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Prove or disprove that if f(n) = (g(n)), then 4f(n): = (49(n)).

Prove or disprove that if f(n) = (g(n)), then 4f(n): = (49(n)).

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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**Question: Analyze the Big-Theta Notation in Exponential Expressions** **Objective:** Prove or disprove the following statement in the context of algorithm analysis and computational complexity: **Statement:** Given that \( f(n) = \Theta(g(n)) \), determine whether \( 4^{f(n)} = \Theta(4^{g(n)}) \). **Detailed Explanation:** - **\( f(n) = \Theta(g(n)) \):** This means that the function \( f(n) \) grows asymptotically at the same rate as \( g(n) \). In mathematical terms, there exist positive constants \( c_1 \), \( c_2 \), and \( n_0 \) such that for all \( n \geq n_0 \), \[ c_1 g(n) \leq f(n) \leq c_2 g(n). \] - **Exponential Function Transformation:** The statement asks us to prove or disprove whether transforming the functions into the exponential format \( 4^{f(n)} \) and \( 4^{g(n)} \) preserves their asymptotic bounds. Specifically, it questions whether \[ 4^{f(n)} = \Theta(4^{g(n)}). \] To analyze this, one would typically need to consider properties of exponential functions and their growth rates.
Transcribed Image Text:**Question: Analyze the Big-Theta Notation in Exponential Expressions** **Objective:** Prove or disprove the following statement in the context of algorithm analysis and computational complexity: **Statement:** Given that \( f(n) = \Theta(g(n)) \), determine whether \( 4^{f(n)} = \Theta(4^{g(n)}) \). **Detailed Explanation:** - **\( f(n) = \Theta(g(n)) \):** This means that the function \( f(n) \) grows asymptotically at the same rate as \( g(n) \). In mathematical terms, there exist positive constants \( c_1 \), \( c_2 \), and \( n_0 \) such that for all \( n \geq n_0 \), \[ c_1 g(n) \leq f(n) \leq c_2 g(n). \] - **Exponential Function Transformation:** The statement asks us to prove or disprove whether transforming the functions into the exponential format \( 4^{f(n)} \) and \( 4^{g(n)} \) preserves their asymptotic bounds. Specifically, it questions whether \[ 4^{f(n)} = \Theta(4^{g(n)}). \] To analyze this, one would typically need to consider properties of exponential functions and their growth rates.
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