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IMO SL-7(2009)
Posted: Fri Mar 09, 2012 10:22 pm
by Sazid Akhter Turzo
Find all functions f from the set of real numbers into the set of real
numbers which satisfy for all real x, y the identity
$f (x f (x+y)) = f (y f (x))+x^2$.
I'm trying to solve it for 5 hours but still couldn't solve (
). I've prove its surjectivity but couldn't prove that it's injective.
Re: IMO SL-7(2009)
Posted: Sat Mar 10, 2012 12:03 am
by nafistiham
Sazid Akhter Turzo wrote:Find all functions f from the set of real numbers into the set of real
numbers which satisfy for all real x, y the identity
f (
x f (x+y)) = f (y f (x))+
x2.
I'm trying to solve it for 5 hours but still couldn't solve (
). I've prove its surjectivity but couldn't prove that it's injective.
I think the first red part means multiplying $x,f(x+y)$ and the second one means $x^2$
[using latex would be good
]
Re: IMO SL-7(2009)
Posted: Sat Mar 10, 2012 12:33 am
by *Mahi*
Re: IMO SL-7(2009)
Posted: Sat Mar 10, 2012 2:13 am
by zadid xcalibured
plugging x=0 we get f(o)=f(yf(o)).this implies the functional equation is constant.but how?this is an sl prblm
Re: IMO SL-7(2009)
Posted: Sat Mar 10, 2012 9:35 am
by *Mahi*
zadid xcalibured wrote:plugging x=0 we get f(o)=f(yf(o)).this implies the functional equation is constant.but how?this is an sl prblm
What if $f(0)=0$?Then all we get is $f(0)=0$.
Re: IMO SL-7(2009)
Posted: Sat Mar 10, 2012 10:56 am
by zadid xcalibured
oh sorry this only implies f(o)=o