Consider the following for the two (02) items that follow: The function f(x) satisfies f((x)/(y)) = (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?

Correct answer: C. 9 Given f((x)/(y)) = (f(x))/(f(y)) for all x, y > 0 . Substituting x = 1 and y = 1 , we get: f(1) = (f(1))/(f(1)) f(1) = 1 Substituting x = 4 and y = 2 , we get: f((4)/(2)) = (f(4))/(f(2)) f(2) = (f(4))/(f(2)) f(4) = (f(2))^2 Since f(2) = 3 , we have f(4) = 3^2 = 9 . Therefore, f(1)f(4) = 1 × 9 = 9 . Answer: 9

NDA 2025

Exam: NDA
Year: 2025
Subject: Mathematics
Chapter: Functions

NDA 2025 · Mathematics · Functions

Consider the following for the two (02) items that follow: The...

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?

Options

A $4$
B $8$
C $9$
D $18$

Correct answer

C. $9$

Step-by-step 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

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