Tìm tất cả các hàm $f: \mathbb{R} \to \mathbb{R}$ sao cho $$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x)$$
#2
Posted 17-05-2016 - 12:35
Zaraki
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#3
Posted 17-05-2016 - 12:41
baopbc
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Himura Kenshin
#5
Posted 22-05-2016 - 22:26
superpower
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Sĩ quan
Tìm tất cả các hàm $f: \mathbb{R} \to \mathbb{R}$ sao cho $$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x)$$
$R -> R $ thì đơn giản hơn
Ta có $x=y=z=0 => f(0) = 2f(0) => f(0)=0$
$z=0 => f(x+y)=f(x)+f(y) (1)$
$y=0 => zf(x) = f(xf(z)) => f$ đơn ánh $=> f(1) =1 => f(f(z))=z $
Mặt khác $f(z)f(x)=f(xz) (2)$
Từ $(1);(2)$ suy ra $f(x)=x $
Edited by baopbc, 25-05-2016 - 22:18.
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