Problem 40 - Precalculus

Question

Write the given function as a composition of two or more non-identity functions. (There are several correct answers, so check your answer using function composition.) $$w(x)=\frac{x^{2}}{x^{4}+1}$$

Step-by-Step Solution

Answer

The function can be written as a composition: \( w(x) = f(g(x)) \) where \( f(u) = \frac{x^2}{u} \) and \( g(x) = x^4 + 1 \).

Step 1: Identify the Inner Function

First, let's identify a part of the function as an inner function, which can be represented as another function. Observing the given function, notice the denominator can be separated. Let's choose:\( g(x) = x^4 + 1 \)This function represents the denominator part of the original function.

Step 2: Decide on the Outer Function

Given \( g(x) = x^4 + 1 \), the remaining part of the original function is the full expression itself, which is a division of \( x^2 \) by \( g(x) \). Thus, the outer function can be:\( f(u) = \frac{x^2}{u} \)where \( u = g(x) \). This function takes the result of \( g(x) \) as its input and outputs the original function value.

Step 3: Verify the Composition

To ensure correctness, compose the functions: Substituting \( g(x) \) into \( f(u) \), gives:\( f(g(x)) = \frac{x^2}{x^4 + 1} \)This matches the original function, \( w(x) \). Therefore, the composition is correct.

Key Concepts

Inner Function

In function decomposition, an *inner function* is the part of a composite function that is encapsulated, or nested inside another function. Let's consider the original function, \( w(x) = \frac{x^2}{x^4 + 1} \). Here, the inner part can be seen as the denominator \( x^4 + 1 \), which depends on the variable \( x \). We represent it as:

• \( g(x) = x^4 + 1 \)

The inner function is crucial because it simplifies the larger function into manageable pieces. By identifying \( g(x) \), we can more easily substitute and manage the variables in the function composition process.
This approach allows us to work with complex functions more fluently.

Outer Function

The *outer function* in a composite relation is the function that takes the result of the inner function as its input. After isolating our inner function, \( g(x) = x^4 + 1 \), we focus on the remaining structure of \( w(x) \). We need to express the outer function, which essentially forms the directive frame of the composite function:

• \( f(u) = \frac{x^2}{u} \)

Here, \( u \) represents the output of our inner function, \( g(x) \).
The outer function process is essential because it defines how we manipulate or transform the result of the inner function. In essence, the outer function "wraps" around the inner function's result, determining how that value will be used in the overall expression.

Function Decomposition

*Function decomposition* involves breaking down a complex function into simpler, more manageable parts or sub-functions. In the context of \( w(x) = \frac{x^2}{x^4 + 1} \), this process helps in understanding and managing the function more efficiently.
By choosing:

• \( g(x) = x^4 + 1 \)
• \( f(u) = \frac{x^2}{u} \)

Thus, by substituting \( u = g(x) \) into \( f(u) \), we efficiently compose:

• \( f(g(x)) = \frac{x^2}{x^4 + 1} \)

Through decomposition, complex function analysis becomes significantly simpler, allowing for rapid verification of correctness and deeper analysis when tackling mathematical problems.

Non-identity Functions

In the process of breaking down a function, we focus on creating *non-identity functions*. An identity function would simply return the input given, like \( h(x) = x \). In our example with \( w(x) \), the functions we derived, \( g(x) = x^4 + 1 \) and \( f(u) = \frac{x^2}{u} \) are non-identity functions. They perform specific operations on their input values, transforming them to new outputs.
Creating non-identity functions helps capture meaningful transformations within the function's behavior. This ensures that each segment of our decomposed function contributes uniquely to the overall operation of the composite function. Understanding how these non-identity functions work allows us to easily customize or adjust parts of a function without altering its critical structure or losing functionality.