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(16) equal to?

Correct answer: D. 81 Given f((x)/(y)) = (f(x))/(f(y)) Substituting x = 1 and y = 1 , we get f(1) = 1 Substituting x = 1 , we get f((1)/(y)) = (f(1))/(f(y)) = (1)/(f(y)) Replacing y with (1)/(y) in the given equation, we get: f(xy) = (f(x))/(f((1)/(y))) = 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

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(16)

equal to?

Options

A $18$
B $27$
C $54$
D $81$

Correct answer

D. $81$

Step-by-step 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

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