Problem 29 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

Let \(f(x) = \frac{4}{x^2}\) and \(g(x) = x+2\).

Step 1: Understand the Problem

We need to express the function \(h(x) = \frac{4}{(x+2)^2}\) as a composition of two functions: \(f(g(x))\). This means we need to find two separate functions \(f(x)\) and \(g(x)\) such that their composition results in \(h(x)\).

Step 2: Identify Inner Function \(g(x)\)

Let's first focus on the expression inside the denominator of \(h(x)\). The term \((x+2)^2\) indicates a transformation from \(x\) to \(x+2\), which suggests a possible choice for \(g(x)\). So, we choose \(g(x) = x+2\).

Step 3: Identify Outer Function \(f(x)\)

With \(g(x) = x+2\), the function \(h(x)\) becomes \(\frac{4}{(g(x))^2}\). This suggests setting the outer function \(f(x)\) to be the function that takes an input and returns \(\frac{4}{x^2}\). So, let \(f(x) = \frac{4}{x^2}\).

Step 4: Verify the Composition

Compose \(f(g(x))\) using the functions we identified. We have \(g(x) = x+2\), so \(f(g(x)) = f(x+2) = \frac{4}{(x+2)^2}\). This matches the original function \(h(x)\), confirming that our choices for \(f(x)\) and \(g(x)\) are correct.

Key Concepts

Inner Function

The concept of the "Inner Function" is crucial when working with composite functions. In our exercise, we're trying to express the function \( h(x) = \frac{4}{(x+2)^2} \) as a composition of two functions: \( f(g(x)) \). To do this, we begin by identifying the part of the function that's being transformed first - this is our inner function.
When examining \( h(x) \), notice the transformation inside the denominator: \((x+2)^2\). This transformation hints at our choice for the inner function \( g(x) \). Choosing \( g(x) = x + 2 \) isolates the transformation applied directly to \( x \) before it is passed to the rest of the function.
Understanding and identifying the inner function correctly is essential, as it often simplifies finding the outer function and ensures the correctness of function composition.

Outer Function

After determining the inner function, it's time to focus on the "Outer Function". This is the function applied after the inner transformation. In the given problem, the transformation defined by \( g(x) = x+2 \) simplifies the representation of \( h(x) \) to \( \frac{4}{(g(x))^2} \).
The role of the outer function \( f(x) \) is to take this result of \( g(x) \) and transform it into the final form of \( h(x) \). Here, \( f(x) = \frac{4}{x^2} \) becomes our outer function. It effectively takes any input and applies the operation to result in \( \frac{4}{(x+2)^2} \) when put together with \( g(x) \).
The outer function provides the broader framework within which the inner function operates, finalizing the computation started by the inner function. Without both being correctly identified, recreating the original function through composition would be difficult.

Verification of Composition

Once you've identified functions for the inner and outer stages, it's crucial to verify that these selections accurately recreate the original function through composition. This step ensures that the chosen functions are correct and that they compose properly.
For our example, once we have identified \( g(x) = x+2 \) and \( f(x) = \frac{4}{x^2} \), we compose these to check: compute \( f(g(x)) = f(x+2) = \frac{4}{(x+2)^2} \).
Doing this computation shows that our composed function \( f(g(x)) \) matches the original \( h(x) \). This type of verification is not just a formality; it is essential for confirming the logical accuracy of function composition. It acts as a check against any mistakes in identifying \( f(x) \) and \( g(x) \), ensuring the correctness of the problem-solving approach.