Problem 3 - Brief Applied Calculus

Question

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \left(x^{2}-x\right)^{-3} $$

Step-by-Step Solution

Answer

Let \(g(x) = x^{2} - x\) and \(f(u) = u^{-3}\).

Step 1: Identify the Inner Function

To solve this, observe the expression \((x^{2} - x)^{-3}\). A reasonable choice for the inner function \(g(x)\) is the expression inside the brackets: \(g(x) = x^{2} - x\). This simplifies the composition, allowing you to then consider the outer function.

Step 2: Determine the Outer Function

With the inner function identified as \(g(x) = x^{2} - x\), now we need to determine the outer function \(f\). The composition \(f(g(x))\) should represent the expression \((x^{2} - x)^{-3}\). Therefore, let \(f(u) = u^{-3}\), where \(u = g(x)\).

Step 3: Verify the Composition

Verify that the composition \(f(g(x))\) is equal to \((x^{2} - x)^{-3}\). Substituting for \(g(x)\) and \(f(u)\), we find:\[f(g(x)) = f(x^2 - x) = (x^2 - x)^{-3}\] This is the same as the given function, thus confirming that our choices for \(f\) and \(g\) are correct.

Key Concepts

Inner Function

In the composition of functions, identifying the **inner function** is a crucial first step. The inner function, denoted as \(g(x)\), is essentially the component that we substitute into another function. Think of it as the initial input or the inside layer in a multi-layered process.

In the example given, we look at the expression \((x^2 - x)^{-3}\). Here, the this task requires finding a sensible expression to simplify. We choose \(g(x) = x^2 - x\), as it represents the core operation being performed on the variable \(x\) before any further transformation.

Why is this choice intuitive? Consider that \(x^2 - x\) is enclosed within parentheses, indicating it should be handled separately before applying subsequent operations. By first setting \(g(x)\) to this expression, everything outside of the parentheses becomes much simpler to tackle in the next step of the composition.

Outer Function

The **outer function**, denoted as \(f(u)\), is what processes the result of the inner function \(g(x)\). It takes the output from \(g(x)\) and then applies additional operations to yield the final result.

Once the inner function is identified as \(g(x) = x^2 - x\), the next step is to determine the appropriate form for the outer function. In this problem, the outer operation is raising the result to the power of \(-3\). Hence, we define \(f(u) = u^{-3}\), where \(u = g(x)\). This transformation was chosen because it matches the operation we see in the original expression: \((x^2 - x)^{-3}\).

The outer function effectively "wraps" around the inner function, transforming its output to fit the desired form. It's like adding the final touch to an already prepared base.

Function Verification

Verifying the composition of functions is an essential step to ensure that our selected functions \(f\) and \(g\) correctly represent the original expression. In essence, **function verification** is about confirming accuracy and correctness in your composition.

After substituting \(g(x) = x^2 - x\) and \(f(u) = u^{-3}\) into the composition \(f(g(x))\), we compute as follows:

• First, determine \(g(x)\): \(g(x) = x^2 - x\).
• Next, substitute \(g(x)\) into \(f\): \(f(g(x)) = f(x^2 - x) = (x^2 - x)^{-3}\).

This matches exactly with the original function \((x^2 - x)^{-3}\), proving that our choice of functions is indeed correct. Verification provides confidence that our deconstruction and reconstruction of the function correctly employs the composition process, aligning perfectly with the original expression.