5This is Discreet math please answer questions with proper formatting for proofs For each part below, determine whether:f(x) is O(g(x))• g(x) is O(f(x))• J(r) is O(g(x)) and g(x) is O(f(x))none of the above.(a) f(x) = 3r+log(r), g(x) = x + (log(x))²(b) f(x)=r!, g(x) = 1²(c) f(x) = g(x) = log(r)(d) f(x)=x20 +¹.9 +¹8..., g(x) = x².1

Answered: For each part below, determine whether:…

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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This is Discreet math please answer questions with proper formatting for proofs

For each part below, determine whether:
f(x) is O(g(x))
• g(x) is O(f(x))
• J(r) is O(g(x)) and g(x) is O(f(x))
none of the above.
(a) f(x) = 3r+log(r), g(x) = x + (log(x))²
(b) f(x)=r!, g(x) = 1²
(c) f(x) = g(x) = log(r)
(d) f(x)=x20 +¹.9 +¹8..., g(x) = x².1

Expert Solution

Step 1 forrg for (a) f(x) = 3x + log(x), g(x) = x + (log(x))^2

(a) f(x) is O(g(x)) and g(x) is not O(f(x)) To show that f(x) is O(g(x)), we need to find positive constants c and k such that f(x) <= c*g(x) for all x > k. Taking c = 4 and k = 1, we have:

3x + log(x) <= 4*(x + (log(x))^2) for all x > 1

which is true. Therefore, f(x) is O(g(x)).

To show that g(x) is not O(f(x)), we need to show that for any positive constant c and any k, there exists an x > k such that g(x) > c*f(x). Let c = 1 and k = 1. Then for any x > e, we have:

x + (log(x))^2 > 3x + log(x)

Therefore, g(x) is not O(f(x)).

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