Question

Prove that if \(f, g\), and \(h\) are functions from \(R^{+}\)to \(R^{+}\)such that \(f(x)=O(g(x))\) and \(g(x)=O(h(x))\), then \(f(x)=O(h(x))\).

Short answer

If \(f(x) = O(g(x))\) and \(g(x) = O(h(x))\), then \(f(x) = O(h(x))\).

Step-by-step solution

Understand Big O Notation

Big O notation, written as \(f(x) = O(g(x))\), means that there exist positive constants \(c\) and \(k\) such that for all \(x > k\), \(|f(x)| \leq c|g(x)|\). This implies that \(f(x)\) grows no faster than \(g(x)\) up to a constant factor.

Analyze Given Information

We are given that \(f(x) = O(g(x))\) and \(g(x) = O(h(x))\). This means there exist constants \(c_1, k_1\) such that \(|f(x)| \leq c_1|g(x)|\) for all \(x > k_1\), and constants \(c_2, k_2\) such that \(|g(x)| \leq c_2|h(x)|\) for all \(x > k_2\).

Setting Up Compound Inequality

To prove \(f(x) = O(h(x))\), we need \(|f(x)| \leq c_3|h(x)|\) for some positive constants \(c_3\) and \(k_3\). Using the provided inequalities, we substitute: \(|f(x)| \leq c_1|g(x)|\) and \(|g(x)| \leq c_2|h(x)|\). Therefore, \(|f(x)| \leq c_1c_2|h(x)|\).

Define Constants for Final Inequality

From the compound inequality \(|f(x)| \leq c_1c_2|h(x)|\), define \(c_3 = c_1c_2\). This leads to \(|f(x)| \leq c_3|h(x)|\). We also need \(x > k_3\), where \(k_3\) is the maximum of \(k_1\) and \(k_2\), to ensure both conditions are satisfied.

Conclusion of the Proof

By choosing \(c_3 = c_1c_2\) and \(k_3 = \max(k_1, k_2)\), we have shown that \(f(x)\) satisfies the condition for \(O(h(x))\), completing the proof that \(f(x) = O(h(x))\).

Key concepts

Asymptotic Analysis

Asymptotic Analysis is a powerful method used to describe the behavior of functions as they tend to infinity. It is particularly useful in computer science and mathematics for understanding the efficiency of algorithms and the growth of functions.
The idea is simple: we want to focus on the dominant parts of functions that determine their behavior for large values. This means that the constant factors and lower-order terms, which have less impact as inputs grow, are often ignored.
When we say a function \( f(x) \) is \( O(g(x)) \), we mean there is a ceiling limit to how fast \( f(x) \) can grow compared to \( g(x) \). This helps us to compare and analyze the efficiency of different algorithms and mathematical functions when inputs are very large.

• \( O(g(x)) \): Big O notation indicates an upper bound on the function's growth.
• Focuses on worst-case scenarios, which is essential for algorithms' time complexity analysis.

Through Asymptotic Analysis, we simplify complex expressions and focus only on growth rates that matter, making it easier to manage and understand large-scale computations.

Transitive Property in Functions

The Transitive Property is an important concept in mathematics that states if one element relates to a second, and that second relates to a third, then the first element is related to the third.
In the context of Big O Notation for functions, it can be understood as follows: if \( f(x) = O(g(x)) \) and \( g(x) = O(h(x)) \), then it follows that \( f(x) = O(h(x)) \).
This happens because the relationship expressed by \( O \) notation is based on inequalities that can be compounded. We can infer the relation between \( f \) and \( h \) from the known relations:

• \( |f(x)| \leq c_1 |g(x)| \) for some \( c_1 \) for sufficiently large \( x \).


• \( |g(x)| \leq c_2 |h(x)| \) for some \( c_2 \) similarly large.

Combining, we get \( |f(x)| \leq c_1c_2 |h(x)| \), using the new constant \( c_1c_2 \) as the multiplicative factor, effectively proving the transitive property.
This concept is crucial in algorithm analysis, where complex problems can be broken down into simpler terms by chaining known behaviors.

Function Growth Rates

Understanding Function Growth Rates helps in distinguishing how different functions behave as their input size increases. This is crucial in computer science for time complexity analysis and in mathematics to evaluate limits and rates of change.
With Big O Notation, we categorize functions based on how quickly they grow:

• Logarithmic: The slowest growth rate. Often appears in divide-and-conquer algorithms.


• Linear: Growth corresponding directly with input size. Typical of simple loops.


• Polynomial: Includes quadratic, cubic, and higher powers. Growth accelerates with degree.


• Exponential: Grows very quickly; often seen in combinations and permutations.

Analyzing Function Growth Rates allows us to predict how a function behaves, especially for very large inputs. Recognizing patterns helps in designing algorithms better and in more effective resource management.
Being able to express and compare these rates using Big O notation equips you with a sharper analytical tool, potentially saving time and computational power in complex systems.