:
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)
蠠 x 1= x a, y 1 = y (a > 0). (2)
, M 1 M Ox a (. 4) (. 4).
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) )
. 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).
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) )
. 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).
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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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, .