Question

Let \(f\) be a function defined by \[ f(xy)=\frac{f(x)}{y} \] for all positive real numbers \(x\) and \(y\). If \(f(30)=20\), then \(f(40)=\)

• A. \(10\)
• B. \(15\)
• C. \(25\)
• D. \(17\)

Hint:

In functional equations, try to rewrite the required value in terms of the given value so that the functional relation can be applied directly.

Answer 1

The Correct Option is B

Step 1: Write down what we know.
We are given the functional equation $ f(xy) = \frac{f(x)}{y} $ for all positive real $ x, y $. We also know $ f(30) = 20 $. We want to find $ f(40) $.
Step 2: Rewrite 40 in terms of 30.
Express $ 40 = 30 \times \frac{4}{3} $. So if we set $ x = 30 $ and $ y = \frac{4}{3} $, then $ xy = 40 $.
Step 3: Apply the functional equation.
Using $ f(xy) = \frac{f(x)}{y} $ with $ x = 30 $ and $ y = \frac{4}{3} $: \[ f(40) = f\!\left(30 \times \frac{4}{3}\right) = \frac{f(30)}{\frac{4}{3}} \]
Step 4: Substitute the known value.
Since $ f(30) = 20 $: \[ f(40) = \frac{20}{\frac{4}{3}} = 20 \times \frac{3}{4} = 15 \]
Step 5: Verify the answer makes sense.
The functional equation says that multiplying the input by $ y $ divides the output by $ y $. Multiplying 30 by $ \frac{4}{3} $ (making it larger) divides the output by $ \frac{4}{3} $ (making it smaller). So $ f(40) = 15 < 20 = f(30) $, which is consistent.
Step 6: State the final answer.
\[ \boxed{15} \]