Problem 5 - Brief Applied Calculus

Question

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \frac{x^{3}+1}{x^{3}-1} $$

Step-by-Step Solution

Answer

Functions are \( f(u) = \frac{u + 1}{u - 1} \) and \( g(x) = x^3 \).

Step 1: Understand the Composition

We are given the function \( h(x) = \frac{x^3 + 1}{x^3 - 1} \). Our goal is to express this as the composition of two functions \( f \) and \( g \), such that \( h(x) = f(g(x)) \). That means \( g(x) \) should be a function to which \( f \) can be applied to yield \( h(x) \).

Step 2: Choose an Inner Function

Choose \( g(x) = x^3 \) as a natural candidate for the inner function because both the numerator and the denominator refer to \( x^3 \). This reduces the complexity of the expression we need to compose.

Step 3: Derive the Outer Function

With \( g(x) = x^3 \), then \( g(x) + 1 = x^3 + 1 \) and \( g(x) - 1 = x^3 - 1 \). The remainder of \( h(x) \) can be described as a function \( f(u) = \frac{u + 1}{u - 1} \), where \( u = g(x) \).

Step 4: Express the Composition

Express the given function as the composition \( h(x) = f(g(x)) \). We have \( f(u) = \frac{u + 1}{u - 1} \) and \( g(x) = x^3 \). So, \( h(x) = f(g(x)) = \frac{g(x) + 1}{g(x) - 1} = \frac{x^3 + 1}{x^3 - 1} \), confirming our composition is correct.

Key Concepts

Composite Functions

Composite functions involve combining two functions into one, where the output of one function becomes the input for another. This concept is crucial in mathematics as it allows us to break down complex functions into simpler parts.
When dealing with composite functions, the notation used is typically \(f(g(x))\). This conveys that the function \(g\) is applied first, and the result is then used by the function \(f\). This layered approach can help simplify and manipulate complex expressions.

• For example, in the original exercise, our composite function is represented as \(h(x) = f(g(x))\).
• Understanding and finding the right functions for \(f\) and \(g\) simplifies the expression \(\frac{x^3+1}{x^3-1}\).

The idea is to solve the function piece by piece, ensuring \(g(x)\) simplifies the expression into something manageable for \(f\). This constructive thinking is the heart of function composition.

Inner and Outer Functions

When composing functions, the terms "inner function" and "outer function" are used to describe the order and structure of function application.
The inner function is applied first, and its output is the input for the outer function. This nested mechanism helps divide and conquer more difficult mathematical problems.
In our example, the inner function is \(g(x) = x^3\).

• We choose \(g(x)\) carefully as it simplifies both the numerator and denominator of \(\frac{x^3 + 1}{x^3 - 1}\) by focusing directly on the term \(x^3\).

The outer function \(f(u) = \frac{u + 1}{u - 1}\) then deftly handles the result from \(g(x)\). The outer function takes the transformation achieved by \(g(x)\) and completes it by handling the algebraic operation needed to return to the initial function.

• This division of labor between inner and outer functions streamlines the solving process.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a value. Mastering them is essential to understanding functions and compositions in mathematics.
They can vary in complexity, and recognizing patterns within them can make solving problems easier.
In the context of our exercise, \(\frac{x^3 + 1}{x^3 - 1}\) is an algebraic expression. It appears daunting at first, but by using composition, we've broken it down to \(f(g(x))\).

• The expression \(x^3\) served as an anchor for both the numerator and the denominator.
• This allowed us to identify \(g(x)\) quickly, simplifying the higher-order terms.

Recognizing similar structures can reduce complex expressions into a series of simpler steps. This not only saves time but also limits the potential for mistakes in calculation.