find-all-function-f-R-R-such-that-x-y-R-f-x-f-y-f-f-y-xf-y-f-x-1-

Question Number 194688 by CrispyXYZ last updated on 13/Jul/23

find all function f : R \to R such that \forall x, y \in R,

f (x - f (y)) = f (f (y)) + x f (y) + f (x) - 1 .

Answered by Tinku Tara last updated on 13/Jul/23

p u t x = f (y)

f (0) = f (x) + x^{2} + f (x) - 1

f (0) = c

f (x) = \frac{1 + c - x^{2}}{2}

p u t y = 0

f (x - c) = f (c) + c x + f (x) - 1

\frac{1 + c - {(x - c)}^{2}}{2} = \frac{1 + c - c^{2}}{2} + c x + \frac{1 + c - x^{2}}{2} - 1

\frac{1 + c - x^{2}}{2} + c x - \frac{c^{2}}{2} = \frac{1 + c - c^{2}}{2} + c x + \frac{1 + c - x^{2}}{2} - 1

\frac{1 + c}{2} - 1 = 0 \Rightarrow c = 1

f (x) = 1 - \frac{x^{2}}{2}

Commented by CrispyXYZ last updated on 14/Jul/23

Thank you!