. , , ,

,,,

. .. .

: .

: ..

: ..

.

2005.

:

       酅.3

       ..3

       腅.4

       腅..4

       酅....6

) .......7

) ..8

) 9

) ..10

) ..11

o   Y=sin x.11

o   Y=cos x13

o   Y=tg x..14

o   Y=ctg x15

) ..16

o   Y=arcsin x.16

o   Y=arccos x17

o   Y=arctg x..18

o   Y=arcctg x.19

       ..20

.

(), xⁿ, , 1g , sin , s , tg , ctg x, rcsin , rccos , rctg .

, , .

, = sin (xⁿ) .

.

, :

 

- x, .

● - .

● - .

● - ,

.

● - , .

● ( )- , .

● - f(x)=f(-x).

● - f(-x)=-f(x).

● - 1 2, , 1< 2, f(1)2).

● - 1 2, , 1< 2, f(1)>f(2).

:

● , , . =f(x), f(x) - . , .

● . , .

.

, , , , .

x y X Y. , x X - . y x X, x X y Y.

x .

: -, X x ( ) , -, x y ( Y , ).

, , x X , y ( ). , , .

, y x, :

y=f (x), y=g (x), y=F (x) ..

f, g, F, , x, y. , x, , .

f , ; y: y=y(x). ࠠ , ,

, y=f(x), , x, f(

F (x)=, g (t)=, f(1) f(x) x=1, .. g(5) 2, . .

, , .

, , x, , y. .

, , . , . , , E(x) x. ,

E (1)=1, E (2,5)=2, E ()=3, E (-)=-4 . .,

, E(x), .

, , . C (f (x) = C).

f (x) () X, < , f (f (f () > f ()).

f(x) , X , , x X f(-x)=f(x).

Oy.

f(x) , X , , x X f(-x)=-f(x).

.

() ().

, y(x)=f(x) + g(x). , f(x) g(x) , y (-x) = f(-x) + g(-x) = f (x) + g (x) = y (x). f (x) g (x) , y (x) , y (-x) = f (-x) + g (-x) = -f (x) g (x) = -[f (x) + g (x)] = -y (x). ( ).

, .

, y (x) = f (x)*g (x) f (x) g (x) , y (-x) = f (-x)*g (-x) = f (x)*g (x) = y (x); f (x) g (x) , y (-x) = f (-x)*g(-x) = [-f (x)]*[-g(x)] = y (x); f (x) , g (x) , y (x) = f (x)*g (-x) = f (x)*[-g (x)] = -y (x).

f (x) , x f (x - T) = f (x) = f (x + T). T . T , T, f (x-T) = f [(x - T) +T] = f (x).

T , kT, k (k=f (x 2T) = f [(x)T] = f (x) = f (x), f (x 3T) = f [(x 2T) ] = f (x 2T) = f (x 2T) = f (x); , .

.

 

:

 

        y=kx+b;

        y=xⁿ;

        ;

        (0

        (0 < a1);

        : sin x, cos x, tg x, ctg x;

        : arcsin x, arccos x, arctg x, arcctg x.

.

y = kx + b

1. R , kx+b x

2. k¹0 R

3. , , f (-x) = -kx + b .

4. , , y=b.

5. .

6. k>0, k<0.

7. .

8. y=kx+b . , , , A(0; b) B(-b/k; 0), k¹0. y=kx+b y=kx. k , y=kx Ox, k . k>0, , k<0 ; k=0 Ox.

9. .

10. .

y=kx+b (k<0) y=kx+b (k>0)

.

y=xn,

n- .

1. : D(f)= R;

2. : E(f)= (0;+∞);

3. , .. f(-x)=f(x);

4. : y=0 x=0;

5. x(-∞;0];

6. x[0;+ ∞);

7.               

a)


b)

8. n-, x=0

n-, 򠠠

9. n-,

n-, x=0

10. :

a) n=2, ;

b)蠠 = 3, = 3.   ;

c) 堠 ,  1,     , = 3.

[1]

n

n -

[2]

(1):

1.  : D(f)= R;

2.   [0,+∞];

3.   , .. f(-)=f();

4.  蠠 :  = 0 = 0;

5.   (-∞;0), (0;+∞).

6.   : [1]

:

1.  : D(f)= R;

2.   : E(f)= R;

3.   , .. f(-)=-f();

4.  蠠 :  = 0 = 0;

5.   .

6.   : [2]

.

Y = ax

1.    : -∞ < < +∞

2.    : 0 < y < +∞

3.    , , f(-x) = a-x

4.    .

5.    :

,

= 0

6.    > 1, -∞ < x < +∞ ( .1);

7.    0 < a < 1, -∞ < x < +∞ ( . 2);

8.    (0; 1) .

9. .

10. .

[1]

.

Y = logax

1.     : 0 < x < ∞

2.     : -∞ < y < +∞

3.     , , f(-x) = loga(-x)

4.    

5.     :

= 0

6.     a > 1, 0 < x < +∞ ( .1);

0 < a < 1, ( .2);

7.     (1; 0) 蠠

.

8. .

9. .

[2]

[1]

 

.

y=sin x


y=sin x:

1.     : D(f)=R;

2.     : E(f)=[-1;1];

3.     , .. sin(-x) = - sin x;

4.     2π;

5.     : sin x = 0 x = πk, kZ;

6.     : sin x>0 xπk; π+2πk), kZ;

7.     : sin x<0 xπ+2πk; 2π+2πk), kZ;

8.     [-1;1] xπk; πk], kZ;

9.     [1;-1] xπk; πk], kZ;

10.                    , 1, x=πk, kZ;

11.                    , -1, x=πk, kZ;

12.          

a)

b)


13. .

y=sinx

y=cos x

y=cos x:

1.     : D(f)=R;

2.     : E(f)=[-1;1];

3.     , .. cos (-x) = cos x;

4.     2π;

5.     : cos x = 0 x = πk, kZ;

6.     : cos x>0 xπk; πk), kZ;

7.     : cos x<0 xπk; πk), kZ;

8.     [-1;1] xπ+2πk; 2πk], kZ;

9.     [1;-1] xπk; π+2πk], kZ;

10.                    , 1, x=2πk, kZ;

11.                    , -1, x=π+2πk, kZ;

12.          

a)


b)

13.                    :

y=cosx

y=tg x

y=tg x:

1.     : D(f)=R , x =πk, kZ;

2.     : E(f)=R;

3.     , .. tg (-x) = - tg x;

4.     π;

5.     : tg x = 0 x = πk, kZ;

6.     : tg x>0 xπk; πk), kZ;

7.     : tg x<0 xπk; πk), kZ;

8.     (-xπk ; πk ), kZ;

9.    

a) x=+ πn


b)

10.                                                                                                            :

y=tgx


y=ctg x

y=ctg x:

1.     : D(f)=R , x = πn , nZ;

2.     : E(f)=R;

3.     , .. ctg (-x) = - ctg x;

4.     π;

5.     : ctg x = 0 x = πn, nZ;

6.     : ctg x>0 xπn; πn), nZ;

7.     : ctg x<0 xπn; π +πn), nZ;

8.     (πn ; π +πn), nZ;

9.     a) x= πn x=0

b)

10.                                                                                                                       

: y= ctgx

.

y=arcsin x

y=arcsin x:

1.     : D(f)=[-1;1];

2.     : E(f)=[-

3.     , .. arcsin (-x) = - arcsin x;

4.     : arcsin x = 0 x = 0;

5.     [-1;1];

6.     x=1;

7.     x= -1;

8.    

a)


b)

9.     y = arcsin x:

y=arcsinx

y=arccos x

y=arccos x:

1.     : D(f)=(-1;1);

2.     : E(f)=[0; π];

3.     , ;

4.     : arccos x = 0 x = 1;

5.     (-1;1);

6.     π x =-1;

7.     0 x= 1;

8.     a) x=-1 x=1


b)


9.     y = arccos x:

y=arccosx

y=arctg x

y=arctg x:

1.     : D(f)=R;

2.     : E(f)= (-

3.     , .. arctg (- x) = - arctg x;

4.     : arctg x = 0 x = 0;

5.     R;

6.     a)

b)    y= πn

7.     y = arctg x:

y=arctgx

y=arcctg x

y=arcctg x:

1.     : D(f)=R;

2.     : E(f)= (0; π );

3.     , ;

4.     : arctg x = 0 x =

5.     a)

b) y= πn

6. R;

7. y = arcctg x:


:

ü ..

ü ..

ü .. , .. , ..

. .. . : .

 

 

 

! , , , .
. , :