Problem 41 - College Algebra

Question

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). $$h(x)=\sqrt{\frac{2 x-1}{3 x+4}}$$

Step-by-Step Solution

Answer

\( f(x) = \sqrt{x} \) and \( g(x) = \frac{2x-1}{3x+4} \).

Step 1: Identify the Outer Function

The given function is \( h(x) = \sqrt{\frac{2x-1}{3x+4}} \). To decompose this into two functions \( f(x) \) and \( g(x) \), we need to recognize the outermost operation, which is the square root. Therefore, \( f(x) \), the outer function, is \( f(x) = \sqrt{x} \).

Step 2: Identify the Inner Function

The expression inside the square root is \( \frac{2x-1}{3x+4} \). This forms the inner function, \( g(x) = \frac{2x-1}{3x+4} \).

Step 3: Express h(x) in Terms of f and g

Using the functions identified in the previous steps, express \( h(x) \) as \( f(g(x)) \). Substituting \( g(x) \) into \( f(x) \) gives \( f(g(x)) = \sqrt{\left( \frac{2x-1}{3x+4} \right)} \), which matches \( h(x) \). Thus, \( h(x) = f(g(x)) \).

Key Concepts

Outer Function

The outer function in function composition is the operation you perform last. Imagine it as a layer that wraps around something else. In our specific example, the given function is \( h(x) = \sqrt{\frac{2x-1}{3x+4}} \). You first want to look at the operation furthest out, which is the square root. Thus, the outer function \( f(x) \) is defined as \( f(x) = \sqrt{x} \).

This function takes the result of another function and processes it with a square root. Understanding the role of the outer function is crucial because it helps you understand the effect it has on the entire expression.

Remember, in any function composition, the outer function changes whatever is returned by the inner function without altering its process. This concept is particularly useful when you're working out transformations and simplifying complex expressions.

Inner Function

The inner function is what sits inside the outer function. In our example, we have identified the expression \( \frac{2x-1}{3x+4} \) as this component. Hence, this makes \( g(x) = \frac{2x-1}{3x+4} \) the inner function.

The key task of the inner function is to prepare or modify the input before it goes into the outer function. Think of it as the groundwork that sets the stage for further operations.

When breaking down a complex function, being able to spot the inner function allows you to understand the initial transformation involved. This grasp on the concept lets you tackle each piece of a function separately and efficiently.

Decomposition of Functions

Decomposition of functions involves breaking down a complex function into its simpler parts, typically through inner and outer functions. The function in question, \( h(x) = \sqrt{\frac{2x-1}{3x+4}} \), is reconstructed by recognizing the simpler operations within it.

Decomposition is extremely valuable because it aids in understanding intricate operations step-by-step. Here's what decomposition entails in a nutshell:

• Identify the fundamental sub-functions (inner and outer).
• Understand how each function interacts or transforms its input.
• Reconstruct the original function from these smaller components.

By practicing decomposition, you will develop the ability to tackle more complex mathematical problems and arguments with confidence.