# finding the value of $f(2001)$ if.....

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?

• You should always post an attempt at your own solution here. Where exactly did you fail? Aug 3 '14 at 8:39
• 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. Aug 3 '14 at 9:40
• @Aaron Please also note that $f(1) =1$. Aug 3 '14 at 9:52
• Replacing $x$ with $xy$ suggests $f(xy) = f(x)f(y)$. Aug 3 '14 at 11:59
• I changed my question slightly. Now is it possible? #Aaron Aug 4 '14 at 9:01

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 $$f(2001)={2001\over2000}.$$