CMI 2024 | Data Science | Part A | Question: 18

Question

Let $f$ be a function on the positive real numbers such that $f(x y)=f(x)+f(y)$. If $f(2024)=2$ then which of the following statement(s) is/ are true?

• A. $f\left(\frac{1}{2024}\right)=1$
• B. $f\left(\frac{1}{2024}\right)=-1$
• C. $f\left(\frac{1}{2024}\right)=-2$
• D. $f\left(\frac{1}{2024}\right)=2$

   

Answer 1

Given:

$ f(xy) = f(x) + f(y) $

put $ x = y = 1 $:
$ f(1) = f(1) + f(1) \Rightarrow f(1) = 2f(1) \Rightarrow f(1) = 0 $

$ f(2024 \cdot \frac{1}{2024}) = f(2024) + f\left(\frac{1}{2024}\right) $

So,
$ f(1) = 2 + f\left(\frac{1}{2024}\right)  $

$ \Rightarrow f\left(\frac{1}{2024}\right) = -2 $