Find-all-f-R-R-such-that-f-x-f-x-f-y-f-y-f-x-x-f-y-f-f-y-for-all-x-y-R-

Question Number 26576 by gunawan last updated on 27/Dec/17

Find all f : R \to R such that

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

for all x, y \in R

Commented by prakash jain last updated on 27/Dec/17

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

There is bracket missing on RHS .

Is it at the end ?

Commented by gunawan last updated on 27/Dec/17

i^{'} m sorry sir

Commented by prakash jain last updated on 27/Dec/17

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

y = x = 0

f (0 + 2 f (0)) = f (0 + f (0)) + 0 + f (0) - f (f (0))

f (2 f (0)) = f (0)

a s s u m e f (0) = 0

f (f (y)) = f (y) + f (y) - f (f (y))

f (f (y)) = f (y)

f (x) = x (A)

c a s e f (0) \neq 0 and f (a) = 0 for some a

x = y = a

f (a + f (a) + f (a)) = f (a + f (a)) + a + f (a) - f (f (y > a))

0 = 0 + a + 0 - f (0)

f (0) = a

f (2 f (0)) = f (0) \Rightarrow f (2 a) = a

x = a, y = 0

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

f (a + f (a) + f (0)) = f (0 + f (a)) + a + f (0) - f (f (0))

f (2 a) = f (0) + a + f (0) - f (a)

f (2 a) = a + a + a - 0 = 3 a

i f f (a) = 0 for some a \Rightarrow a = 0

c a s e f (x) \neq 0 \forall x \in R

c o n t i n u e

Commented by gunawan last updated on 28/Dec/17

thanks Sir

Yes

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