.


:




:

































 

 

 

 


2 :




:

1. f () f (- ) )

2. f(x) f(x a) ( )

3. f (x) f (x)+ b ( )

4. f() f(k) ( () )

5. f() mf() ( () )

1. f () f (- )
( )

, (; ) 1( 1; 1) , + 1 = 0( = 1), y = y 1. , : + 1 = 0 y = y 1, (; ) 1( 1; 1) (. 1). f ( x), f (x).

. 1 . 2

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

. :

1) = 2 ; 2) = 2 ; 3) = 2 .

. :

3 2 1 0 1 2 3
y 8 4 2 1

, . , f () = 2 x, f ( ) = 2 x , f () = 2 x. , (. 3).

 


= 2- 4 = 2

 

1

 

0

 

= 2

. 3

2. f(x) f(x a)
( )

M (x; y) M 1(x 1; y 1), :

x 1= x + a, y 1 = y (a > 0) (1)

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

, M 1 M Ox a (. 4) (. 4).

 

 

 


) )

. 4

, (; ) (a > 0) , M 1(x 1; y 1) (1) (2) (. 4).

f (x) f( + ) f ( ). , (; ) f (x) 1( 1; 1) f ( + ) , = 1 .. f () = f (x 1 ++ ), = 1 + x 1 = . , 1( 1; 1) f (x + ), (; ) f() a . , = f( + ) = f (x) .

= f (x ) = f () (. 5).

f ( + ) f ( ) ( > 0) , f (), , , , = f ( ), , = f ( +a).

. .

f ( ) ( > 0). , , '', ' '' . = f (), . = f (x + ) ( > 0).

.

, , = ( + 2)2 (. 6).

. 2 , . '' = 2. = ( + 2)2 O. . = 0, = 22 = 4; y = 0, ( + 2)2 = 0, x = 2.

 

 

, A (0; 4) ' (2; 0) Ox (. 5).

3. f (x) f (x)+ b
( )

(; ) M 1(x 1; y 1), :

x 1 = , 1 = + b. (1)

, 1 b , b > 0 (.7), , b < 0 (. 7).

 

 


) )

. 7

, , (; ) b (b > 0) (b < 0), M 1(x 1; y 1) (1).

, M 1(x 1; y 1) = f()+b (; ) y = f() b , b > 0, , b < 0.

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

, = f () + b = f () b , b > 0, , b < 0 (. 8).

.

, = f () + b. () , b , b > 0(b < 0), , . ''. y = f (x), . b < 0.

. = 2 + 1.

, 1 , . '' = 2. (. 9).

 

. 8 . 9

4. f() f(k)
( () )

(; ) M 1(x 1; y 1), :

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

, M 1 (0 < k < 1) k (k > 1) . , (0 < k < 1) k (k > 1), (; ) M 1(x 1; y 1) (1).

M 1(x 1; y 1) = f(k) (0< k <1) k (k > 1) (;) = f (). , 1 = f (k 1) = f (), k 1 = x, 1= , .. (1). , = f(k) (. 10) = f () x, (0< k <1), k , k > 1 ( Oy, Oy).

 

 

 


. 10

k < 0 () Oy.

5. f() mf()
( () )

(; ) M 1(x 1; y 1), :

x 1 = , 1 = . (1)

, 1 ( > 1) (0 < < 1) . , () m > 1 ( 0 < m < 1) , (: ) M 1(x 1; y 1) (1).

M 1(x 1; y 1) = mf (x) (m > 1) (0 < m < 1) (; ) = f (x). , x 1= x, y 1 = mf (x 1) = mf (x) = m, .. (1).

, = f (x) (. 11) y = f(x) m Oy, m > 1 , 0 < m < 1( Ox, Ox).

m < 0 () Ox.

. y = 1,5 sin x.

. = sin , 1,5 (. 12).

 

 

c.11 . 12

3
: f () f ( + ) + b

:

1. f () f ( + ) + b

2. f(x) mf(x + a) + b, mf(kx), mf(kx + a) + b

3.

4.

1. f () f ( + ) + b

= f ( + ) + b .

I . (. 1). = f () x || > 0 < 0, | b | b > 0 b < 0, .. .

, , , .

II . (. 2), , .. , . 'O'' = f (), . , ¢ (- , b), .. xO O (0; 0), ' ( ; b). , ¢ 'O'' (0, 0).

 

 

. 1 . 2

2. f(x) mf(x + a) + b, mf(kx), mf(kx + a) + b

= f (x), = f ( + ) + b : , ' ( , b), , '' 'y' . = f(), .

= f (k) : = f (x), () | k | | m |. m < 0 (k < 0) () ().

= b = 0, f (k + ) + b
= f (k). . , b .

(1)

:

(2)

(2) : , ''', .. ' () '' '', . = f (k). (1) .

, .





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