Q245 / 269 · NDA 2025 (Phase 1) · Mathematics
Consider the following for the two (02) items that follow:
The function $f(x)$ satisfies $f\left(\dfrac{x}{y}\right) = \dfrac{f(x)}{f(y)}$ for all positive real values of $x$ and $y$, and $f(2) = 3$.
What is $f(1)f(4)$ equal to?
The function $f(x)$ satisfies $f\left(\dfrac{x}{y}\right) = \dfrac{f(x)}{f(y)}$ for all positive real values of $x$ and $y$, and $f(2) = 3$.
What is $f(1)f(4)$ equal to?
- A$4$
- B$8$
- Correct answerC$9$
- D$18$
Solution
Given $f\left(\dfrac{x}{y}\right) = \dfrac{f(x)}{f(y)}$ for all $x, y > 0$.
Substituting $x = 1$ and $y = 1$, we get:
$f(1) = \dfrac{f(1)}{f(1)} \Rightarrow f(1) = 1$
Substituting $x = 4$ and $y = 2$, we get:
$f\left(\dfrac{4}{2}\right) = \dfrac{f(4)}{f(2)}$
$f(2) = \dfrac{f(4)}{f(2)} \Rightarrow f(4) = (f(2))^2$
Since $f(2) = 3$, we have $f(4) = 3^2 = 9$.
Therefore, $f(1)f(4) = 1 \times 9 = 9$.
Answer: $9$
Substituting $x = 1$ and $y = 1$, we get:
$f(1) = \dfrac{f(1)}{f(1)} \Rightarrow f(1) = 1$
Substituting $x = 4$ and $y = 2$, we get:
$f\left(\dfrac{4}{2}\right) = \dfrac{f(4)}{f(2)}$
$f(2) = \dfrac{f(4)}{f(2)} \Rightarrow f(4) = (f(2))^2$
Since $f(2) = 3$, we have $f(4) = 3^2 = 9$.
Therefore, $f(1)f(4) = 1 \times 9 = 9$.
Answer: $9$
Hence, the correct answer is option: 'C'.