Q244 / 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 $f(16)$ 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 $f(16)$ equal to?
- A$18$
- B$27$
- C$54$
- Correct answerD$81$
Solution
Given $f\left(\dfrac{x}{y}\right) = \dfrac{f(x)}{f(y)}$
Substituting $x = 1$ and $y = 1$, we get $f(1) = 1$
Substituting $x = 1$, we get $f\left(\dfrac{1}{y}\right) = \dfrac{f(1)}{f(y)} = \dfrac{1}{f(y)}$
Replacing $y$ with $\dfrac{1}{y}$ in the given equation, we get:
$f(xy) = \dfrac{f(x)}{f\left(\dfrac{1}{y}\right)} = f(x)f(y)$
Using this property, $f(16) = f(2^4) = (f(2))^4$
Given $f(2) = 3$, we have $f(16) = 3^4 = 81$
Answer: $81$
Substituting $x = 1$ and $y = 1$, we get $f(1) = 1$
Substituting $x = 1$, we get $f\left(\dfrac{1}{y}\right) = \dfrac{f(1)}{f(y)} = \dfrac{1}{f(y)}$
Replacing $y$ with $\dfrac{1}{y}$ in the given equation, we get:
$f(xy) = \dfrac{f(x)}{f\left(\dfrac{1}{y}\right)} = f(x)f(y)$
Using this property, $f(16) = f(2^4) = (f(2))^4$
Given $f(2) = 3$, we have $f(16) = 3^4 = 81$
Answer: $81$
Hence, the correct answer is option: 'D'.