AMC12 2001 A

AMC12 2001 A · Q9

AMC12 2001 A · Q9. It mainly tests Functions basics, Manipulating equations.

Let $f$ be a function satisfying $f(xy) = \frac{f(x)}y$ for all positive real numbers $x$ and $y$. If $f(500) =3$, what is the value of $f(600)$?

设 $f$ 是一个满足 $f(xy) = \frac{f(x)}y$ 对于所有正实数 $x$ 和 $y$ 的函数。若 $f(500) =3$，则 $f(600)$ 的值为多少？

(A) 1 1

(B) 2 2

(D) 3 3

(E) $\frac{18}{5}$ $\frac{18}{5}$

Answer

Correct choice: (C)

正确答案：（C）

Solution

Letting $x = 500$ and $y = \dfrac{6}{5}$ in the given equation, we get $f(600) = f(500\cdot\frac{6}{5}) = \frac{3}{\left(\frac{6}{5}\right)} = \boxed{\textbf{(C) } \frac{5}{2}}$.

在所给等式中令 $x=500$ 且 $y=\dfrac{6}{5}$，得到 $f(600)=f\left(500\cdot\frac{6}{5}\right)=\frac{3}{\left(\frac{6}{5}\right)}=\boxed{\textbf{(C) } \frac{5}{2}}$。

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