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Chapter 3: Problem 16

Show that if \(f(x)\) is \(O(x),\) then \(f(x)\) is \(O\left(x^{2}\right)\)

Short Answer

Expert verified
If f(x) is O(x), then f(x) is also O(x^2).

Step by step solution

01

Understand Big-O Notation

Recall that a function f(x) is said to be O(g(x)) if there exist positive constants C and k such that \[ |f(x)| \leq C|g(x)| \text{ for all } x \geq k \]
02

Express f(x) as O(x)

Given that f(x) is O(x), there exist positive constants C1 and k1 such that \[ |f(x)| \leq C1|x| \text{ for all } x \geq k1 \]
03

Relate O(x) to O(x^2)

We need to show that there exist positive constants C2 and k2 such that \[ |f(x)| \leq C2|x^2| \text{ for all } x \geq k2 \]
04

Determine constants C2 and k2

Notice that \[ |x| \leq |x^2| \text{ for all } x \geq 1 \] So, we can take C2 to be the same as C1 and k2 to be \( \max(k1, 1) \)
05

Combine the results

By combining the previous steps, we have \[ |f(x)| \leq C1|x| \leq C1|x^2| \text{ for all } x \geq \max(k1, 1) \] Thus, f(x) is O(x^2)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Asymptotic Analysis
Asymptotic analysis helps us study the efficiency of algorithms as the input size grows. It focuses on the behavior rather than the exact values. When we use asymptotic analysis, we describe the approximate performance instead of precise details. In our exercise, we use asymptotic analysis to show how a function behaves for large inputs.

The notation used in asymptotic analysis includes Big-O, Big-Omega, and Big-Theta. For this problem, we focus on Big-O notation, which describes an upper bound on the time complexity. This means it tells us the worst-case scenario for an algorithm's growth rate.
Time Complexity
Time complexity measures the amount of time an algorithm takes to run as a function of the input size. We often express time complexity using Big-O notation.

In this problem, we are given that a function f(x) has a time complexity of O(x). This means that there are constants C1 and k1 such that:o [ f(x) 鈮� C1 * x for all x 鈮� k1 ]

We need to show that f(x) is also O(x^2). For this, we must find constants C2 and k2 so that:

o [ f(x) 鈮� C2 * x^2 for all x 鈮� k2]

Since o [ x 鈮� x^2 for all x 鈮� 1] we can conclude that O(x) is indeed O(x^2), with appropriate constants.
Mathematical Proof
Mathematical proofs help us validate statements with logical reasoning and steps. For this exercise, we follow a series of steps to prove the relation between O(x) and O(x^2).

First, we know that f(x) is O(x). This implies that
o[ |f(x)| 鈮� C1 * x for all x 鈮� k1 ]

Next, we need to show that f(x) is also O(x^2).

To prove this, we notice that: o [ |x| 鈮� |x^2| for all x 鈮� 1 ]
So we can take the same constant C1 and a new constant k2 = max(k1, 1). Then we have:
o [ |f(x)| 鈮� C1 * x 鈮� C1*x^2 for all x 鈮� k2 ]

This completes the proof and shows that f(x) is indeed O(x^2). Thus, we have used a logical sequence of steps to prove the required relationship.

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Most popular questions from this chapter

Big- \(O,\) big-Theta, and big-Omega notation can be extended to functions in more than one variable. For example, the statement \(f(x, y)\) is \(O(g(x, y))\) means that there exist constants \(C\) , \(k_{1},\) and \(k_{2}\) such that \(|f(x, y)| \leq C|g(x, y)|\) whenever \(x>k_{1}\) and \(y>k_{2} .\) Define the statement \(f(x, y)\) is \(\Theta(g(x, y))\)

Compare the number of comparisons used by the insertion sort and the binary insertion sort to sort the list \(7,4,\) \(3,8,1,5,4,2 .\)

Devise an algorithm that finds the sum of all the integers in a list.

Show that the problem of determining whether a program with a given input ever prints the digit 1 is unsolvable.

What is the largest \(n\) for which one can solve within a day using an algorithm that requires \(f(n)\) bit operations, where each bit operation is carried out in \(10^{-11}\) seconds, with these functions \(f(n) ?\) $$ \begin{array}{llll}{\text { a) } \log n} & {\text { b) } 1000 n} & {\text { c) } n^{2}} \\ {\text { d) } 1000 n^{2}} & {\text { e) } n^{3}} & {\text { f) } 2^{n}} \\ {\text { g) } 2^{2 n}} & {\text { h) } 2^{2^{n}}}\end{array} $$
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