06 my-net-land.blogspot.gr

7/22/2019 06 my-net-land.blogspot.gr

1/14

KE 6

6.1.

1. i) x 1 0, x 1.

: (1 ,) = {1} (1, +).

ii) x2 4x 0 x (x 4) 0 x 0 x 4.

, :

(0 ,) = {0,4} (0, 4) (4, ).

iii) x2

+ 1 0 . , -

fi .

iv) |x| + x 0 |x| x x > 0.

, (0, +).

2. i) : x 1 0 2 x 0 1 x 2.

, [1, 2].

ii) x2 4 0 x 2 x 2

x2 4 2 2.

, (, 2] [2, +).

iii) O, [1, 3]

1 3.

iv) 0 x 0 x 1.

,

[0, +) {1} = [0, 1) (1, +).

3.

f(5) = (5)3

= 125.

f(0) = 2 0 + 3 = 3.

f(6) = 2 6 + 3 = 15.

4. i) x fi fi. fi,

:

x 1x 1


2/14

x x + 1 (x + 1) 4 (x + 1) 4 + x2.

E, f(x) = (x + 1)4 + x2

= x2

+ 4x + 4 = (x + 2)2

f(x) = (x + 2)2, x . (1)

f(0) = 22

= 4, f(1) = 32

= 9, f(2) = 42

= 16 f(3) = 52

= 25.

ii) x > 0, :

f(x) = 36 (x + 2)2

= 62

x + 2 = 6 x = 4.

f(x) = 49 (x + 2)2

= 72

x = 5.

f(x) = 100 (x + 2)2

= 102

x = 8.

f(x) = 144 (x + 2)

2

= 122

x = 10.

5. i) x 1 :

f(x) = 7

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

ii) x 0, 4 :

g(x) = 2

x + 4 = 2x x = 4, .

iii) x :

h(x) = x2

+ 1 = 5 x2

= 4 x = 2 x = 2.

6.2.

1. T fi

.

2. 2 < x < 5 1 < y < 6.

1

x2

+ 1=

1

5

1

5

x + 4

x= 2

(x 4)(x + 4)

x(x 4)= 2

x2

16

x2

4x= 2

4

x 1= 2

4

x 1+ 5 = 7

6: 80

+ 1 4 + x2


3/14

3. fi (1, 3),

i) xx (1, 3)

ii) yy (1, 3)

iii) fi x y

(3, 1)

iv) fi

(1, 3).

4. fi -

(x1, y

1) B(x

2, y

2),

i)

ii)

iii)

iv)

5. i)

, (AB) = (A), fi AB .

ii)

fi ()2

= 8.

fi ()2

= 18.

fi ()2 = 26.

fi ()2

= ()2

+ ()2.

, .

6.

() = (5 2)2

+ (1 5)2

= 5.

() = (4 + 1)2 + (2 1)2 = 26,

(A) = (4 1)2

+ (2 + 1)2

= 2 32

= 3 2,

(AB) = (1 1)2

+ (1 + 1)2

= 2 22

= 2 2,

(B) = (3 4)2

+ (5 + 2)2

= 2 72

+ = 7 2.

(A) = (3 1)2

+ (5 2)2

= 42

+ 32

= 5.

(AB) = (4 1)2

+ (2 2)2

= 32

+ 42

= 5.

(AB) = 02

+ (4 + 1)2

= 5.

(AB) = (1 + 3)2

+ 02

= 4.

(B) = (3 + 1)2

+ (4 1)2

= 42

+ 32

= 25 = 5.

(O) = 42

+ (2)2

= 20 = 2 5.

() = (x2 x1)2

+ (y2 y1)2

6.2. 81


4/14

fi , fi

fi.

fi: fi

fi,

.

7.

i) f(2) = 6 22

+ k = 6 k = 2.

ii) g(2) = 8 k(2)3

= 8 k = 1.

iii) h(3) = 8 k k = 4.

8. i) f fi .

y = 0 x = 4, ofi y = f(x) xx

(4, 0).

x = 0 y = 4, fi y = f(x) yy

(0, 4).

ii) g fi

xx 1(2, 0) 2(3, 0)

yy B(0, 6).

iii) H h fi

xx fi (1, 0).

yy B(0, 1).

iv) H q fi

xx.

yy (0, 1).

v) H [1, +), fi

xx fi fi (1, 0)

yy.

vi) H (, 2] [2, +), fi

xx , 1(2, 0) 2(2, 0).

yy.

4 = 8

(A) = (2 + 1)2

+ (5 1)2

= 5.

() = (1 2)2

+ (1 + 3)2

= 5.

(B) = (2 5)2

+ (3 1)2

= 5.

6: 82


5/14

9. i) x = 0 f(0) = 1. C f yy (0, 1).

y = 0 x2 1 = 0 x = 1 x = 1.

Cf xx 1(1, 0) 2(1, 0).

ii) f(x) > 0 x2 1 > 0 (x + 1)(x 1) > 0 x < 1 x > 1.

10. i) f(x) = g(x) x2 5x + 4 = 2x 6 x

2 7x + 10 = 0

x =

x = 5 x = 2.

x = 2, g(2) = 4 6 = 2.

x = 5, g(5) = 4.

Cf Cg (2, 2) (5, 4).

ii) f(x) < g(x) x2 5x + 4 < 2x 6 x

2 7x + 10 < 0

(x 2)(x 5) < 0 2 < x < 5.

6.3. f(x) = x +

1. fi, y = x +

: = , fi y = x +

xx. ,

i) = 1, fi = 45.

ii) = , fi = 60.

iii) = 1, fi = 135.

iv) = , fi = 120.

2. x = x2 x

1 y = y

2 y

1, :

i)

ii)

iii)

iv)

3. fi y = x + .

i) = 1 = 2, : y = x + 2.

ii) = 45 = 1 = 1, : y = x + 1.

=y

x=

1 3

2 1=

2

1= 2.

= yx

= 1 11 2

= 0.

=y

x=

1 2

2 1= 1.

=y

x=

3 2

2 1= 1.

3

3

7 9

2=

7 3

2

6.3. f(x) = x + 83


6/14

iii) y = 2x 3

, fi = 2. -

: y = 2x + fi (1, 1)

1 = 2 1 + fi = 1. ,

y = 2x 1.

4. 2, fi -

, fi y = x + .

i) = 1, y = x +

fi (1, 2) 2 = 1 +

fi = 1. y = x + 1.ii) = 1, y = x +

fi (1, 2) 2 = 1 +

fi = 3. : y = x + 3.

iii) = 0, y =

fi (2, 1), y = 1.

iv) = 2, y = 2x +

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

= 5. : y = 2x + 5.

5. H C = F + fi -

0C 32F, 0 =

32 + . (1)

, , fi 100C 212F,

100 = 212 + . (2)

(1) (2) 100 = 180, fi

. ,

fi , fi

= ( 32) 9 = 5 5 32 4 = 5 32 = 40.

40F 40C.

6. f :

fi y = x + 2

x (, 0].

fi y = 2

x [0, 1]

5

9

C =5

9(F 32).C =

5

9F

5

9 32

= 5

9 32 =

5

9

6: 84


7/14

fi y = x + 1 -

x [1, +).

7. i) O f(x) = 1 y = f(x)

y = 1, 1 1.

f(x) = x

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

ii) f(x) < 1

y = f(x) fi y = 1,

x (, 1) {1}.

f(x) x y = f(x) fi y = x

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

8. i) f(x) = |x|

g(x) = 1 fi

.

O |x| 1

y = |x|

fi

y = 1 ,

x [1, 1].

O |x| > 1 y = |x|

fi y = 1, x (,1) (1, +).

ii) fi fi > 0

|x| x .

|x| > x < x > .

|x| 1 1 x 1.

|x| > 1 x < 1 x > 1.

1. i)

f(6) = 1, f(5) = , f(4) = 0, f(3) = , f(2) = 1, f(1) = 0.

f(0) = 1, f(1) = 1, f(2) = 1, f(3) = 0, f(4) = 1, f(5) = 2.

ii) O f(x) = Cf

.

f(x) = 0 4, 1 3.

f(x) = 1 2 4.

f(x) = 1 fi 6 fi -

[0, 2].

1

2

1

2

6.3. f(x) = x + 85


8/14

iii) y = x + -

fi (2, 1) (2, 1) 1 = (2) +

1 = 2 + .

fi, fi , fi

= 0 = 0,5.

y = 0,5x.

E, f(x) 0,5x

f

fi y = 0,5x, . fi

x [2,5] {2}.

2.

(1, 0) -

(.)

x = 1.

,

fi

(1, 0) (2, 1), fi

.

y = x + , x 1

, fi fi (1, 0)

(2, 1). 0 = + 1 = 2 + , fi -

= 1 = 1.

: y = x 1, x 1.

3. i) ) (t) fi fi

t, fi (t) = 2000 100t

(t) 0 2000 100t 0 t 20.

, (t) = 2000 100t, 0 t 20.

) (t) fi

t, fi (t) = 600 + 100t, 0 t 20.

6: 86


9/14

ii) -

.

fi (t) = (t),

2000 100t = 600 + 100t 200t = 1400 t = 7.

t = 7min.

4. fi fi -

fi

.

=

=

= 12 2x (4 x) = x + 8.

E, f

f(x) = x + 8, 0 x 4.

, .

(0, 8) (4, 4).

5. i) . k1 h = t + -

fi (3, 0) (0, 20) 0 = 3 + 20 = ,

fi = 20. , . k1

, 0 t 3.

1

, 0 t 3. (1)

, fi

2

h2(t) = 5t + 20, 0 t 4. (2)

h1(t) = 20

3t + 20

h = 20

3t + 20

= 20

3

4 + 2

2 4

x 4

2

(4 x) 2

2

6.3. f(x) = x + 87


10/14

ii) T k2

fi k1

h2(t) = 2h

1(t). fi:

h2(t) = 2h1(t)

3t + 12 = 8t + 24

5t = 12 t = 2,4.

, k2

fi k1

t = 2,4h.

iii) A fi i) fi

h1(t) = , 0 t 3.

h2(t) = , 0 t 4.

fi, h2(t) = 2h

1(t)

t = 2,4.

fi k2

fi k1

-

t = 2,4h, k1

k2.

6.4. Kfi - fi fi

1. 4.3, (x) = |x|,

fi fi x y x y. H

f(x) = |x| + 2 fi fi fi y = |x|,

2 , f(x) = |x| 2

fi fi fi y = |x|, 2

().

1

4t + 1 = 2

1

3t + 1

4t + = 2

3t +

4t +

3t +

1

4t + 1 =

2

3t + 2

1

4t + 1 = 2

1

3t + 1

20

4t + 20 = 2

20

3t + 20

6: 88


11/14

2. H h(x) = |x + 2| fi fi -

fi y = |x|, 2 ,

q(x) = |x 2| fi fi fi

y = |x|, 2 ().

3. y = |x + 2|, , fi

, fi fi fi y = |x| 2 -

. y = |x + 2| + 1,

, fi , fi fi fi

y = |x + 2| 1 . -

, F(x) = |x + 2| + 1. fi

y = |x|, fi 2

fi 1 ().

, G(x) = |x 2| 1, fi

y = |x|, fi 2

fi 1 ().

4. i)

6.4. fi - fi fi 89


12/14

ii)

iii)

5. i) f(x) = 2(x 2)2 1 + 1 = 2(x 2)

2.

ii) f(x) = 2(x 3)2 1 2 = 2(x 3)

2 3.

iii) f(x) = 2(x + 2)2 1 + 1 = 2(x + 2)

2.

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

2 3.

6.5. - fi -

1. f (, 1] [1, +).

g (, 0], [0, 2]

[2, +).

h (, 1], [1, 0],

[0, 1] [1, +).

6: 90


13/14

2. f fi x = 1, f(1) = 1 -

fi .

g fi fi .

h fi x = 1 x = 1

h(1) = h(1) = 2, fi .

3. i) f(x) f(3).

f(x) f(3) x2 6x + 10 3

2 6 3 + 10 (x 3)

2 0, .

ii) fi g(x) g(1).

g(x) g(1) 2x x2 + 1 0 (x 1)2, .

4. i) H f1 x

f1(x) = 3(x)2

+ 5(x)4

= 3x2

+ 5x4, f1 .

ii) H f2 x

f2(x) = 3|x| + 1 = 3|x| + 1, f2 .

iii) H f3 x

f3(x) = |x + 1|, fi , ,

f3(1) f3(1).

iv) H f4 x f4(x) = (x)

3 3(x)

5= (x

3 3x

5) = f4(x), f4 .

v) H f5 (, 1) (1, +) -

0. , f5 , .

f5(x) = , , .

vi) H f6

x

f6(x) = = f6(x), f6 .

5. i) H f1 x} = * | x 0} x *

f1 .

ii) H f2

o [2, +)

. , .

f1(x) =1

|x|=

1

|x|= f1(x).

2x

(x)2

+ 1=

2x

x2

+ 1=

2x

x2

+ 1

(x)2

1 x=

x2

1 x

2 11

2+ 1

2xx

2+ 1

6.5. - fi - 91


14/14

iii) H f3

o x

f3(x) = |x 1| |x + 1| = |x + 1| |x 1| = f3(x).

f3

.

iv) H f4 o , fi *

, fi i), fi:

v) H f5 , fi f5(x) = f5(x),

x .

vi) H f6 [1, 1] , fi f6(x) = f6(x),

x [1, 1].

6. i) H Cf (0, 0). f .

ii) H Cg yy. g .

iii) H Ch yy,

O(0, 0). h .

7.

i) H f .ii) H g .

iii) H h , .

8. ) C1, C2 C3 yy.

) C1, C2 C3 fi.

f4(x) =

x2

+ 1

x

x2

+ 1=

1

x

6: 92

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06 my-net-land.blogspot.gr