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]
\(\displaystyle f(xy) = \frac{f(x)}{y}\)
Let x = 500 and xy = 600. Then y = 600/500 = 6/5.
Thus
\(\displaystyle f(600) = \frac{f(500)}{\frac{6}{5}} = \frac{3}{\frac{6}{5}} = \frac{5}{2}\)
[/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)?
The function $f(x)$ is a mathematical relationship that maps an input value (x) to an output value (f(x)). In other words, it tells us what value f(x) will be for any given x.
The value of $f(500)$ is 3, as given in the problem statement.
Since we know the value of $f(500)$, we can use the knowledge that the function $f(x)$ is a continuous, smooth curve to estimate the value of $f(600)$. We can also use algebraic methods, such as finding the slope of the curve at $x=500$ and using that to extrapolate to $x=600$.
One limitation is that we are assuming the function $f(x)$ is continuous and smooth, which may not always be the case in real-world scenarios. Another limitation is that our estimation may not be accurate if there are significant changes in the behavior of the function between $x=500$ and $x=600$.
We can verify the accuracy of our solution by plugging in the value of $f(600)$ into the original function and checking if it matches our estimation. We can also use a graphing calculator or software to plot the function and visually confirm if our estimation is accurate.