Question
Download Solution PDFComprehension
The function f(x) satisfies \(f(x), f \left( \frac{x}{y} \right) = \frac{f(x)}{f(y)}.\) for all positive real values of x and y, and f(2) = 3
What is f(1)f(4) equal to?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFCalculation:
Given,
The function satisfies the functional equation:
\( f\left(\frac{x}{y}\right) = \frac{f(x)}{f(y)} \)
Also, we are given that:
\( f(2) = 3 \)
We use the functional equation to calculate f(4) . Substituting x = 4 and y = 2 , we get:
\( \frac{f(4)}{f(2)} = f(2) \)
\( \frac{f(4)}{3} = 3 \quad \Rightarrow f(4) = 3 \times 3 = 9 \)
\( \frac{f(2)}{f(2)} = f(1) \quad \Rightarrow f(1) = 1 \)
\( f(1) \times f(4) = 1 \times 9 = 9 \)
Hence, the correct answer is Option 3.
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