Find-all-f-R-R-such-that-f-x-2-yf-x-xf-x-y-

Question Number 677 by prakash jain last updated on 22/Feb/15

Find all f : R \to R such that

f (x^{2} + y f (x)) = x f (x + y)

Commented by 123456 last updated on 22/Feb/15

s u p o s i n g f (x) = a x + b

f [x^{2} + y f (x)] = x f (x + y)

f [x^{2} + y (a x + b)] = x [a (x + y) + b]

f (x^{2} + a x y + b y) = x (a x + a y) + b x

a (x^{2} + a x y + b y) + b = {a x}^{2} + a x y + b x

{a x}^{2} + a^{2} x y + a b y + b = {a x}^{2} + a x y + b y

i f b = 0 \Rightarrow {a x}^{2} + a^{2} x y = {a x}^{2} + a x y \Leftrightarrow a x y = 0 \lor a = 1, f (x) = x

i f a = 0 \Rightarrow b = b y \Leftrightarrow y = 1 \lor b = 0, f (x) = 0

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