|x|) , x 0g (x) x 1 , x 0

>

1. . . 2. . . 3. . . 4. . . 92. [ / ..2545] :( k r {(x, y) | x k x y k y } R R . k 1 r . k 1 r 1. . . 2. . . 3. . . 4. . .

85. 2 86. 3 87. 1 88. 1 89. 4 90. 3 91. 4 92. 2

5 354 93. [ / 2531] :( 2f(x) x A R , R 1f (A) { x | f(x) A } 1. 1f ([ 25, 0]) {0} 2. 1f ([ 1, 1]) [ 1, 1] 3. 1f ([0, 1]) [ 1, 1] 4. 1f ([4, 9]) [2, 3] 94. [ / 2521] ; 1 1f( x 1) x 12 2 1f (2) 1. 6 2. 4 3. 2 4. 0 5. 1. 4. 95. [ / ..2543] (f g)(x) 3x 14 1f ( x 2) x 23 1(g f)(x) 1. 3x 4 2. 3x 6 3. 3x 8 4. 3x 10 96. [ / 2535] 1 xf (x) x 2 (f g)(x 2) 3x 6 g(2) 1. 56 2.

32 3.

125 4.

2411

97. [ / 2537] f(x) x 1 1 2(g f )(x) 4x 1 g(x) 0 1. [ 4, 1] 2. [ 1, 0] 3. [0, 4] 4. [4,6] 98. [ / 2538] f g R R 3f(x) x 1 3 2(f g)(x) x 3x 3x 2 1(g f )( 7) 1. 1 2. 2 3. 1 4. 3 99. [ / ..2547] 2f (x) ax b g(x 1) 6x c a, b, c f (x) g(x) x 1, 2 (f g)(1) 8 1(f g )(16) 1. 319 2.

619 3. 10 4. 20

355

100. [ / 2536] * R , f : R R g: R R 2x 1f(x) a g(x) bx 5 1(f g )( 2) 27 (f g)(0) 15 3f( 1) 4g(2) 1. 35 2. 33 3. 37 4. 39 101. [ / ..2546] a 0 2f (x) ax , x 0 > 3g(x) x 1(f g)(4) 2 11f (64)g (64)

102. [ / ..2546] f, g fD [0, ) 1 2f (x) x , x 0> 21g (x) (f (x)) 1 , x 0> a 0 f (a) g(a) 19 1 1f (a) g (a) 1. 273 2. 274 3. 513 4. 514 103. [ / 2527] (f g)(x) x , 1g(x) x 33 g(h(x)) 2x 1 1. 1f(x) g (x) h(x) 6x 4 2. 1f(x) g (x) h(x) 6x 6 3. f(x) 3x 9 h(x) 6x 4 4. f(x) 3x 9 h(x) 6x 6 104. [ / ..2544] f (x 1) 3x 2 f (x) g(3x 1) 2x 8 f (0) 1 1g (f (2)) 1. 1 2. 0 3. 1 4. 2 105. [ / ..2546] f g f (x) 0 x 2(g f)(x) 2 [f (x)] 2 f (x) 4 1 x 1g (x) 3 . g f . f (100) g(100) 300 1. . . 2. . . 3. . . 4. . . 93. 4 94. 2 95. 2 96. 2 97. 3 98. 1 99. 4 100. 3 101. 0.5 102. 1 103. 2 104. 1 105. 2

5 356 106. [ / 2533] f : R R R g: R R 2(g f)(x) 3(f(x)) 2f(x) 1 2g(x) x x 2 1. (g f)(1) 2 2. (g f)(1) 2 3. g( )(1) 2f 4. (g f)(1) 2 107. [ / ..2545] f, g 1f (g(x)) x 2 x R . f (2x) g(2(x 1)) x R . 1g (f (x)) R 1. . . 2. . . 3. . . 4. . . 108. [ / 2538] ; f(x) (3 x)(2 x) 1g(x) x 3 fg 1. 2. ( , 2] 3. ( 3, 2) 4. ( 3, 2] 109. [ / 2534] ; 21f(x) 3x 12 , g(x) 3 x , 2h(x) x 5x 6 gu h f uR D 1. ( 4, 1) 2. ( 1, 5) 3. (2, 7) 4. (4, 8) 110. [ / 2529] 1. f A B g B C g f A C 2. f(x) x 2g(x) x gof fogD D 3. 2f(x) x 4x 3 g(x) x fog gofR R 4. 2x 1f(x) 3 3 2g(x) x 3x 3x 1 1 1 1f g (1) g f (1) 111. [ / 2527] 2f(x) x 25 , g(x) 2x 2h(x) f(x) g(x) (x 25)(2x) (g h)(x) 1. { x | x 5 }> 2. { x | 5 x 0 < < x 5 }> 3. { x | x 5< x 5 }> 4. { x | x 0 }>

357

112. [ / 2541] 2f {(x, y) | y x 2x 1} R R 2

1g {(x, y) | y }1 x R R

h (g f) fg h 1. { x | x 1} 2. { x | x(x 2) 0 } 3. 2{ x | (x 1)(x 2) 0 } 4. 2{ x | x(x 1)(x 2) 0 } 113. [ / ..2544] 2f (x) 4 x 21g(x) 9 x gofR 1. 12 2.

14 3.

18 4.

114

114. [ / 2531] * xf(x) 10 , 2g(x) 1 x r {(x, y) | y (f g)(x)} R R 1. rD [ 1, 1] , rR [0, 1] 2. rD [0, 1] , rR [1, 10] 3. rD [ 1, 1] , rR [1, 10] 4. f g 115. [ / ..2547] * xf (x) 10 2g(x) 100 3x gofR 116. [ / ..2543] 2f (x) (x 1) g(x) x 1

fog gofD R ' 1. [0, 1) 2. [0, 2) 3. [1, ) 4. [2, ) 117. [ / ..2542] xf (x) 1 x 2g(x) x 1 gofA D gB D (A B') 1. { 1, 1} R 2. ( 1, ) 3. 1( , 1) (1, )2 4. ( 1, 1) (1, )

106. 4 107. 1 108. 4 109. 2 110. 1 111. 1 112. 4 113. 3 114. 3 115. 9 116. 1 117. 4

5 358 118. [ / ..2544] * xf (x) 2 sin 2 2g(x) x 1 f g gof(R D ) R 1. { 1, 1} 2. { 2, 2} 3. [2, 3] [1, 2] 4. [ 2, 1] ( 3, 2] 119. [ / ..2545] :( f (x) 5 g(x) g(x) 5 2x fogD [a, b] 4(a b) 1. 15 2. 20 3. 25 4. 30 120. [ / 2531] :( A f : A A f 1. f(n) n , n N , N 2. f(n) 2n , n N , N

3. n ,f(n) n 1 ,

n n 4.

(n 1)2n2

f(n),,

n n

121. [ / 2529] :( { x | x 0 } >R R N {0, 1, 2, 3, ...} f : R R f(x) 2x g(0) 1 , g(n 1) f(g(n)) , n N 1. g N R 2. f g N R 3. g fogR R 4. g(n) 2 , n N 122. [ / ..2541] I f g f (x) 2x g(x) x 1 x I (f g) f 1. x{ x | 2 I } 2. x{ x | 2 I } 3. 4.

359

123. [ / 2536] I f : I I g: I I f(x) 2x x I

0g(x) x/2

x x F: I I F g f f F 1. 2. 3. 4. 124. [ / ..2545] I f, g I I f(x) 2x x/2 , xg(x)

x , x

g f f I I 1. 2. 3. 4. 125. [ / ..2543] :( f, g : R R xf (x) x 1 g(x) x ( g(1.01) 2 , g( 6) 6 , g( 7.99) 7 ) F(x) (f g)(x) G(x) (g f)(x) 1. FD ( , ) 2. FR (0, 1) 3. G(x) 1 x 0 4. G(x) 0 x 0 118. 4 119. 120. 4 121. 3 122. 1 123. 4 124. 1 125. 2

5 360

x 2 O

y

(2,4) y = 2x

x = 2

A x 2 O

y

y = 4

x = 2

4 y/2 = x

B

x 3 O

y

3 y = x 3

(1) .. x 0, y 0> > x y a x 0, y 0> x y a x 0, y 0 > x y a x 0, y 0 x y a 4 1.

(2) f; 2y 4 x

2 2y 4 x 2 2y x 4 ( y ) 2 g h 1. (f g) {(2, 0)} (f g) h .. 2. (f g) h {(2, 0)} h .. 2. 3. (f h) {( 2, 0)} (f h) g {( 2, 0)} g .. g ( 2, 4) 4. (f g) h .. x 2 y

(3) 1r 2 2x y 2 < 2 (0,0) 2r y x> (0,0) 1 2r r 4.

(4) 1. A (B C) A C {(0, 4)} 2. A B 3. (B A) C C B A (0, 4) 3. 4. C B

(5) A B 1.

(6) 1.

2 -2 -2

f g h

B

0 4 -4

4 A 4

4 -4

-4

C

4 -4

4

361

1

-1 x y 1

2y x

(7) 1. .. 2. .. 1 (0,2) ( 2 ) {(0,2)} 3. .. (0.49, 0.50) ( 0.50 0.49 0.50 0.49 ) 4. .. (1, 2)

(8) 2 2x y 04 9 2 29x 4y 0 (3x 2y)(3x 2y) 0 3x 2y 3x 2y ( 3y x2 3y x2 )

2. 2 3r {(x, y) | y x }2 2r r R R .. 2.

1. 1r 2y x3 3y x2 1 2r {(x, y) | y x }3 11r r R R .. 1. 3. 3r 13r 3 3r {(x, y) | y x , 1 x 1}2 < < 4. .. (0,0) 5. .. 4r 323

2

x x 0y x x 0

>

(9) 1r x 3> 2r x 0< 1 2r r 2 1r r ( 1. 2. ) 11r R 12r x 3> 3. 4. 1(4, 0) r 12(4, 0) r 3. .. 4.

(10) A B A ' B A ' B A ' B 1. 2. .. A ' B (0, 3) 4. 3. 3. 2y x 3 2y x 3 .. 3 3( , )2 4

-3 -3/2 3

B

A 33

5 362

+ + 2 0

(11) S { 5, 4, 3, 2, 1, 0, 1,2, 3, 4,5} f(1) 0 2 21 1 4 a a 4 01 b 4 b 5 a 2 -2 b -5 () a b S 2 10 .. 2 10 20

(12) r A P(A) .. a {0, 1,2, 3} B P(A) B {0, 1,2, 3} a 2 B {0} {1} {0, 1} .. 4 a 3 B {0} {1} {2} {0, 1} {0,2} {1,2} {0, 1,2} .. 8 (a,B) r 12

(13) S; 2x 8x 20 0 <

(x 10)(x 2) 0 < .. S [ 2, 10] A {2, 3,5, 7} B { 1, 1, 3,5, 7, 9}

n(A B) 4 6 24 (A B) (B A) .. A B {3,5, 7} A B B A 3 3 9 [ (3, 3) (3,5) (3, 7) (5, 3) (5,5) (5, 7) (7, 3) (7,5) (7, 7) ] 24 9 15

(14) r1D ; 3 x 0 x 3 r2D ; 1 x 3 0 x 3 1

x 2 4

A ( , 3] B { 2, 4} R A B 3.

(15) 2 2x (y 6y 9) 1 (y 3) 1 y x 1 .. 1rD R 1rR [1, ) 3.

(16) 2y x 1 x 1 0 x 1 x 1, 1 fD { x | x 1 x 1}

; 2x 1 y 2x 1y 2 1 0y > 2 y 0y >

y ( , 2] (0, ) fR { x | x 2 < x 0} 4.

(17) 1r r; 3 2 2 22x 3xy x y 0 2 2 3(3x 1)y x 2x 2 32 x 2xy 3x 1

13x 1 0 x 3 2 3x 2x 03x 1 >

2x (2x 1) 03x 1 < .. ( 1/3, 0] [0, 1/2]

1 1( , ]3 2

363

(18) rD 2 29 x 0 x 9 0 > < .. rD [ 3, 3]

rR 2x 0> 29 x 9 < 20 9 x 3 < < .. rR [0, 3]

1sR sD 2x 9 0 ( 0)

.. 1sR ( , 3) (3, ) 1sD sR 2x 9 0

221x 9 0 0

x 9

.. 1sD (0, ) . 1r sD R . . 1r sR D [0, ) .

(19) A; 2 2y 1 x 21 x 0 > 2x 1 1 x 1 < < < .. A [ 1, 1]

B; 2 1y 1 x 1 2 1x 1 y 1 2 1x 1y 1

1 1 0y 1 > y 0y 1 > y 0y 1 < .. B ( 1, 0]

A B { 1} (0, 1] 2.

(20) 24 2 y(x 1) 4

2 4(x 1) 42 y

4 4 02 y > 4 8 4y 02 y >

4(y 3) 0y 2 >

.. rR ( ,2) [3, )

(21) ..

2 29(x 2x 1) 4(y 4y 4) 11 9 16

2 2(x 1) (y 2) 14 9 ..

(1, 2) , b 2 , a 3 .. rD [1 2, 1 2] [ 1, 3] fR [ 2 3, 2 3] [ 5, 1] r fD R [ 1, 1]

(22) 221 xy 1 x 21 x 0 > ( 21 x 0 ) 2x 1 1 x 1 < < < rD [ 1, 1] ( 1 rrD R ) 22 2 2 2 221 xy y x y 1 x1 x

22 2 2 2 2

21 yx x y 1 y x 1 y

2 221 y 0 1 y 01 y > > 2y 1 1 y 1 < < < y 0> 1rD [0, 1] 2.

(23) . y 4 x ..

22x 44 4 x 8 x 4x 2

2x 4 x 4 0

x A 4A 4A 4 0 () x . . 1 rrR D x 2 0 () x 0> ()

x 2 x 4 .. [0, 4) (4, ) . 3.

5 364 (24) fR ; 2 x 3 < < 0 x 3< < 0 f(x) 6< < 3 x 8 2 f(x) 3 fR [0,6] ( 2, 3) ( 2,6] 1 ggD R ; 2 x 0 < 4 g(x) 2 < 0 x 4 < 0 4 x 4 < 0 g(x) 2< 1 ggD R ( 4, 2] [0,2) A B' A B ( 2, 0) [2,6] ( 1.)

(25) 2y x 2x 2 bx 12a fR [ 3,6] (1) , (2)

.. [ 1, 1) h 1h 11 (3) 4.

(26) r 1 rrR D ( , 0] [2, )

(27) 1r ; 2x (y 2) < 2r ; 2 0y x e 1 > .. 2y x 1 > 2x y 1 < 2y x 1 < 2x y 1 > 1 2r r .. A, B 3 1( , )2 2 1 2r r 1( , ] [1,2]2

(28) 2 2y x 2x 3

2 2 2 23 x 2x y 3 1 (x 2x 1) y 2 2(x 1) y 14 4

( 1, 0) a b 2 () x y> .. (3/2,3/2) rD [1, ) r 3R ( , ]2 . , .

(-1,-3)

(-3,1) (2,6)

-1

(0,0)

(2,4)

2y x y 2x

r1 2

r2 1 -1

A 1 -1

2

B

(1,0)

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

x=y

365

O

2

3

5 (3,5)

(4,0)

4

x+y-4=0

x-y+2=0

(29) 2 2x y 2x 6y 8 < 2 2(x 2x 1) (y 6y 9) 8 1 9 <

2 2(x 1) (y 3) 0 < 0 (x 1 y 3)(x 1 y 3) 0 <

(x y 2)(x y 4) 0 < 0 x, 0 y 5< < <

rD [0, 4] . (3, c) r c 5 .. . 4.

(30) . rD ; () 2 1 0x 1 >

1 0(x 1)(x 1) > rD ( , 1) (1, ) .

. 1 221 1r ; x y 1 xy 1 2 1 x1y 1 yx x

.. 1 xx . [ x x 0.5 ] 2.

(31) 1. 1r {(1, 1),(4,2)} 2. 2r {(1, 1),(2, 4)} 1, 2, 5. .. 3. 4. (1,1) (4,1)

(32) 11

1. 1 3{(2, 1),(5, 0),(6, ),(7, ),(8, 1)}2 2 2. {(2, 0),(5, 3),(6, 4),(7,5),(8,6)} 3. {(2, 4),(5,5),(6, 12),(7,21),(8, 32)} ( 11) 4. {(2,2),(5, 1),(6,2),(7, 3),(8, 0)} 11 4.

(33) A; 28 x 2 8

26 x 10 A { 3, 2, 1, 0, 1,2, 3}

B; x 1 0x B ( , 1) (0, ) A B { 3, 2, 1,2, 3}

A B B A B ( 2. 4.) B 4.

(34) 1. .. x y ! 2. () 3. y 2 B ( A B ..) 4. ( 3. )

(35) 1. {(1,2),(1,3),(2,3)} (3,1) (3,2) ..

2. 2y x 1 2 1y x 1(4, )2 1(4, )2 ..

x

y

2

5 366 3. x y 2 y x 2 x y 1 3. 4. {( 2, 0),( 1, 1),(0, 2),(1, 1),(2, 0)} (0, 2) (0,2) ( 1, 1) ( 1, 1)

(36) 11 A B .. n(A) n(B) 1.

(37) 1. n(A A) 2 2 4 .. A A 42 16 2. A A 2 2 4 3. f(x) .. 2 {(1,1),(2,1)} {(1,2),(2,2)} 4. .. {(1,1),(2,2)} {(1,2),(2,1)}

(38) n(A A) 10 10 100 n((A A) A) 100 10 1000 .. A A A n((A A) A) 10002 2

(39) 11 A B A B 1. 4. 5.

(40) n(B A) 6 3 3 n(A B) 8 3 5 .. 11 (B A) (A B) 5 4 3 60 ()

(41) n(A B) 4 n(A B) 3 2 6 .. 11 (A B) (A B) 6 5 4 3 360 S 360

(42)

(4 4 4) (4 3 2) 40 40

(43) A B A B

2 2 2 2 2 2 2 2 72 2 128 2 126

A B 2 a b

(44) ( 2 a b ) 2 2 2 2 2 2 30

(45) A 11

.. 1 2 4 5 2 4 3, 4, 5 3, 2, 1

2 4 3 2 1 48 S 48

367

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

8 1 8

(46) 2 2 1 2 1 2 1 2 0 5 () 1 2 2 1 2 2 2 1 2 2 5 2 2 80

(47) 1 2 1 2 2 3 1, 2 3 3 4 () 4 4 () 2 3 4 4 96

(48) 1 4 (2, 3, 4, 5) 2 3 (3, 4, 5) 3 3 (3, 4, 5) 4 2 (3, 5) 5 1 (3) 4 3 3 2 1 72

(49) {1} , {1,2} , {1,2, 3} , {1,2, 3, 4} 1, 2, 3, 4, 5, 6 (f(x) x)

{1} 5 (26) {1,2} 4 (36) {1,2, 3} 3 (46) {1,2, 3, 4} 2 (5,6) 5 4 3 2 120

(50) x f(x) x f(1) 1 f(2) 2 .. f(1) 1 ; 1 9 9 f(2) 2 ; 9 1 9 f(1) 1 f(2) 2 ;

1 1 1

( ) 9 9 1 17 ( 17)

(51)

.. A B A B f(1) 2 f(2) (5 5 5 5) (4 2 5 5) 425

(52)

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

702

(53) f(2) 3 3 a sin2 2b cos 2 4 .. a sin2 2b cos 2 1 f( 2) a sin( 2) 2b cos( 2) 4 a sin2 2b cos 2 4 ( 1) 4 5 sin( 2) sin2 cos( 2) cos 2

(54) 1. x 3y 2x 1 y = x y 3x 2y 1 .. 2xy x y 3 2xy y x 3

5 368 x 3y 2x 1

x 31 2x .. 1.

2. 2x 25g(x) x 5x 5 x 5 ( g(x) x 5 ) .. f g .. 2.

3. f( )(x)g x 3 x 3 ( 2x 2x 3 (x 3)(x 1) ) .. 3.

4. x 3 xx 3 3 x x 33

>

(55) x 0> g( x x) g(x x) g(0) 0 g x 0 g( x x) g( x x) g( 2x) 2x () 2g( x x) ( x x)

2 2x 2 x x x 22x 2 x x 2x(x x )

(56) x 1< .. 1 x 2 .. 2f( x ) ( x 1) 4

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

.. x 2> .. f( x ) x 1 4

x 3 x 3 3 () { 3, 3} 3.

(57) f(3 h) (3 h) 1 f( h) f( h ) f(h) h 1 f( 2) f(f( 1)) f(f(f(0))) f(f(1)) f(2) 3 (3 h 1) (h 1) 3 13 3

(58) ( 2) f(3) ( 2) f(1 ( 2))

( 2) ( 1 ( 2)) ( 2) (1) 2f(1 ( 2) ) f( 3) f(1 4)

1 4 3 1.

(59) ..

2 21y 36 4x 3y 36 4x3 2 2 2 29y 36 4x 4x 9y 36

2 2x y 19 4 y 0> ()

.. y {0, 1,2, 3} x 5 A 5 21 36 4x3 0, 1, 2, 3 x

(60) 0 x 1< < y 1 x x 1

2y 1 x 1 (y 1) x 1 y 1 .., (1, 1)

-3 0 3

1 2

1

1 O (1/2,1/2)

(2,2)

369

. .. (2,2) 1f(2) f (2) . .. 1 1 1f ( )2 2 1f (2) 2 ( 2 ) 3.

(61) 1. 2F(x) x 4x 4 x 2x 3x 4 0 (x 4)(x 1) 0 x 4 1 x 1. 2. x y x y < x, y () 3. ( A B) 4. .. n(A) 7 , n(B) 1 5. ( g f )

(62) 1. .. f : A B B ( f f B B ) 2. 1f f 1f f ( 1f f f .. 1f f 1f f) f {(1,2)} 1f {(2, 1)} 1f f {(1, 1)} 1f f {(2,2)} 3. 11 4. f(1) e f( 1) e

(63) f {(2,5),(4, 4),(6, 3),(8,2),(10, 1)} f f {(4, 4),(8,5)} 1.

(64) f fD R R g gD R [0, ) h hD R {0} R 1. .. f h h fR D 2. .. h g g hR D ( (h g)(0) ) 3. .. g f f gR D 4. .. h f f hR D

(65) f : R R g : R R u : R R R v : R R R 1. f(a) R .. u(f(a)) 2. v(a) 3. u(a, a) R .. f (u(a, a)) R 4. g f u ( (a,a) )

(66) 1g(x) f(x) 1 1(g f)(x) f(f(x)) 1 1 x 1 x 1 > 1 1(g f)(x) 1 1 x 2 x

(67) f; 25 xy 2 .. 25 xf(x) 2 g; y 2x 3 .. g(x) 2x 3

2 25 x 5 x(g f)(x) g( ) 2( ) 32 2

2 2(5 x ) 3 2 x 2y 2 x 2x y 2 1.

5 370 (68) 1. .. 21(f g)(x) 6(x 3) 2. .. 2 21 1(g f)(x) (x 6) 3 x 3 3. .. 2x 0> f(x) 6> fR [6, ) 4. g(x) x 3 .. x 3 0 x 3 5. ( 4.)

(69) 2g(1) 1 1 2 (h g)(1) h(g(1)) h(2) 2(2) 3 1 f (h g)(1) f(1) 3(1) 3

(70) I I(x) x () 1(f f)(x) 1( ) 1x 1

x 11 x 1

x 1x 2

1(f I)(x) xx 1 2x x 1x 1

2x 1 x x 1g(x) ( )( )x 2 x 1

2x x 1x 2

2(x 1) xx 2

(71) n

nf(n) 2 1 (g f)(n) g 2n(g f)(n) 2 20 (g f)(n) 0 .. 2n2 20 0 2n2 20 n n 3

(72) 1. 2. f g ; 21(f g)(x) 11 x

2 22 2 2

x1 1 x x1 x 1 x 1 x

2. 1. f g 2xy 1 x 1(f g) 2yx 1 y

3. 2 21 1(g f)(x) 1 (x 1) 2 x

4. 1 21g {(x, y) | x }1 y 1 2 21g {(x, y) | x }1 y 5. g/ f g fD D D f(x) 0 gD ; 2 2 g1 x 0 x 1 D ( 1, 1) fD ; 2 2x 1 0 x 1 > > fD ( , 1] [1, ) g fD D ( 5. )

(73) (1) 3 3f (x) a x 33 33f f (x) a (a x ) x x 3 3f f f (x) a x f f f f (x) x ( (1) ) (2) f(x) x x 0 .. x 0 f(x) ( (2) ) (3) y x y x .. 1f (x) x f(x) ( (3) ) 3.

371

(74) 1. x c> (f g)(x) x 2c x c (f g)(x) x (f g)(x) x ( 1. ) ( c 0) 2. ( 1.) x 2c x x c () f+g y x c 11 2. 3. f g f gD D D R ( 3. ) 4. ( 2.) f+g 11 f+g ()

(75) 1(f g)(3) ; g(3) 3 3 1 1(f g)(3) f (3) x 3 32 x 3 .. f(3) 3 1f (3) 3 .. 1(f g)(3) 3 1(f g)(2) ; g(2) 2 2 1 1(f g)(2) f (2) x 3 22 x 1 f(1) 2 1f (2) 1 .. 1(f g)(2) 1

3 1 23 2

(76) f {(1, 1),(2, 3),(3, 4),(4,5)} 1f {(1, 1),(3,2),(4, 3),(5, 4)} 1. 1f f {(1, 1),(2,2),(3, 3),(4, 4)} A B ( 2 B ) 2. f f {(1, 1),(2, 4),(3,5)} A A ( A ) 3. ( 1.) A A .. 3. 4. 1f f {(1, 1),(3, 3),(4, 4),(5,5)} B A ( 5 A )

(77) (f g)(x) (g f)(x) 2 24 x 1 4x 1

8 2 8(4x 1) 2(x 1)x 1 4x 1 30x 6 .. x 0.2

(78) g(k) 5 f(k) 2 5 f(k) 3

22 , k 1f(k) (k 1) , 1 k 2k 1 , k 2

k f(k)=2 3 k 0 f(0) 1 , k 1 f(1) 0 k k 2 f(2) 3 , k 3 f(3) 4 k f(k)>3 3 (g f)(3) g(4) f(4) 2 7

(79) 3x 2 2 4x 3 4f( ) 23 .. 1 4f (2) 3

42x 7 3 17x 6 17 4g( )6 3 .. 1 4 17g ( )3 6

1 1 1 4 17(g f )(2) g ( )3 6

(80) 1g (7) 3 2x 3x 3x 9 7

3(x 1) 8 7 3(x 1) 1 (x 1) 1 x 2

g( 2) 7 .. 1g (7) 2 1(f g )(7) f( 2) 1

5 372 (81) 1 1 (1 x) xf(x) 11 x 1 x 1 x

f : 1 1 ..

xy 1 x .. x y yx 1 y x xy y x y xy

x y1 x f 1 1 1 xf (x) 1 x 2. 1. 11 x 1 xf (x) 1 x x

2. 1 1 xf (x) 1 1 x 1 x

(82) 1f (x) 1g (x)

1 yf (x); x x xy y1 y

x xxy y x y x 1 1 x

1 xf (x) (x 1)1 x g(x)

.. 1g (x) f(x) 1. 1 1 1 1(f g) (x) (g f )(x) g (g(x)) x 2. 1 1 1(f g )(x) f (f(x)) x 3. 1 1

xx 1 x(f g)(x) f x1 x 1 1 x

x x1 x x 1 2x .. 3.

4. 1 1x

x 1 x(g f)(x) g x1 x 1 1 x

x x1 x x 1 2x

4. 1x 2

(83) r 2x y 1 y x 1 x 1>

1r 2y x 1

s x y y x x 0>

1s y x 1. 2.

1 1r s 2y x 1 2x 1 1 1 1r s r ( 2. )

1 1s r 2y x 1 2x 1 ( 1. ) 3. 1 1r r 2 2y (x 1) 1 1r 3. 4. 1 1s s y x x ( 4. )

(84) x 0> xy 1 x yx 1 y

x xy y xy 1 x

x 0 xy 1 x yx 1 y

x xy y xy 1 x x

1 ( x)

1 xf (x) 1 x

(85) r 22x x 0y x x 0

>

x 0> y 0> x 0 y 0 1r x x 0y x x 0

>

373

(86) f(x) x (y 0) > 1 2f (x) x (x 0) > 2A {x | x x 2 0} {1, 2 R } ( x 2 1f x 0> ) A {1} . 21 1 6 0 . 21 2 3 0 3.

(87) . 2y (x 1) x 1 y 0< < ....(1) 2 2x (y 1) x (y 1)

x y 1 y 1< ( x 1< ) x .. x y 1 y 1 x x 0< ( 1) x x

1f (x) 1 x x 0< .

. 1 21 1f ( ) (x 1)4 4 1 1x 1 x2 2

1 1 1f ( )4 2

1 1 1 3g ( ) 1 x x2 2 4 1 1 11 1 3(g f )( ) g ( )4 2 4 .

(88) . 1 2f (x) x x 0> 2 21 2 2x ; 0 x 1g f (x) x 1; x 1

x [0, ) .. [0, ) .

. 1 x ; 0 x 1g (x) x 1 ; x 2

( gR ) 1 x ;0 x 1f g (x) x 1 ; x 2

x [0, 1) [2, ) .. [0, 1) [2, ) .

(89) 1. (g f)(x) 1 2. 1(g f)(x) (f f)(x) x 3. 2(g f)(x) x 4. 33(x 1) ; x 0(g f)(x) (x 1) ; x 0

>

x 0 4.

(90) . 2xy 1 x 2yx 1 y

2 2x xy y xy y x 0 x 0

21 1 1 4xf (x) y 2x

.. f(x) x f(x) x .. . .

222 2 2 2

x 1(1 x )(1) (x)( 2x)f(x) (1 x ) (1 x )

.. .

(91) gR x 1 x x0 10 10 10a a(10 ) 0 gR ( 10a, 0)

f(x) f-1(x)

5 374 x 1> 3 3x 1 x 1 0 > > gR [0, ) .. 110 a 2.5 a 4

. 1 1 3g (a 1) g ( )4 3 ( 2.5, 0)4 x1 3(10 )4 4

x10 3 x log 3 . . 1g (x) gR ( 2.5, 0) [0, ) 1 3log 4 x , 2.5 x 0g (x) x 1 , x 0 > .

(92) . x x y y

y y (x x) 0 1 1 4(x x)y 2 1 4x 4 x 1y 2

1 (2 x 1) x2 1 x

y 1 x y x

. x x y y

y y (x x) 0 1 1 4(x x)y 2

1 4x 4 x 1y 2

1 (2 x 1) x2 1 x

2 ( x=1 y=1 , y=0 ) 2.

(93) 1f (A) x 2x A

1. A [ 25, 0] 2x [ 25, 0] x 0 ()

2. A [ 1, 1] 2x [ 1, 1] x [ 1, 1] ()

3. A [0, 1] 2x [0, 1] x [ 1, 1] ()

4. A [4,9] 2x [4,9] x [ 3, 2] [2, 3]

(94) 1 1 1f ( x 1) x 12 2 .....(1) 1 x 1 22 x 6 x 6 (1) 1f (2) 4

(95) 1f( x 2) x 23 1A x 23 .. x 3(A 2) f(A) 3(A 2) 2 3A 8

f(x) 3x 8 f(g(x)) 3(g(x)) 8 (f g)(x) 3x 14 .. 3(g(x)) 8 3x 14

1g(x) x 2 g (x) x 2 1(g f)(x) (3x 8) 2 3x 6

(96) (f g)(x 2) 3x 6 .....(1) x 2 2 x 0 0 x (1) (f g)(2) 3(0) 6 6 f(g(2)) 6 1f (6) g(2) 1 6 6 3f (6) 6 2 4 2 3g(2) 2

375

(97) f(x) x 1 1f (x) x 1 1 2(g f )(x) 4x 1 2g(x 1) 4x 1 x x 1 2g(x) 4(x 1) 1 0

2 1(x 1) 4 1x 12

1 3{ , }2 2 3.

(98) 1f ( 7) 3x 1 7 x 2 f( 2) 7 .. 1f ( 7) 2 g( 2) x 2 (f g)(x) ..

3 2f(g( 2)) ( 2) 3( 2) 3( 2) 2 0 1g( 2) f (0) 3x 1 0 x 1 1f (0) 1 1 1(g f )( 7) g( 2) f (0) 1

(99) f(1) g(1) f(1) g(1) 8 f(1) 4 a b 4 .....(1) g(1) 4 ( x 2 g)

12 c 4 c 8 f(2) g(2) ( x 3 g) 4a b 18 8 10 .....(2) .. (1), (2) a 2 , b 2

2f(x) 2x 2, g(x 1) 6x 8 1g (16) 1g (6x 8) x 1 x 4 1g (16) 3 1f(g (16)) f(3) 2(9) 2 20

(100) (fg)(0) 15 1a 5 15 a 3

1(f g )( 2) 27 1 1f (27) g ( 2) .....(1)

2x 1f(x) 3 2x 127 3 x 1 f(1) 27 1f (27) 1 (1); 1g ( 2) 1 g(1) 2 b 5 2 b 7 13 f( 1) 4g(2) 3(3 ) 4( 7(2) 5) 1 36 37

(101) 3g(4) 4 64 1g (64) 4 .....(1) 1 1 1(f g)(4) f (g(4)) f (64) 2 .....(2) 11f (64) 2 0.5g (64) 4

a 1f (64) 2 f(2) 64 2a(2 ) 64 a 16

(102) 1 2f (x) x f(x) x 1 2g (x) ( x) 1 x 1 g(x) x 1 f(a) g(a) a a 1 19

a a 20 0 ( a 4)( a 5) 0 .. a 16 1 1 2f (16) g (16) 16 16 1 273

(103) 1f ( ) f ( ) f(g(x)) x 1f (x) g(x) .. 1f(x) g (x) ( 1. 2.)

1g(x) x 33 1g(h(x)) h(x) 33 g(h(x)) 2x 1 1 h(x) 3 2x 13 h(x) 6x 6 2.

5 376

1 2

4 5 6

A B C 7 8 9

f g

(104) f(2) f(0) x 0 f(1) 3(0) 2 f(0) 2 1 3 x 1 f(2) 3(1) 2 f(1) 3 2 3 8 1 1g (f (2)) g (8) 1g (2x 8) 3x 1 .. 2x 8 8 x 0 .. 1g (8) 1 1 1g (f (2)) g (8) 1

(105) 1 x 1g (x) 3 g(x) 3x 1 (g f)(x) 3 f(x) 1 2(g f)(x) 2[(f(x)] 2f(x) 4 .. 22[(f(x)] 2f(x) 4 3f(x) 1

3(2f(x) 3)(f(x) 1) 0 f(x) 2 1 f(x) 0 f(x) 1 . (g f)(x) g( 1) 4 () . . f(100) g(100) ( 1) (300 1) 298 .

(106) 2g(x) x x 2 2(g f)(x) (f(x)) f(x) 2 2(g f)(x) 3(f(x)) 2f(x) 1 2 23(f(x)) 2f(x) 1 (f(x)) f(x) 2

22(f(x)) f(x) 1 0 (2f(x) 1)(f(x) 1) 0

1f(x) 2 f(x) 1 f : R R f(x) 1 1. (g f)(1) g(1) 2 () 2. (g f)(1) g(1) f(1) 2 1 2 () 3. g g(1) 2( )(1) 2f f(1) 1 () 4. (g f)(1) g(1) f(1) 2 1 1

(107) 1f (g(x)) x 2 f(x 2) g(x) x x 2 f(x) g(x 2) .....(1) . (1) x 2x f(2x) g(2x 2) .. . . (1) 1g (f(x)) x 2 R () .. .

(108) fg f gD D D f; (3 x)(2 x) 0 >

(x 3)(x 2) 0 < 3 x 2 < < g; x 3 0 ( 0 )

x 3 fgD [ 3,2] ( 3, ) ( 3,2]

(109) f; 23x 0> 23x 1 1 >

23x 1 1 > 1f(x) 2 > f 1R [ , )2 u; u g hD D D h(x) 0

gD ; 3 x 0 > x 3 < hD ; 2x 5x 6 0 > (x 6)(x 1) 0 <

1 x 6 < < h(x) 0 1 x 6 uD ( 1, 3] f u 1R D [ , 3]2 2.

(110) 1. g f C

377

2. gof fD D [0, ) fogD R 3. ..

2 2(f g)(x) (g(x)) 4(g(x)) 3 (g(x) 2) 1 g(x) 0> 2(g(x) 2) 0 > fogR [ 1, )

(g f)(x) f(x) 2f(x) (x 2) 1 f(x) 1 > f(x) 0 >

gofR [0, ) 4. f(1) 1 , g(1) 1 1f (1) 1 , 1g (1) 1 1 1f g (1) 1 1 1g f (1) 1

(111) (g h)(x) 2 h(x) h(x) 0> .. x 25 2x 0 2 > 0

goh hD D f gD D

fD 2x 25 0 > fD ( , 5] [5, )

gD 2x 0> gD [0, ) gohD [5, ) ( 1.)

(112) h gof fg gof f gD D D D D D fD R ( x ) gD { 1, 1} R ( 0) gofD .. 21(g f)(x) 1 (f(x)) f(x) 1 1

2x 2x 1 1 2x 2x 1 1 2(x 1) 1 () x(x 2) 0

gofD {0, 2} R hD {0, 2, 1, 1} R ( 4.)

(113) gofR f; 2 2 2x 0 4 x 4 0 4 x 2 > < < <

20 f(x) 2 0 [f(x)] 4 < < < < 25 9 [f(x)] 9 < <

21 1 1

59 9 [f(x)]

< <

1 1(g f)(x) 59 < 2x 1 <

rD [ 1, 1] Rr; g .. 2x 0>

21 x 1 < 20 1 x 1 < < 0 g(x) 1< < 0 110 (f g)(x) 10< <

rR [1, 10] 3.

(115) 2(g f)(x) 100 3(f(x)) xf(x) 10 f(x) 0

2 2f(x) 0 3(f(x)) 0 2100 3(f(x)) 100

2gof0 100 3(f(x)) 10 R [0, 10) <

9

(116) fogD 2f(x) (x 1) (x) f

2(f g)(x) (g(x) 1) g(x) fogD gD .. fogD [0, )

gofR f(x) f(x) 0> (g f)(x) f(x) 1 1 >

gofR [1, ) fog gofD R' [0, 1)

5 378 (117) gD 2x 1 0 (x 1)(x 1) 0 > > gB D ( , 1] [1, )

gofD .. 2(g f)(x) (f(x)) 1 f(x) ( , 1] [1, ) x x 1 x1 01 x 1 x < <

1 0x 1 > .. x 1 x x 1 x1 01 x 1 x > >

2x 1 0x 1 < .. 1/2 x 1< gofA D [1/2, 1) (1, )

A B' ( 1, 1) (1, )

(118) fR [ 2,2] ( sin 1 2)

gD 2 gx 1 0 D ( , 1] [1, ) > f gR D [ 2, 1] [1,2]

gofR 22 f(x) 2 0 [f(x)] 4 < < < < 2

gof0 [f(x)] 1 3 R [0, 3] < < f g gof(R D ) R [ 2, 1] ( 3,2]

(119) g(x) 5 2x f(x) 5 g(x) 5 5 2x (f g)(x) 5 5 2 5 2x fogD 3 55 2x 0 x 2 > > 5 2 5 2x 0 > .. 5 2x 0 > 5 5 2 5 2x 0 >

5 2 5 2x 5 < 5 2 5 2x 25 < 5 2x 10 5 2x 100 < <

95x 2 < .. fog 5 95D [ , ]2 2 904(a b) 4( ) 1802 ()

(120) p q p q f A A ( = ) 11 1. f {...,( 2, 2),( 1, 1),(0, 0),(1, 1), ...} f 11 2. f {...,( 2, 4),( 1, 2),(0, 0),(1,2), ...} f ( f1 D f1 R ) 3. f {(1, 1),(2, 3),(3, 3),(4,5),(5,5), ...} f ( f2 D f2 R ) 4. f {(1, 1),(2, 1),(3,2),(4,2),(5, 3), ...} f 11 ( (3,2) (4,2) f)

(121) g(0) 1 , g(1) f(g(0)) f(1) 2 , g(2) f(g(1)) f( 2) 2 2 , g(3) f(g(2)) f( 2 2) 2 2 2 , .. 1. .. a b g(a) g(b) g {(0, 1),(1, 2),(2, 2 2),(3, 2 2 2), ...} 2. .. f g 11 f g {(0, 2),(1, 2 2),(2, 2 2 2), ...} 3. .. 1. 2. gR 1 fogR 1 4. .. g(n) 2 2 2... 2

a 2 2 2... 2a a2 2a 2a 0

(a)(a 2) 0 a 2 ..

379

(122) (f g)(x) f(x) 2(x 1) 2x 4x 2 x I f g f {...,( 1, 6),(0, 2),(1,2),(2,6), ...} .. fog fR { 2, 6, 10, 14, ...} 1.

(123, 124) fR .. f(x)(g f)(x) 2 2x x2 () .. F(x) (g f f)(x) x 2x x F {...,( 2,2),( 1, 1),(0, 0),(1, 1),(2, 2), ...}

(125) f(x) ; x ( x 1 ) xx 1 -1 1 ( fR ( 1, 1) ) g(x) ; x g(x) ( gR I ) 1. fogD R .. x f g 2. fogR (0, 1) .. ( 1, 1) 3. (g f)(x) 1 x 0 .. f(x) 0.1

1 4. (g f)(x) 0 x 0 .. f(x) 0.1

0 2.

.. 5

126. [ / ..2547] 40 4000 30 8000 f (x) f (x) x 1. f (x) 400 x 12000 2. f (x) 200 x 4000 3. f (x) 400 x 20000 4. f (x) 200 x 12000 127. [ / 2543] 2 x, y y y x y f(x) f( 8) f(2) f(4) 1. 4 2. 5 3. 6 4. 7 128. [ / 2534] x y 2 129. [ / 2538] y 1 x 2 x 0 1. 2 . 2. 4 .

3. 8 . 4. 16 . 130. [ / 2545] 1r {(x, y) | 2 x y 4 } R R 1 2r r

126. 3 127. 4 128. 8 129. 2 130. 3

5 382

() f 0.01 x 0.01

1g x 1

131. [ / 2547] R A A {(x, y) | x y x y 2}

383

f(x)

x

18 16 14 12 10 8 6 4 2

0 1 2 3

135. [ / ..2548] 15,000 2,000 ( 1,000 ) 8 136. [ (.) / 2547] h h 2 h(x) f(x) x [ 1, 1) h(100) 137. [ / 2536] f (x) 1. xf (x) 2 x 2. 2f (x) 2x 1 3. 2f (x) x x 1 4. 3 2f (x) x 2x 3x 1 138. [ (.) / 2538] 0 p 1< < p(1 p) 1. 0 2. 1 3. 12 4.

14

5. 1. 4. 139. [ (.) / 2543] 2 22 1 2 1(11 )(7 )x x2x 2x 140. [ (.) / 2547] P (t, 0) 110 t 3 PBA y t AB

131. 4 132. 3 133. 100 134. 3 135. 2625 136. -3 137. 4 138. 4 140. 11/6 t p

O B

A y=x(4x)

y=x/3

x

y

5 384 141. [ / 2535] 2f(x) x 4x 3 1. g(x) f(x) f(x) 2. h(x) 2f(x) 3. k(x) f(x) f(x) 4. p(x) 1 f(x) 142. [ / 2535] 1. 1f {(x, y) | x y 2 } 21f {(x, y) | y (x 2) } 2.

21

x x 1g {(x, y) | y }x 1

31 2

x 1g {(x, y) | y }x 1

3. 1h {(x, y) | xy x 1} 1 1h {(x, y) | x }y 1 4. 1 1F {(x, y) | y }1 x 1 1 xF {(x, y) | y }1 x 143. [ / 2544] * 1. 1r {(x, y) | x y } R R 2. 2r {(x, y) | xy 0 } R R 3. 23r {(x, y) | x ln y } R R 4. 4r {(x, y) | x y 1} R R 144. [ / 2534] 1. 1 1r {(x, y) | y x }x R R

2. 22 2r {(x, y) | x 5 8 2y y }3 R R

3. 2 2

3x yr {(x, y) | 1 , xy 0 }4 9 R R

4. 4r {(x, y) | y x x } R R 145. [ / 2546] 2 2r {(x, y) | 9y 4x 8x 36y 4 0 } R R . r rD R ( , 4) . r 1. . . 2. . 3. . 4. . .

385

146. [ / 2546] 4 2 2r {(x, y) | 16x 16y 5 8x 16y } R R . r . r rD R 1. . . 2. . 3. . 4. . . 147. [ . / 2540] r {(x, y) | y x 9 } r rD {3, 12, 36} r 148. [ . / 2545] A { x | 0 x 1 2 } < 149. [ . / 2538] 2r {(x, y) | (x 1)y x 5x 6 } R R 1. [2, 3] 2. ( 1, 2] [3, ) 3. ( , 2] [3, ) 4. ( , 1) ( 1, 2] [3, ) 150. [ . / 2546] 2f(x) x 1 f fA { x | x x D R } I A 151. [ / 2535] xf(x) x 1 A f B [ 1, 1] A B 1. ( 1, 1) 2. ( 1, 0] 3. [0, 1] 4. ( 1, 1] 141. 4 142. 3 143. 1 144. 1 145. 1 146. 1 148. 3 149. 4 150. 0 151. 3

5 386 152. [ . / 2537] xf(x) 1 x f fD R 1. R 2. {0}R 3. [0, 1) 4. ( 1, 1) 153. [ / 2545] 2r {(x, y) | x y xy 1 0 } R R A { x | x 5 }

387

158. [ . / 2543] A {(x, y) | y x } < 2B {(x, y) | y x 6 } > A B 1. ( 5, 5) 2. ( 7, 4) 3. ( , 0) 4. ( 2, ) 159. [ (.) / 2542] 1r , 2r 1r {(x, y) | x y 2 } R R 1 2r r r 1r , 2r 1r 160. [ / 2545] * 21r {(x, y) | ln(y x ) 0 } > 2 xy 1r {(x, y)| 3} r1 r2R R 1. { y | 0 y 2}< < 2. { y | 0 y 3}< < 3. { y | 1 y 3}< < 4. { y | 1 y 4}< < 161. [ / 2544] 2r {(x, y) | y x xy 1} >

5 388 164. [ / 2547] f(x) x , 4g(x) 1 x A { x |(g f)(x) 0 } B { x | g(x) 0 } A B 1. 2 2. 4 3. 8 4. 16 165. [ (.) / 2538] f {(a, 1),(b, 1),(c, 2),(d, 3),(e, 4),(f, 2)} A {a, b, c} Af {(x, y) f | x A } Af 166. [ / 2544] A {1, 2, 3, ..., 10} r A r {(a, b) | a ba } r 167. [ / 2545] A {0, 1,2, ..., 9} B {(a, b) A A | a b } C {(a, b) A A | a b } > D B D C 1. 202 2. 352 3. 452 4. 552 168. [ / 2534] 2f(x) x 4x A { x | f(f(x)) f(x)} R P(A) 169. [ (.) / 2543] A {1, 2, 3, 4} f : A A x 1 , x 3f(x) 1 , x 4

<

g : A A g(1) 3 g f f g 170. [ / 2540] f 2f(3x 5) 18x 57x 48 2(f f)( 1) f (1) 1. 3 2. 5 3. 2 4. 1

389

171. [ / 2549] 2f(2x 1) 4x 14x f(x) 0 1. 94 2.

94 3. 5 4. 5

172. [ (.) / 2543] f 2 4 2f(x 1) x 5x 3 x 2f(x 1) 173. [ / 2549] f : I I (1) f(f(n)) 4n 15 (2) n n 1f(2 ) 2 5 n I f(1659) 174. [ / 2547] f g f(x) x 2 2 2g(x) (x 4) h gh f H { x | h h(x) R } H 175. [ / 2543] 2 3r {(x, y) | (y x y 0) (y x y 0)} > 1r ( 4) 1. 16 2. 16 3. 64 4. 64 176. [ / 2538] f(x) x 2 1 2(f g)(x) 3x 5 g(x) 0 1. [ 1, 1] 2. ( , 1) (1, ) 3. ( 1, 1) 4. ( , 1] [1, ) 177. [ / 2545] 3f(x 1) x 1 1(g f )(x) 3x g(x) 3 1. {1} 2. {0} 3. { 1} 4. 164. 2 166. 45 167. 3 168. 16 170. 1 171. 3 173. 3323 175. 4 176. 3 177. 1

5 390 178. [ / 2546] f g 2f(x) x x 2 , 2(f g)(x) 4x 2x 2 . g . 2g(f(x)) 4 2x 2x 1. . . 2. . 3. . 4. . . 179. [ (.) / 2544] f 1 2f {(x, x 1) | x 0 } > g g f (x) 2x 3 x fD f g(x) 180. [ (.) / 2547] f g (1, 1) , (f g)(1) 5 (f g)(2) 3 f(x) , g(x) , 1f (x) (g f)(x) 181. [ / 2548] f x 1f(x) x 1 x {1, 1} R 1 11 1(f f )(x)f f 1. 0 2. 1 3. 1x 1 4.

1x 1

182. [ / 2547] f f(n) (x,y) x 2y n x, y n g t 3 2g(t) 2t t 2t 3 1(g f)(5) 183. [ (.) / 2549] f(x) g(x) f(x g(y)) 2x y 5 x y g(x f(y)) x y

391

184. [ / 2534] x

2 x[f(x 1)] k (k ) 12y229 y[f( )]y 1. k 2. 2k 3. 2k 4. y k 185. [ / 2544] f : R R 3f(x) ax b 3 f(1) 1 < < 1 f(2) 4 < < 1. 4 f(3) 3 < < 2. 3 48 f(3) 177 7 < <

3. 1 f(3) 23 < < 4. 5 52 f(3) 37 7 < < 186. [ / 2548] g h 2g(x) 5x 1 h(x) 2x 1 1. goh 1D [ , )2 2. hogD ( , ) 3. gohR [1, ) 4. hogR [0, ) 187. [ / 2547] f g f(3x 2) 2x 3 23x x 0g(x) x 1 x 0

>

fog fogD R 1. ( 4, 2) 2. ( 3, 1) 3. ( 2, 0) 4. ( 1, 1) 188. [ / 2546] f : R R 2 3x f(x) f(1 x) 1 x x x R 189. [ / 2547] R f : R R 2x f(x) f(1 x) 2x x xR 1 f(2004) f(2546)2545

178. 4 181. 1 182. 1.5 184. 3 185. 3 186. 4 187. 2 189. 2003

5 392 190. [ / 2548] 12 f( ) 4 f(2x) 10x 2x x f(5) 191. [ / 2539] f : {0} R R x {0}R

1 1x f( x) f(x ) x f(x) 192. [ / 2544] f : R R x 2 x xx f( ) x f( ) x 13 3

2f( )3 1. 310 2.

510 3.

710 4.

910

193. [ / 2534] af (n) n(n 1)(n 2)...(n a) a, n a n 31

2

f (27)f f (26)

194. [ / 2543] 2f(x) 2x , xg(x) 4 , h(x) (f g)(x) ,

2h (x) (h h)(x) n n 1h (x) (h h )(x) n n 2 20h (x) 1. x 2. 20x 3. 50x 4. 20

x2

195. [ (.) / 2546]

2x 9f(x) x 3 , g(x) x 3

x 4 x 50h(x) h(h(x 8)) x 50

>

f g(1) h(4) 196. [ / 2549] f : I R 1 f(x 1)f(x) 1 f(x 1)

x I f(1) 3 f(2548) f(2549)

393

f(x)

x

f(x)

x

f(x)

x

f(x)

x

197. [ / 2549] f : I I f( 100) 15,000 f(n) f(n 1) 3n 2 n f(0) 198. [ / 2534] 1 1f(x) 1 x n n n 1(f f f )(x) 1 n 1 8 2f (1) f (3) 1. 211 2. 0 3.

15 4. 1

199. [ (.) / 2544] n n n 1 n 2 n 3 3 2f(n) 1 2 3 4 ... (n 2) (n 1) n f(n 1)f(n) n 6> 200. [ / 2549] f

n n 2f(n)2n 3 n

f(f(n)) 37 201. [ (.) / 2542] f : R R f(a b) f(a) f(b) a, b R f(2) 4 f(0) , f(1) , f( 1) 202. [ / ..2544] f R R (linear function) (i) f (x y) f (x) f (y) (ii) f (x) f ( x) 1. 2. 3. 4. 190. -29/3 192. 3 193. 702 194. 4 195. 53 196. 3.5 197. 50 198. 1 199. 732/209 200. {7, 34, 148} 201. 0, 2, -2 202. 3

5 394

x

y y

(126) 8000 4000m 40030 40 ( 3. ) .. y 4000 400(x 40)

f(x) y 400x 20000 3.

(127)

2 2 y (x/2) 2 21 (x) y (x/2) 22 x 8 2 22 y ( 2) 2 y 2 x 2 2 21 y (1) 2 y 5 x 4 2 22 y (2) 2 y 5 f( 8) f(2) f(4) 2 5 5 7

(128) x y 1 x y 2 2 x y 2 3 x y 2 4 x y 2 1 4 42 8

(129) y 1 x 2 (y 1) (x 2) (y 1) (x 2) (2, 1)

1 4 22 4

(130) 1r 2r 1 2r r 5 3( , )3 2 4

1 34 ( 1 )2 2 3

(131) 1; x y (x) (y) (x y) 2 <

x y 1 < 3; x y (x) (y) (x y) 2 <

x y 1 >

O 2

2

-2

-2

2

3

-1

2 -2

4

-4

1 -1

r1 r2

395

O

n 2,000

2000n

50000+1975n

2; x y x y 0 > ( x) (y) (x y) 2 < y 1 < x y 0 ( x) (y) (x y) 2 < x 1 > 4; x y x y 0 > (x) ( y) (x y) 2 < x 1 < x y 0 (x) (y) (x y) 2 < y 1 >

1, 2, 3. 4.

(132) 2 33x 2x

3 2 22x 3x 0 x (2x 3) 0 x 0 3x 2 (0, 0) 3 27( , )2 4 27/4 0m 4.53/2 0 .. t 1 2 1 t 3 < < < 1. 20 8 8 4.5

2. 2 2t 8 4.5 t 12.5 1 t 3 < < 2t 12.5 3. 1 t 3 < < 2t 3t 8 23[ , 12]4 18

4. 1 t 3 < < 2t t 8 31[ , 14]4 18

(133)

10.01x 0.01 x 1 2100x 1 (x 1) 100x 1

x 1 10 () x 11 .. f g 0.1

100

(134) n

1,000n kWh .. 2 1,000n 50,000 1,975n 2,000n 50,000 1,975n

n 2,000 . ( 2,000 . )

x+y=0

5 396 (135) a 15,000 2a 8a 15,000 2a 8a < a 2,500> 2,500 3 21,000 ()

21,000 2,6258 ( 2,626 3,000 )

(136) x n h 2 .. h(100) h(100 50(2)) h(0) h(100) h(0) f(0) 3

(137) x 3 1. 3f(x) 2 3 11 2. 2f(x) 2(3) 1 19 3. 2f(x) 3 3 1 13 4. 3 2f(x) 3 2(3) 3(3) 1 19 1. 3. x 2 2. 2f(x) 2(2) 1 9 4. 3 2f(x) 2 2(2) 3(2) 1 7 4.

(138) 2y p(1 p) p p y B 1 1p 2A 2( 1) 2 .. 1 1 1(1 )2 2 4

(139) 22 1A x 2x (11 A)(7 A) 2(11 A)(7 A) 77 4A A 4A 22 (11 2)(7 2) 81 22 1A 2x 2x

24x 4x 1 0 2(2x 1) 0 1x 2

(140) A BAB y y t(4 t) t/3 211 t t3 11/3 11t 2 6 ( d(AB) 0dt )

(141) f(x) B 4x 22A 2 (2, 1) 1. f(x) f(x) f(x) 0 f(x) 0

(2,-1)

397

2. 2f(x) 2 3. f(x) f(x) f(x) 0 f(x) 0 4. 1 f(x) f(x) 1 ( 1) 2 4.

(142) 1. .. x 2> ( ) x ( ) 2. .. x 1 .. x 1 x 1 ( 0) x 1 0 0 1

3. .. y 1 x 0 4. .. 1 x ( 2.) .. x 1 x 1 ( 0)

(143) 1. x y 1 ( x 0> y 0< ) 2. x 0 y 3. x 0 y 1 1 4. x 0 y 1 1

(144) 1. 1x y y .. x 2.5 y 2 y 0.5

2. 22y 5 8 2x x3 23(y 5) 2 8 2x x

2 29(y 5) 4(8 2x x ) 2 29(y 5) 4(x 1) 36

2 2(y 5) (x 1) 14 9

( y 5> )

3. 2 2y x 14 9 (0,0) xy 0 1 (x,y ) 3 (x,y )

(2,-2)

(2,-2)

(2,2)

5 398 x, y ( xy 0 ) .. 4. x y y 2 2y y 0x y y 0

>

(145) ;

2 29(y 4y 4) 4(x 2x 1) 4 36 4

2 29(y 2) 4(x 1) 36 2 2(y 2) (x 1) 14 9

(1,2) (1, 4) (1, 0) R R ( y 0) . r rD R [4, ) ( , 4) R .. . r () ..

(146) ;

4 2 2(16x 8x ) (16y 16y) 5 4 2 2116(x x ) 16(y y) 52

4 2 21 1 116(x x ) 16(y y ) 5 1 42 16 4 2 2 21 116(x ) 16(y ) 04 2

2 2 21 1(x ) (y ) 04 2 0 2 1x 4 1y 2 1 1 1 1r {( , ),( , )}2 2 2 2 .. . .

(147) x 9 .. x 3 (y) 3 .. x 12 (y) 3 .. x 36 (y) 0 r {(3, 3),(12, 3),(36, 0)} r {(3, 3),(3, 12),(0, 36)}

(148) A [ 1, 1] B ( , 3] [3, ) A B ( , 3] [ 1, 1] [3, ) 1. x 1 0 x 1 > > x 1 0 x 1 fD [ 1, 1) (1, ) .. 1. 2. 2y 3 x y 3 0 y 3 > > gR [ 3, ) .. 2. 3. 2 2(x 9)(x 1) 0 >

(x 3)(x 3)(x 1)(x 1) 0 > ( , 3] [ 1, 1] [3, ) .. 3. 4. x(x 3)(x 1) 0 > [ 1, 0] [3, ) .. 4.

(149) 2x 5x 6y x 1 2

(1) 2x 5x 6 0 (x 2)(x 3) 0 > > x ( ,2] [3, )

(2) x 1 0 x { 1} R () ( , 1) ( 1,2] [3, )

399

(150) .. 2y x 1 x 1 0 .. x 1 1 fD { 1, 1} R

.. 2x 1y 2 1 0y > 2 y 0y > fR ( , 2] (0, ) .. f fD R ( 2, 0] f fD R A A 0

(151) xy x 1 x 0> xy x 1

yxy y x x y 1

y y0 0y 1 y 1 > < f1R [0, 1)

x 0 xy x 1 yxy y x x y 1

y y0 0y 1 y 1 f2R ( , 1) (0, )

f1 f2A R R ( , 1) [0, ) 3.

(152) x ( 0 ) fD R 2 x 0> xy 1 x yx 1 y

y y0 0 y [0, 1)1 y y 1

x 0 xy 1 x yx y 1 y 0 y ( 1, 0)y 1 .. () fR ( 1, 1) f fD R ( 1, 1)

(153) 1rD rR .. y 0 1 0 y 0 y 0 2y y 4yx 2y 2y 4y 0 >

y ( , 4] (0, )

1r rR ( , 4] (0, ) D 1rD A { 5, 4, 1,2, 3, 4,5} 7

(154) 2. 2 yx y 4 x y ( 0) r R ( 2. ) .. 2(x)y (1)y (4x) 0 x 0 y 0 (0, 0) r ( 3. ) x 0 21 1 16xy 2x ( 4. )

5 400 ( 0) 21 1 16xy 2x .. 21 16x 0 >

216x 1 0 (4x 1)(4x 1) 0 < < 1 1x4 4 < < .. 1.

(155) 2xy y x x 1

20 x (y 1)x (y 1)

; 2(y 1) (y 1) 4(y 1)x 2 y 2(y 1) 4(y 1) 0 >

2y 2y 1 4y 4 0 > 2y 2y 3 0 > (y 3)(y 1) 0 > fR ( , 1] [3, )

(156) A; xg(x) log( 2 2 ) x2 2 0

x2 2 1x 2

g 1 1A D ( , )2 2

B; 2 2x x 1f(x) x 1 2x 11 1 1x 1 x x

1x x ( , 2] [2, ) ( 2 x 1 x , 2 x 1 x ) 1 1 1[ , 0) (0, ]1 2 2x x 1 1 3f(x) 1 [ , 1) (1, ]1 2 2x x f 1 3B R [ , 1) (1, ]2 2 3. ( A ' B' 1 )

(157) 1 1r sx D D r sx R R r; 21 x 1 < 20 1 x 1< < 21 x1 10 10< < rR [1, 10] s; 0 0 2x 4x y 2 0

2 2y 2 x 4x 4 (x 2) y 2 0 < .. y 2 < .. x 2 x 2 2y 2 4(2) 2 14

sR ( ,2] { 14} r sx R R [1,2] A B (1, 3) (2, 1) 2 2AB 1 2 5

(158) A B

2x x 6 (x 3)(x 2) 0 x 3 (3, 3) A B [ 6, 3] 2.

-6

-6

A B

401

(159) 1r x y 2 < 2 y x > x 0> 2 y 0 >

y 2 2 y 2 < < < r1R [ 2,2] 2r 2y x 1> 2y 1 x > 2x 0> y 1 0 >

y 1 > r2R [ 1, ) 1r , 2r 1r r 1 rrD R [ 1,2]

(160) 2ln(y x ) 0 > 2 0y x e 1 > 2y x 1> ... y 1> ( log 2y x 0 2y x 1 > ) r1R [1, ) xy 1 3 xy 1 3 ... y 1 3 <

3 y 1 3 < < 2 y 4 < < r2R [ 2, 4]

r1 r2R R [1, 4] 4.

(161) .. 2y x< (0,0) .. xy 1> 1 3 r 1 rrR D ( , 0) [1, )

(162) y x x x 0> y x x 2x x 0 y x x 0 1. .. J ( , 0) y ( 0) 2. .. I [0, ) 2 3. .. fR [0, ) 4. ..

f fD R [0, ) ( , 0) R

(163) n(A) 3 3n(P(A)) 2 8 8s 3 3 3 ... 3 3 6561 t 8 7 6 336 s t 6225

2 -2

2

-2

1 -1

r1 r2

-1

2

-1

r

(1,1)

y=0 y=2x

5 402 (164) (g f)(x) 0 21 x 0

2x 1 x 1 ( x f ) .. A { 1}

g(x) 0 41 x 0 4x 1 x 1 1 .. B { 1, 1} A B 1 22 4

(165) Af (a, 1),(b, 1),(c,2) ( x A)

Af {(a, 1)} Af {(b, 1)} Af {(c,2)} Af {(a, 1),(c,2)} Af {(b, 1),(c,2)}

(166) a ; a 2, 3, 5, 7 .. b (a,b) 4 10 40 a ba ; (a,b) (1,2), (1,3), (1,5), (1,7), (4,8) .. 5

r 40 5 45

(167) B (0,0), (1,1), (2,2), , (9,9) 10 C (0,0), (1,0), (2,0), (9,0) (1,1), (2,1), (3,1), (9,1) (2,2), (3,2), (4,2), (9,2) (8,8), (9,8), (9,9) 10 9 8 ... 1 55

.. D (0,0), (1,1), (2,2), , (9,9) 10 45 () .. D 452

(168) f(f(x)) f(x) 2(f(x)) 4(f(x)) f(x)

2(f(x)) 3(f(x)) 0 f(x)(f(x) 3) 0 f(x) 0 f(x) 3

2x 4x 0 2x 4x 3 x 0, 4, 3, 1 A {0, 4, 3, 1} P(A) 42 16

(169) f f {(1,2),(2, 3),(3, 4),(4, 1)} g {(1, 3),(2, a),(3, b),(4, c)} .. g f {(1, a),(2, b),(3, c),(4, 3)} .....(1) f g {(1, 4),(2, f(a)),(3, f(b)),(4, f(c))} .....(2) g f f g .. (1) (2) a 4 b f(a) f(4) 1 c f(b) f(1) 2 g {(1, 3),(2, 4),(3, 1),(4,2)}

(170) 2f(3x 5) 18x 57x 48 .....(1) 3x 5 1 4x 3 4x 3 (1) f( 1) 4 3x 5 4 x 3 x 3 (1) f(4) 39

(f f)( 1) f(f( 1)) f(4) 39 3x 5 1 x 2 x 2 (1) f(1) 6 2 2(f f)( 1) f (1) 39 6 3

403

(171) A 2x 1 .. A 1x 2 2f(2x 1) 4x 14x 2 2A 1 A 1f(A) 4( ) 14( ) A 5A 62 2

.. 2f(x) x 5x 6 2x 5x 6 (x 6)(x 1) 0 6 1 6 1 5

(172) 2 2 2 2f(x 1) (x ) 5(x ) 3 2x 2x 2 2 2 2 2f(x 1) (x 2) 5(x 2) 3 4 2 2x 4x 4 5x 10 3 4 2x x 3

(173) (2) n n 1 nf(2 ) 2 5 2 2 5 nx 2 f(x) 2x 5 f(f(x)) 2(2x 5) 5 4x 15 (1) .. f(1659) 2(1659) 5 3323

(174) 2 2 2g(x) (x 4) x 4 2x 4h(x) x 2x 2

.. x 2 .. h(x) x 2 h h(x) 2

(1) x 2 ( h(x) ) (2) h(x) 2 .. x 0 4 ( h(h(x)) ) H { 4, 0,2}

(175) 1r ( 4) a r (a) 4 .. x y 4 y 3y x 3( 4) x 64 x

x 64 ( r (64) 4 ) 1r ( 4) 64

(176) f(x) x 2 1f (x) x 2 1 2(f g)(x) 3x 5 2 2g(x) 2 3x 5 g(x) 3x 3 23x 3 0 3(x 1)(x 1) 0 ( 1, 1)

(177) 1(g(f (x)) 3x 1 1g (3x) f (x) .....(1) x g(x) 3 .. 1x g (3) x 1 (1) 1 1g (3) f (1) 3f(x 1) x 1 1 3f (x 1) x 1 x 0 1f (1) 1 1 1x g (3) f (1) 1 1.

(178) 2f(x) x x 2 2f(g(x)) (g(x)) g(x) 2 .. 2f(g(x)) 4x 2x 2 2 2(g(x)) g(x) 2 4x 2x 2

2 2(g(x)) g(x) (4x 2x) 0 21 1 4(4x 2x)g(x) 2

21 (4x 1) 1 4x 1g(x) 2 2

5 404

1 (4x 1) 2x 1 x 1/42g(x) 1 ( 4x 1) 2x x 1/42

>

1 (4x 1) 2x x 1/42g(x) 1 ( 4x 1) 2x 1 x 1/42

>

.. . g(x) V . g(f(x)) 2 ( 2 )

(179) 1 2f (x) x 1 ( x 0> ) f(x) x 1 g f (x) 2x 3 .. x 1f (x) 1 1g f (f (x)) 2 f (x) 3

2 2g(x) 2(x 1) 3 2x 5 2 2f g(x) (2x 5) 1 2x 4

(180) f g (1, 1) f(1) 1 g(1) 1 f(x) ax b () f(1) 1 a b 1 .....(1) (f g)(1) f( 1) 5 a b 5 .....(2) a 2 b 3 .. f(x) 2x 3 1 x 3f (x) 2 g(x) cx d () g(1) 1 c d 1 .....(1) (f g)(2) 1 g(2) 3 1 2c d 3 .....(2) c 3 d 4 .. g(x) 3x 4 (g f)(x) 3(2x 3) 4 6x 13

(181) x 1y x 1 y 1x y 1

xy x y 1 xy y x 1 x 1y x 1

1 x 1f (x) x 1 f(x) 1 11 1(f f )(x) 0f f

(182) f(5) (x,y) x 2y 5 x, y 0> (1,2) , (3, 1) , (5, 0) .. f(5) 3 1 1(g f)(5) g (3) 3 22t t 2t 3 3

3 22t t 2t 6 0 2(2t 3)(t 2t 2) 0

.. g(t) 3 3t 2 1 1 3(g f)(5) g (3) 2

(183) f(x g(y)) 2x y 5 .....(1) y 0 f(x g(0)) 2x 5 x x g(0) f(x) 2x 2g(0) 5 .....(2) x x g(y) f(x g(y)) 2x 2g(y) 2g(0) 5 (1) .. 2x y 5 2x 2g(y) 2g(0) 5 yg(y) g(0) 2 .....(3) f g (2) (3)

405

x f(y)g(x f(y)) g(0) 2 .....( 3)

x 2y 2g(0) 5g(0) 2 ...( 2)

x 5y2 2

(184)

12 12y y2

2 29 y 9[f( )] [f( 1)]y y

2 x[f(x 1)] k x 3y

3y29[f( 1)] ky

3 12y y2 2

2 29 9[f( 1)] [f( 1)] ky y

2k

(185) f(1) a b f(2) 8a b f(3) 27a b 26 19f(3) f(2) f(1)7 7 () 26 26 104f(2)7 7 7 < < 19 19 57f(1)7 7 7< < 1 f(3) 23 < < f(1) a b f(2) 8a b f(2) f(1) 7a f(2) 8f(1) 7b .. f(3) 27a b 27 1[f(2) f(1)] [f(2) 8f(1)]7 7

(186) 2(g h)(x) 5(h(x)) 1 h(x) (g h)(x) .. goh h 1D D [ , )2 ( 1. )

h(x) 2x 1 0 25(h(x)) 1 1 > .. gohR [1, ) ( 3. ) (h g)(x) 2 g(x) 1 g(x) 1/2 (h g)(x) 2 15x 1 2 > x .. hogD ( , ) R ( 2. ) 2g(x) 5x 1 1 2 g(x) 1 1 > .. hogR [1, ) 4.

(187) A 3x 2 A 2x 3 f(3x 2) 2x 3 A 2 2 5f(A) 2( ) 3 A3 3 3 2 5f(x) x3 3

23

2 5x , x 03 3(f g)(x) 2 5(x 1) , x 03 3

>

fogD R x fogR .. x 0> .. 2x 0>

22 5 5x3 3 3 >

fogR 5[ , )3 x 0 .. 3x 0 3x 1 1 32 5 7(x 1)3 3 3

fogR 7( , )3

fog 7 5R ( , ) [ , )3 3 fog fog 7 5D R [ , )3 3 .. 2.

5 406 (188) 2 3x f(x) f(1 x) 1 x x .....(1) x 1 x

2 3(1 x) f(1 x) f(x) 1 (1 x) (1 x) 2 3(1 x) f(1 x) f(x) 1 x 2x x .....(2)

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

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

.. 2 3 422 2x x x xf(x) 1 x x 22 x

(189) x f(x) f(1 x) x(2 x) .....(1) x 1 x (1 x) f(1 x) f(x) (1 x)(1 x) .....(2) (1 x) (1) (2)

2(x x 1) f(x) (1 x)[ x(2 x) (1 x)] 2 2(x x 1) f(x) (1 x)(x x 1)

f(x) 1 x

1 ( 2003)( 2545) 20032545

(190) 12 f( ) 4 f(2x) 10x 2x .....(1) x 12x

1 52 f(2x) 4 f( ) 2x x .....(2) 2 (1) 56 f(2x) 20x 6x x 2.5 f(5)

50 2 6 29f(5) 6 3

(191) 1 1f( x) f( ) xx x .....(1) x 1x

1 1x f( ) f( x)x x .....(2) (1) (2) x (1) 21f( x) x f( ) xx (2) 2 12f( x) x x

2x 1f( x) 2 2x x x 2x 1f(x) 2 2x

(192) 2 x xx f( ) x f( ) x 13 3 .....(1) x x

2 x xx f( ) x f( ) x 13 3 .....(2) x (1) 3 2 2x xx f( ) x f( ) x x3 3 (2) .. 3 2x(x x) f( ) x 2x 13

23

x x 2x 1f( )3 x x

2 32 2 2(2) 1 7f( )3 102 2

2f( )3 x 2 2 2 24 f( ) 2 f( ) 33 3 2 24 f( ) 2 f( ) 13 3

(193) 3

2

f (27) 27 26 25 24 27f (26) 26 25 24

31 12

f (27)f f(27) 27 26 702f (26)

407

(194) 2x xh(x) 2( )4 2 ( x 0> ) 2 2( ) xh (x) (h h)(x) 2 4

x

3 2 4( ) xh (x) (h h )(x) 2 8x

.. n nxh (x) 2 20 20xh (x) 2

(195) 5f g(1) f(g(1)) f(2) 51 h(4) h(h(12)) h(h(h(20)))

5h h h h(28) h h ... h(36)

6 7 h h ... h(44) h h ... h(52)

6 7 h h ... h(48) h h ... h(56)

6h h ... h(52)

x 52 52 (52 48 56 52 ) h 1 5 ..

h(52) 48 5 48 53

(196) f(1) 3

1 f(1) 1 3f(2) 21 f(1) 1 3

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

1 1/31 f(3) 1f(4) 1 f(3) 1 1/3 2

1 1/21 f(4)f(5) 31 f(4) 1 1/2

f(6) f(2) 2 f(7) f(3) 1/3 .. 4 f(2548) f(2549) f(4) f(1) 3.5

(197) f(n 1) f(n) 3n 2 f( 99) f( 100) 3( 100) 2 15000 3( 100) 2

f( 98) f( 99) 3( 99) 2 15000 3( 100 99) 2(2)

f( 97) f( 98) 3( 98) 2 15000 3( 100 99 98) 2(3) .. f(0) 15000 3( 100 99 ... 1) 2(100) f(0) 15000 3( 5050) 2(100) 50

(198) n n n 1(f f f )(x) 1 n n 1(f (x))(1 f (x)) 1 n

n 1

1f (x) 1 f (x)

.. 1 1 1f(1) 1 1 2 2

1 121 1 2f (1) 1 f(1) 31

32 231 1 3f (1) 1 f (1) 51

43 351 1 5f (1) 1 f (1) 81

.. ( ) 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 8 34f (1) 55

.. 1 1 1f(3) 1 3 4 2

1 141 1 4f (3) 1 f(3) 51

8 2 34 4 2f (1) f (3) 55 5 11

5 408 (199) 1f(1) 1 1

2 1f(2) 1 2 3 3 2 1f(3) 1 2 3 8 4 3 2 1f(4) 1 2 3 4 22 5 4 3 2 1f(5) 1 2 3 4 5 65 6 5 4 3 2 1f(6) 1 2 3 4 5 6 209 7 6 5 1f(7) 1 2 3 ... 7 732 8 7 6 1f(8) 1 2 3 ... 8 2780

f(2) 3 3f(1) 1 f(3) 8 2.67f(2) 3 f(4) 22 2.75f(3) 8 f(5) 65 2.95f(4) 22 f(6) 209 3.22f(5) 65 f(7) 732 3.50f(6) 209 f(8) 2780 3.80f(7) 732 f(n 1)f(n) n 6> f(n 1)f(n) f(7) 732f(6) 209

(200) f(f(n)) 37 f(n) 74 f(n) 17 f(n) 74 n 148 (OK) n 35.5 () f(n) 17 n 34 (OK) n 7 (OK) {7, 34, 148}

(201) f(a b) f(a) f(b) a 0 b 2 f(2) f(0) f(2) .. f(0) 0 f(a b) f(a) f(b) a b 1 f(2) f(1) f(1) 4 2 f(1) .. f(1) 2 f(a b) f(a) f(b) a 2 b -1 f(1) f(2) f( 1) 2 4 f( 1) .. f( 1) 2 f(0) 0 , f(1) 2 , f( 1) 2

(202) (i) x 0, y 0 f(0) f(0) f(0) 2f(0)

f(0) 0 1. 3. (ii) f( x) f(x)

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

.. 10

1. i 1 a b i ( a b ) ( C ) a b z z a bi a Re (z) b Im (z) 3 4 i , 2 i , 2 i , 3 i 3, 2 , , 0 4, 1, 2 , 3 5 5 0 i , 2 2 0 i , 0 i 0 2. (a, b) (a, b) z a bi 1 () 2 () (3, 2) 3 2 i 3. i ( 2i 1 )

im

re O 3

(3,-2) -2

10 72 a bi c di a c b d (a bi) (c di) (a c) (b d)i () 2(a bi) (c di) ac ad i bc i bd i (ac bd) (ad bc)i (x 4) (y 2)i 3 x 4 3 y 2 0 x 1 y 2 1z 3 2 i 2z 4 i 1 2z z (3 2 i) (4 i) 7 i 1 2z z (3 2 i) (4 i) 1 3 i 21 2z z (3 2 i) (4 i) 12 3 i 8 i 2 i 14 5 i ** (1) , (2) a b ab a, b 4. , , , ,

0 0 0 i (0, 0) 1 1 0 i (1, 0) () z a bi z a bi z a bi 1 1 1z a biz a bi 2 22 2 2 2aa bi b1 a ba bi a b a b i z 3 2 i z z 3 2 i z 1 1 1z z 3 2 i 2 2 3 213 13

3 2 i 3 2 i1 i3 2 i 3 2 i 3 2

73

1z 3 2 i 2z 4 i 1

2

3 2 i 3 2 i 4 izz 4 i 4 i 4 i

2

210 1117 17

12 3 i 8 i 2 i 10 11i i1716 4 i 4 i i

21

4 i 4 i 3 2 izz 3 2 i 3 2 i 3 2 i

2

2 210 1113 13

12 8 i 3 i 2 i 10 11i i133 2

1 1 11 2 1 2(z z ) z z n 1 1 n n(z ) (z ) z

z 1 2 i 25 z

2 2 25 5 55 z (1 4) 4 iz (1 2 i) 3 45 5

5( 3 4 i)5 i3 4 i 25

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

26 27 28i i i 2 3 4i i i ( 1) ( i) (1) i

1210(1 i)(1 i) 1 2 2(1 i) 1 2 i i 2 i 2 2(1 i) 1 2 i i 2 i 12 66 5510(1 i) (2 i) 64 i 2 i( 2 i) 32 i(1 i)

2 1 i 1 i 1 i 2 i i1 i 1 i 1 i 2 1012 210(1 i) 1 i (1 i)1 i(1 i)

1110(i) (2 i) 2 i 2 i ( 11 3i i i )

10 74 6. a bi a bi a bi 2 2(a b ) z z z a bi z a bi 1z 2 3 i 1z 2 3 i 2z 1 i 2z 1 i 3z 3 i 2 3z 3 i 2 4z 5 4z 5 5z 5 i 5z 5 i 7. (1) z z z (2) z z (3) 1 1(z ) (z) n n(z ) (z) n (4) 1 2 1 2z z z z (5) 1 2 1 2z z z z 1 2 1 2z z z z (2 3i)( 1 i)(3i 2) (5i) (2 3i)( 1 i)( 3i 2) ( 5i) 12z 1 2 2z z z i 1z 1 2 i 1 2 2z z z i .. 2 1z (z 1) i 2

1

iz z 1

21

iz z 1

1 12 z 1 2 iz 2i i 8. (0, 0) .. 2 2z a bi a b 5 5 ( (5,0) ) 3 i 3 ( (0,-3) ) .. 5 5 3 i 3

75

O

z (a,b)

re

r

im

a

b

2 3 i 2 22 3 13 ( (2,3) ) 3 4 i 2 23 ( 4) 5 ( (3,-4) ) 1 i 2 2( 1) ( 1) 2 ( (-1,-1) ) .. 2 3 i 13 , 3 4 i 5 , 1 i 2 9. (1) z 0> (2) 2zz z ( 2 2a b ) (3) z z z ( 2 2a b ) (4) 11 zz nn zz n (5) 1 2 1 2z z z z 1 2 1 2z z z z **

z 32(2 2 3 i)(3 4 i)z (2 i) (1 i)