The discussion revolves around determining the value of the function $f(600)$ given the properties of the function $f$ and the specific value $f(500)=3$. The context includes mathematical reasoning and problem-solving related to functional equations.
Participants generally agree on the correctness of Dan's solution, but there is an underlying uncertainty expressed regarding the simplicity of the problem and whether it might overlook critical points.
There is an implicit assumption that the function $f$ behaves consistently under the given functional equation, but this has not been explicitly verified or discussed in detail.
Is this a trick question or did I just stumble upon a quick way to do this? I usually go insane when you post one of these.Albert said:given a function $f$, satisfying :
(1) $f(xy)=\dfrac {f(x)}{y}$ for all $x,y>0$
(2) $f(500)=3$
what is the value of $f(600)$
please don't go insane,your answer is correct,thanks for participating.topsquark said:Is this a trick question or did I just stumble upon a quick way to do this? I usually go insane when you post one of these.
[sp]
[math]f(xy) = \frac{f(x)}{y}[/math]
Let x = 500 and xy = 600. Then y = 600/500 = 6/5.
Thus
[math]f(600) = \frac{f(500)}{\frac{6}{5}} = \frac{3}{\frac{6}{5}} = \frac{5}{2}[/math]
[/sp]
-Dan
I don't usually lack self-confidence but whenever I can quickly answer a problem from you or the POTWs I get the feeling I've overlooked a critical point! Like this one...I wasn't expecting a one line solution so I was unsure about it. Personally I prefer the challenge, though I usually don't post my attempts. I like these. (Yes)Albert said:please don't go insane,your answer is correct,thanks for participating.
By the way ,do you prefer more challenging problems (need a lot of tricks)?