Find-all-functions-f-R-R-for-all-x-y-R-such-that-f-f-x-y-yf-x-xf-y-

Question Number 180759 by depressiveshrek last updated on 16/Nov/22

F i n d a l l f u n c t i o n s f : R \to R f o r a l l x, y \in R s u c h t h a t

f (f (x - y) - y f (x)) = x f (y)

Answered by aleks041103 last updated on 17/Nov/22

x = 0

\Rightarrow f (f (- y) - y f (0)) = 0

y = 0

\Rightarrow f (f (x)) = x f (0)

x = y

\Rightarrow f (f (0) - x f (x)) = x f (x)

\Rightarrow f (f (0) - z) = z

v = f (0) - z \Rightarrow z = f (0) - v

\Rightarrow f (v) = f (0) - v

\Rightarrow f (f (0) + y - x - y f (0) + y x) = x (f (0) - y)

f (0) - f (0) - y + x + y f (0) - y x = x f (0) - y x

x - y + y f (0) = x f (0)

y = 0 \Rightarrow x = x f (0) \Rightarrow f (0) = 1

\Rightarrow x - y + y = x \to t r u e

\Rightarrow f (x) = 1 - x