Problem 4 - Brief Applied Calculus

Question

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \frac{1}{x^{2}+x} $$

Step-by-Step Solution

Answer

\(f(u) = \frac{1}{u}\) and \(g(x) = x^2 + x\) work as the functions for the composition.

Step 1: Identify the Inner Function

To express the given function as the composition of two functions, find an inner function that captures part of the original function's expression. Look for common expressions within the denominator of the fraction. In this case, choose the inner function as \( g(x) = x^2 + x \).

Step 2: Identify the Outer Function

Once the inner function \(g(x)\) is chosen, express the original function in terms of this inner function. The given function is \(\frac{1}{x^2 + x}\), which suggests that the expression can be seen as \(\frac{1}{u}\) when \(u = x^2 + x\). Thus, choose the outer function as \( f(u) = \frac{1}{u} \).

Step 3: Verify the Composition

Verify that the functions \(f\) and \(g\) compose to reproduce the original function. Substitute \(g(x)\) into \(f(u)\): \(f(g(x)) = f(x^2 + x) = \frac{1}{x^2 + x}\). This matches the original function.

Key Concepts

Inner Function

The concept of an **inner function** is essential when dealing with function composition. It acts as the building block of the larger composite function. In our specific exercise, the goal was to express the function \( \frac{1}{x^2 + x} \) as a composition. The inner function serves as the initial layer that needs to be transformed.

For the given function, \( g(x) = x^2 + x \) was selected. This function is chosen because it represents the entire denominator of the fraction. The inner function helps encapsulate a part of the expression, making it easier to apply subsequent transformations or simplifications.

Understanding the role of the inner function is vital as it underpins the structure of the entire composite function. It usually simplifies part of the problem, breaking down complex expressions for further manipulation.

Outer Function

The **outer function** takes the output of the inner function and further alters it to shape the final, desired expression. Once you have identified the inner function, you define the outer function in terms of this variable.

In our example, the original function can be rewritten so that when considering \( u = x^2 + x\), the resulting expression becomes \( \frac{1}{u} \). This suggests the outer function to be \( f(u) = \frac{1}{u} \).

The purpose of the outer function is to utilize the transformation applied by the inner function, and then enclose or 'wrap' it in another layer of function operations. This establishes the final transformation needed to resemble the target expression. The relationship between the inner and outer functions allows for the bridging of simpler to more complex expressions.

Composite Functions

**Composite functions** are constructed by nesting one function inside another. This is fundamental in mathematical function composition, where one function's output becomes the input for another.

To clarify this concept, when we input \(x\) into \(g(x) = x^2 + x\), we get an outcome that then substitutes into \(f(u) = \frac{1}{u}\). In mathematical terms, this is expressed as \( f(g(x)) = \frac{1}{x^2 + x} \).

The composite function \(f(g(x))\) exactly returns us to our original function. This composition process highlights how complex functions can often be broken down and expressed as the composition of simpler ones, making analysis and computation easier. Understanding composite functions is an essential skill that can simplify complex mathematical problems, as it provides a structured approach to tackling intricate equations through layers of operations.