Problem 34 - College Algebra

Question

Find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). $$ h(x)=\left(\frac{8+x^{3}}{8-x^{3}}\right)^{4} $$

Step-by-Step Solution

Answer

Inner function: \(g(x) = \frac{8+x^{3}}{8-x^{3}}\), Outer function: \(f(u) = u^{4}\).

Step 1: Identify Inner and Outer Functions

To express the given function \(h(x)\) as a composition \(f(g(x))\), we need to identify an inner function \(g(x)\) and an outer function \(f(x)\) such that \(h(x) = f(g(x))\). In \(h(x) = \left(\frac{8+x^{3}}{8-x^{3}}\right)^{4}\), the expression inside the power is a good candidate for \(g(x)\).

Step 2: Define the Inner Function

Choose \(g(x) = \frac{8+x^{3}}{8-x^{3}}\). This makes \(g(x)\) the rational expression inside the fourth power.

Step 3: Define the Outer Function

Define the outer function \(f(u) = u^{4}\), where the input \(u\) is the output from the inner function \(g(x)\).

Step 4: Verify the Composition

Substitute \(g(x)\) into \(f(u)\). Calculate \(f(g(x)) = \left(\frac{8+x^{3}}{8-x^{3}}\right)^{4}\). This matches the original function \(h(x)\), confirming that the composition is correct.

Key Concepts

Inner Function

In the context of function composition, the **inner function** plays a crucial role. It is essentially the function that is evaluated first, within the larger composition. In our exercise, the task is to identify what part of the function is serving as the inner function. Here, the inner function is identified as:\[g(x) = \frac{8+x^{3}}{8-x^{3}}\]This expression acts as the inner function because it is evaluated first before any further operations, such as raising it to a power, are performed. Understanding the inner function:

• It simplifies a part of the entire expression, making the outer function's job easier.
• Once determined, the rest of the original function relies on this inner structure to build complexity.

Recognizing this helps in unraveling complex functions into simpler parts, allowing us to piece them together using function composition.

Outer Function

After figuring out the inner function, we focus on the **outer function**. The outer function processes the result of the inner function. In our scenario, the outer function modifies the output of \(g(x)\) from our earlier section. It is given by:\[f(u) = u^{4}\]When plugging in the result of \(g(x)\), it replaces \(u\) in this function, showing how the outer function encapsulates or builds further upon the solutions provided by the inner function.Key points about the outer function:

• It usually involves operations like powers, trigonometric functions, exponentials, etc., applied after the inner function.
• It is responsible for the main transformation of the input into the desired output form.

The composition \(f(g(x))\) visualizes how the outer and inner functions interact, confirming correctness of the composition when it matches the original function \(h(x)\).

Algebraic Expressions

When dealing with function compositions, understanding **algebraic expressions** is fundamental. These expressions can vary widely, ranging from simple polynomials to more complex rational expressions. In our exercise, the function is expressed as a rational expression:- The **numerator** \(8+x^{3}\) and the **denominator** \(8-x^{3}\) together form the fraction \(\frac{8+x^{3}}{8-x^{3}}\).- The power transformation \((...)^{4}\) exemplifies how algebraic expressions are used in operations.Algebraic expressions are vital:

• They provide a way to represent numbers and operations in a structured, logical format.
• They are often manipulated to simplify or express functions in different forms, especially suitable for compositions.
• Understanding their structure helps in breaking down complicated sequences into manageable parts.

In summary, mastering these expressions enhances skills such as factoring, expanding, and simplifying, which are essential in math problem-solving and reasoning.