Problem 11 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find functions \(f\) and \(g\) such that \(h(x)=f(g(x))\) and neither \(f\) nor \(g\) is the identity function, i.e., \(f(x) \neq x\) and \(g(x) \neq x .\) Answers to these problems are not unique. \(h(x)=3 x^{4}+2 x^{2}+3\)

Step-by-Step Solution

Answer

The functions \(f(x)\) and \(g(x)\) that satisfy \(h(x) = 3x^{4} + 2x^{2} + 3 = f(g(x))\) and are not identity functions are \(f(x) = 3x^{2} + 2x + 3\) and \(g(x) = x^{2}\).

Step 1: Recognize the composite structure

The given function \(h(x)=3x^{4}+2x^{2}+3\) is a composite function and can be written as \(h(x)=f(g(x))\) where \(g(x)\) is the inner function and \(f(x)\) is the outer function. We need to find non-identity functions \(f\) and \(g\).

Step 2: Derive the inner function

We first have to locate the innermost component of \(h(x)\), and we can treat \(x^{2}\) as this element. Therefore, we can assume \(g(x) = x^{2}\).

Step 3: Derive the outer function

With \(g(x)=x^{2}\), we substitute \(g(x)\) into \(h(x)\), which gives us \(f(g(x)) = f(x^{2}) = 3x^{4} + 2x^{2} + 3\). From this, we recognize that \(f(x)\) can be written as \(f(x) = 3x^{2} + 2x + 3\).

Step 4: Verify the result

Now we confirm this outcome by checking if \(h(x) = f(g(x))\). Initiate with \(g(x) = x^{2}\) and we can substitute \(g(x)\) into \(f(g(x))\). Checking, \(f(g(x)) = 3(x^{2})^{2} + 2x^{2} + 3 = 3x^{4} + 2x^{2} + 3\). This result matches with the given \(h(x)\), therefore the chosen functions \(f\) and \(g\) are correct.

Key Concepts

Inner Function

When dealing with composite functions like \( h(x) = f(g(x)) \), the inner function \( g(x) \) is the function that you apply first. It's like the onion's core, where you start peeling from the inside. In our case, to determine the inner function, we looked for a simpler expression within the more complex one. Here, we decided to let \( g(x) = x^2 \) be the inner function.

This choice arises because \( g(x) \) represents the simpler operation when plugged into the composite, acting as a building block for \( h(x) \). It is often useful to center the inner function around elements that frequently appear or are repeated within the composite's components, such as powers of \( x \) or common terms. This simplifies both the breakdown and recombination for calculations.

Outer Function

The outer function \( f(x) \) is applied after the inner function \( g(x) \) in compositions of functions. Think of it like the outer layer of an onion that wraps around the core to form the whole. Once we identify \( g(x) = x^2 \), we use it to rebuild the original by setting \( f(x^2) \) equal to \( h(x) = 3x^4 + 2x^2 + 3 \).

By focusing on replacing \( x^2 \) with \( z \), we get \( f(z) = 3z^2 + 2z + 3 \). This indicates how \( f \) acts on the output of \( g \), transforming it into \( h(x) \). The outer function often resolves into expressions that transform the inner function into more complex relations or structures, finalizing the composite process.

Function Decomposition

Function decomposition is the process of breaking down a complex function into simpler parts, specifically into an inner and outer function. By recognizing patterns and commonalities within the original function, we can unravel it into a composite form. It serves as a crucial tool in calculus and algebra to simplify and analyze functions more effectively.

In the example provided, by breaking down \( h(x) \) into \( f(g(x)) \), we discovered \( g(x) = x^2 \) and \( f(x) = 3x^2 + 2x + 3 \). The goal of decomposition is to identify such functions \( f \) and \( g \) that, when combined, precisely recreate the original function \( h \), helping to solve, study, or integrate more complicated expressions through simpler steps.

Non-Identity Function

A non-identity function is simply any function where the output is not just the input repeated, neither \( f(x) = x \) nor \( g(x) = x \). For our decomposition, it was vital that neither \( f \) nor \( g \) was an identity function, ensuring authentic transformation steps were applied in the composition.

For example, picking \( g(x) = x^2 \) ensures \( g \) performs a genuine operation on the input. Similarly, \( f(x) = 3x^2 + 2x + 3 \) results in new outcomes from the input, affirming the function isn't merely reflecting \( x \). Non-identity conditions are essential for establishing enrichment or change in the process, contributing to the meaningful formation of \( h(x) \) beyond trivial transformations.