x-y-R-Find-all-functions-that-satisfy-this-condition-f-x-y-f-x-f-y-f-x-y-1-Find-all-functions-that-satisfy-this-condition-f-f-x-f-x-x-

Question Number 150311 by naka3546 last updated on 11/Aug/21

x, y \in R

F i n d a l l f u n c t i o n s t h a t s a t i s f y t h i s c o n d i t i o n :

f (x + y) = f (x) \cdot f (y) - f (x \cdot y) + 1

F i n d a l l f u n c t i o n s t h a t s a t i s f y t h i s c o n d i t i o n :

f (f (x)) = f (x) + x

Answered by aleks041103 last updated on 11/Aug/21

1 \Rightarrow f (x + 0) = f (x) = f (0) . f (x) - f (0) + 1

(1 - f (0)) f (x) = - (1 - f (0))

I f f (0) \neq 1 t h e n f (x) = - 1 = c o n s t . b u t

t h i s {d o e s n}^{'} t s a t i s f y 2 .

T h i s l e a v e s f (0) = 1, w h i c h t r i v i a l l y s a t i s f i e s

1 . w h e n y = 0 .

T h e n b y 2 .

f (f (0)) = f (1) = f (0) + 0 = 1

f (f (1)) = f (1) = f (1) + 1 \to c o n t r a d i c t i o n

\Rightarrow N o s u c h f u n c t i o n s e x i s t!