if $f (\frac{x}{y}) =\frac{f(x)}{y} $ and $f(2000)=1$ ; then what's the value of $f(2001)$. I tried hard but can't figured out anything. please help me, how can I proceed?
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asked Aug 3, 2014 at 8:38
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$\begingroup$ You should always post an attempt at your own solution here. Where exactly did you fail? $\endgroup$– StijnAug 3, 2014 at 8:39
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3$\begingroup$ Do you know anything else about the function? Is it continuous? Is it defined only for integers? Does it take values in the integers? Without additional information, I don't think there is enough here to determine the value. $\endgroup$– AaronAug 3, 2014 at 9:40
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2$\begingroup$ @Aaron Please also note that $f(1) =1 $. $\endgroup$– William WuAug 3, 2014 at 9:52
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2$\begingroup$ Replacing $x$ with $xy$ suggests $f(xy) = f(x)f(y)$. $\endgroup$– Yiyuan LeeAug 3, 2014 at 11:59
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1$\begingroup$ I changed my question slightly. Now is it possible? #Aaron $\endgroup$– Fazla Rabbi MashrurAug 4, 2014 at 9:01
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1 Answer
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Put $x=y \neq 0$ and we get $$f(1)={f(x)\over x} \Rightarrow f(x)=f(1)x.$$ Since $f(2000)=1$, $f(1)=1/2000$. Therefore $$f(2001)={2001\over2000}.$$
