Problem 38 - Calculus

Question

Determine functions \(f\) and \(g\) such that \(h(x)=f(g(x)) .\) [Note: There is more than one correct answer. Do not choose \(f(x)=x \text { or } g(x)=x\).] $$h(x)=\sqrt{x^{2}+4}$$

Step-by-Step Solution

Answer

One option is to set \( f(x) = \sqrt{x} \) and \( g(x) = x^2 + 4 \).

Step 1: Identifying the Components

We need to express the function \( h(x) = \sqrt{x^2 + 4} \) as a composition of two functions \( f(g(x)) \). In a function composition, \( g(x) \) is the inner function and \( f(x) \) is the outer function.

Step 2: Choosing the Inner Function

Let's select \( g(x) = x^2 + 4 \), since this captures the expression inside the square root of \( h(x) \).

Step 3: Defining the Outer Function

Given that \( g(x) = x^2 + 4 \), we require \( f(x) \) such that \( f(g(x)) = \sqrt{x^2 + 4} \). Therefore, \( f(x) = \sqrt{x} \).

Step 4: Verification

To verify, substitute \( g(x) \) into \( f(x) \): \( f(g(x)) = f(x^2 + 4) = \sqrt{x^2 + 4} \), which matches \( h(x) \). Thus, \( f(x) = \sqrt{x} \) and \( g(x) = x^2 + 4 \) are a valid composition that form \( h(x). \)

Key Concepts

Inner Function

When dealing with function composition, the inner function is a critical starting point. Think of the inner function as the first operation applied in a series. For the problem at hand, our goal is to find two functions, \(f\) and \(g\), such that when they are composed, they form the function \(h(x) = \sqrt{x^2 + 4}\). In this case, we need to unpack \(h(x)\) into two operations. We choose \(g(x) = x^2 + 4\) as the inner function because it represents what is inside the square root.

Choosing the inner function correctly is essential because it simplifies the process of identifying the outer function. It essentially breaks down the original function \(h(x)\) into manageable parts. By selecting the inner function \(g(x) = x^2 + 4\), we target the complexity directly inside the square root, preparing us to find a fitting outer function that operates on that complexity.

Outer Function

The outer function in function composition is the final operation that processes the result of the inner function. It 'wraps around' the inner function, giving the composed function its final form. For the current exercise, with our selected inner function as \(g(x) = x^2 + 4\), the next step is identifying an appropriate outer function. We need \(f(x)\) such that when it takes as input the output of \(g(x)\), it produces \(h(x) = \sqrt{x^2 + 4}\).

For our composition, the outer function \(f(x) = \sqrt{x}\) is clear. This function applies the square root operation to any input it receives. When combined, \(f(g(x))\) starts with \(g(x)\), which outputs \(x^2 + 4\), passed into \(f(x)\), resulting in \(\sqrt{x^2 + 4}\). Each component has its distinct role, ensuring the operations reflect the original composed function accurately.

Thus, the essence of constructing the outer function lies in understanding the transformation needed to achieve \(h(x)\), leveraging the outcome of the inner function.

Function Verification

Verification is an important step to confirm that the functions we've composed truly represent the intended function. It ensures accuracy and validates our choices of \(f(x)\) and \(g(x)\). To verify, we substitute \(g(x)\) into \(f(x)\), following the composition like a flowchart. Starting with \(f(g(x))\), we input \(g(x) = x^2 + 4\) into the outer function \(f(x) = \sqrt{x}\). The goal is to show that this equals \(h(x) = \sqrt{x^2 + 4}\).

Carrying out this substitution, we compute \(f(g(x)) = f(x^2 + 4) = \sqrt{x^2 + 4}\). The result matches exactly with \(h(x)\). This confirms our composition since \(h(x) = f(g(x))\), verifying that \( \sqrt{x^2 + 4}\) can be reconstructed from \(f\) and \(g\).

Through function verification, you ensure the deconstructed functions together still yield the intended complexity of the original problem, allowing confidence in your solution's correctness.