Problem 33 - College Algebra

Question

Find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). $$ h(x)=\sqrt[4]{\frac{3 x-2}{x+5}} $$

Step-by-Step Solution

Answer

Choose \( g(x) = \frac{3x-2}{x+5} \) and \( f(x) = \sqrt[4]{x} \).

Step 1: Identify the Inner Function

To express the function in the form of a composition, first identify the core expression inside the outer function. We see that the expression under the fourth root \( \frac{3x-2}{x+5} \) can be considered as the inner function. Therefore, let's start by defining this inner function, \( g(x) = \frac{3x-2}{x+5} \).

Step 2: Define the Outer Function

Once we have the inner function, the rest of \( h(x) \) is the operation applied to \( g(x) \). In this case, it's raising \( g(x) \) to the power of \( \frac{1}{4} \), or taking the fourth root of \( g(x) \). Thus, we can define the outer function as \( f(x) = \sqrt[4]{x} \).

Step 3: Formulate the Composite Function

Now that we have identified the functions, check that their composition gives \( h(x) \). Substitute \( g(x) \) into \( f(x) \) to form \( h(x) = f(g(x)) = f\left(\frac{3x-2}{x+5}\right) = \sqrt[4]{\frac{3x-2}{x+5}} \). This correctly reconstructs the original function.

Key Concepts

Function Decomposition

Function decomposition is a method used to break down a complex function into simpler, more manageable parts. Instead of dealing with a complicated function directly, we identify two separate functions: one inside, and one outside. These are usually referred to as the inner and outer functions. By decomposing a function, we make it easier to understand its components, analyze it, and even differentiate or integrate it if needed. For example, in the given problem, the function we need to decompose is \( h(x) = \sqrt[4]{\frac{3x-2}{x+5}} \). The goal is to express \( h(x) \) as a composition of two functions, say \( f(x) \) and \( g(x) \), such that \( h(x) = f(g(x)) \). This process involves identifying two smaller functions that when combined, recreate the original function.

Inner Function

The inner function is the part of the original function that you simplify or "plug into" the outer function. It acts like the core or the kernel of the composite function. It is crucial to identify this correctly, as it lays the foundation for breaking down the overall function. In our problem, we identify the expression inside the fourth root as the inner function:

• We observe that the expression \( \frac{3x-2}{x+5} \) lies inside another function \(\sqrt[4]{x}\).
• Thus, we define this expression as \( g(x) = \frac{3x-2}{x+5} \).

The inner function \( g(x) \) captures the essence of the expression that is manipulated further by the outer function.

Outer Function

The outer function represents the overarching operation applied to the inner function. It can be thought of as the 'wrapper' that forms the final structure of a composite function. After identifying the inner function, the rest of the original function becomes the outer function. In the context of our example:

• The expression \( \sqrt[4]{x} \) acts on whatever is inside the fourth root.
• Here, this is our outer function, so we define \( f(x) = \sqrt[4]{x} \).

The outer function \( f(x) \) dictates how the results of the inner function \( g(x) \) are ultimately processed to recreate the original function \( h(x) \). Understanding the outer function as a separate entity helps us to see the composition of functions more clearly and manage complex operations efficiently.