Problem 39 - College Algebra

Question

Find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). $$ h(x)=\frac{1}{(x-2)^{3}} $$

Step-by-Step Solution

Answer

\( f(u) = \frac{1}{u^3} \), \( g(x) = x - 2 \)

Step 1: Identify the Inner Function

To express the function \( h(x) \) as \( f(g(x)) \), identify a part of \( h(x) \) that can serve as an inner function. Look for an expression within the larger function that can be isolated. Notice that the expression \( x - 2 \) is inside another operation: raising to the power of \(-3\). Therefore, we can choose \( g(x) = x - 2 \).

Step 2: Write the Outer Function in Terms of the Inner Function

Once you have selected \( g(x) = x - 2 \) as the inner function, substitute \( g(x) \) into \( h(x) \). The function becomes \( \frac{1}{(g(x))^3} \). Thus, the outer function \( f(u) \) (where \( u = g(x) \)) is \( f(u) = \frac{1}{u^3} \).

Step 3: Verify that \( h(x) = f(g(x)) \)

To ensure that \( h(x) = f(g(x)) \), substitute \( g(x) = x - 2 \) back into the expression for \( f(u) \). This gives \( f(g(x)) = \frac{1}{((x - 2)^3)} \), which confirms that it matches \( h(x) \).

Step 4: Conclude Functions

Conclude with the functions identified: the inner function is \( g(x) = x - 2 \) and the outer function is \( f(u) = \frac{1}{u^3} \). This correctly represents \( h(x) \) as \( f(g(x)) \).

Key Concepts

Inner Function

In function composition, the inner function is the initial function you encounter in a sequence of operations. It works on the input first. You can think of it as a function that prepares or preprocesses the input before it's handed off to another function for further manipulation.
For example, when expressing the given function:

• Let’s take our function, \( h(x) = \frac{1}{(x-2)^{3}} \), and look for an inner function. Here, the expression \( x - 2 \) inside the cube operation can be identified as the inner function.
• Thus, we choose \( g(x) = x - 2 \).

By recognizing the inner function, you set the stage for further transformation or outsourcing operations to a subsequent function.

Outer Function

The outer function is the function that takes the output from the inner function and performs additional operations. This function acts on the result produced by the inner function.
After identifying the inner function, which is \( g(x) = x - 2 \), we need to express the original function in such a way that includes this inner part:

• In our exercise, substitute \( g(x) = x - 2 \) into \( h(x) = \frac{1}{(x-2)^{3}} \).
• This results in \( \frac{1}{(g(x))^3} \) being the operation left for the outer function.
• Thus, the outer function can be described as \( f(u) = \frac{1}{u^3} \), where \( u = g(x) \).

The outer function finalizes the transformation initiated by the inner function, bringing us toward the function composition structure.

Function Decomposition

Function decomposition breaks down a complex function into simpler or more fundamental parts, such as separating \( h(x) = f(g(x)) \) into \( f \) and \( g \). This process is beneficial for simplifying complicated processes into manageable steps.
Decomposing functions frequently involves these steps:

• Identify any inner expressions or sub-processes, as we did with \( g(x) = x - 2 \).
• Recognize how the outer function interacts with this inner part, leading to \( f(u) = \frac{1}{u^3} \).
• Verify that the combination \( f(g(x)) \) yields the original complex function.

By decomposing a function, we can gain clarity on its structure and how each component contributes to its overall operation.

Verify Function Composition

Verification is key in function composition to ensure that the separate functions accurately recreate the original function. It involves re-substituting the inner function into the outer function and checking whether the original expression is retrieved.
Here's how we verify for \( h(x) = \frac{1}{(x-2)^3} \):

• Substitute \( g(x) = x - 2 \) back into \( f(u) = \frac{1}{u^3} \).
• This results in \( f(g(x)) = \frac{1}{((x - 2)^3)} \).
• Notice that this matches the original function \( h(x) \).

Verification successfully proves the accuracy of your decomposition into \( f \) and \( g \), confirming the reliability of the function composition process.