GATE CSE 2022 Q11: Asymptotic relation between f(n²) and (f(n))²

Which one of the following statements is TRUE for all positive functions $f(n)$ ?

Try it: multiple choice: exactly one correct option

Wrong answer costs 1/3 mark in GATE.

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Answer: A (official GATE 2022 final answer key)

Concept used: Compare growth rates by testing typical functions: polynomials, exponentials and logarithms.

First step: For $f(n) = n^k$ , both $f(n^2)$ and $f(n)^2$ equal $n^{2k}$ ; try $f(n) = 2^n$ and $f(n) = \log n$ to rule out the other options.

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Source: official GATE 2022 CS question paper and final answer key, organised by IIT Kharagpur. Question paper · Answer key