Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Decompose the functions by finding functions \(f(x)\) and \(g(x)\), \(f(x) \neq x\) and \(g(x) \neq x\), such that \(h(x)=f(g(x))\). $$ h(x)=\frac{1}{x^{2}+4} $$

Step-by-Step Solution

Answer

The function \(h(x) = \frac{1}{x^{2}+4}\) can be decomposed into two functions, where \(f(x) = \frac{1}{x}\) and \(g(x) = x^{2}+4\).

Step 1: Understanding Function Composition

Function composition is a method of combining existing functions to create new functions. In the problem, it outlines the composition as \(h(x) = f(g(x))\), where \(f(x)\) and \(g(x)\) are the constituent functions of \(h(x)\).

Step 2: Identify the functions

identify the inner function \(g(x)\) and the outer function \(f(x)\) in the given function \(h(x)\). By looking at \(h(x) = \frac{1}{x^{2}+4}\), it's seen that \(x^{2}+4\) is inside the fraction that has composition of functions. Hence, select \(g(x) = x^{2}+4\). Now to determine \(f(x)\), take \(g(x)\) and substitute into \(h(x)\) to get \(f(g(x))\). So, \(f(x) = \frac{1}{x}\).

Step 3: Verify the function decomposition

To make sure the decomposition is correct, substitute \(g(x)\) into \(f(x)\) and the result should be the original function \(h(x)\). Thus, \(f(g(x)) = \frac{1}{x^{2}+4}\), and this matches the given function \(h(x)\).

Key Concepts

Function Decomposition

Function decomposition involves breaking down a complicated function into simpler constituent functions. This is a fantastic way to simplify the process of understanding and manipulating functions. In the given problem, we have the function \[ h(x) = \frac{1}{x^{2}+4} \]which we need to decompose into two functions: \(f(x)\) and \(g(x)\). These functions are combined through composition to form \(h(x)\).

Decomposing functions can be thought of like peeling an onion. Each layer you peel back helps you to understand the bigger picture. The goal is to identify two functions where \(h(x) = f(g(x))\), ensuring that neither function is just \(x\) itself, otherwise decomposition wouldn't be meaningful. By breaking \(h(x)\) down into the simpler functions \(f\) and \(g\), it helps us to understand how the function is operating and might make solving further problems with it easier.

Inner Function

In the context of function decomposition, the inner function is the part of the composite function that serves as the input to another function. For our example, let's examine \[h(x) = \frac{1}{x^{2}+4} \].

What happens inside the main operation? That would be the expression \(x^{2} + 4\). This expression is encapsulated within the larger structure of the function, making it our inner function, denoted as \(g(x)\).

Choose

• \(g(x) = x^{2} + 4\)

This means that all actions performed by \(f(x)\) involve \(g(x)\). Thus, the inner function serves as the foundation for the overall operation of the composite function. Understanding it is crucial because it reveals what is affected by the outer function.

Outer Function

The outer function encapsulates or encloses the inner function in a composition. It determines how the inner function's result will be used and processed. In simpler terms, it acts as an overarching structure that governs the whole process.

For the function \[ h(x) = \frac{1}{x^{2}+4} \], the result of \(g(x) = x^{2}+4\) becomes the argument for \(f(x)\). In our decomposition, the goal is to find \(f(x)\). Since \(g(x)\) will be replaced into \(f\),

• \(f(x) = \frac{1}{x}\)

This means \(f(x)\) encloses the input given by \(g(x)\). After obtaining \(g(x)\), it makes use of this result to compute the final output \(h(x)\). Understanding the role of the outer function helps us further unravel the interaction between \(f(x)\) and \(g(x)\) when building or deconstructing composite functions.

Composite Function

A composite function emerges when you apply one function to the result of another function. It blends the actions of two functions into one streamlined operation, simplifying complex problems into manageable steps. This is conveyed through the notation \((f \circ g)(x)\) or \(f(g(x))\).

In our specific case, \[ h(x) = f(g(x)) = \frac{1}{x^{2}+4} \],we see that the composite function is actually created by using \(g(x) = x^{2}+4\) within \(f(x) = \frac{1}{x}\).

• The beauty of composite functions lies in breaking down a complex task into simpler parts.
• Each function is responsible for its own piece of the puzzle, maintaining clarity.

By understanding composite functions, we decode the intricate process into clear steps. This not only allows for easier analysis but also opens doors for greater flexibility when solving other mathematical problems. It highlights a crucial technique in mathematics, where complexity is managed through intelligent design.