Problem 37 - 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 \(h(x)=f(g(x))\). \(h(x)=\sqrt[3]{x-1}\)

Step-by-Step Solution

Answer

Let \( f(x) = \sqrt[3]{x} \) and \( g(x) = x - 1 \).

Step 1: Define Inner Function

The given function is \( h(x) = \sqrt[3]{x-1} \). To express this as a composition of two functions, first identify the inner function, \( g(x) \). Here, the expression inside the cube root, \( x-1 \), serves as the inner function. Thus, we define \( g(x) = x - 1 \).

Step 2: Define Outer Function

Next, identify the outer function, \( f(x) \), which operates on the result of \( g(x) \). The remaining part of the expression \( h(x) = \sqrt[3]{x-1} \) after substituting \( g(x) = x-1 \) is the cube root, meaning we are taking the cube root of \( g(x) \). Therefore, define \( f(x) = \sqrt[3]{x} \).

Step 3: Verify Composition

Now, verify that the composition \( h(x) = f(g(x)) \) holds true. Substituting, \( f(g(x)) = f(x - 1) = \sqrt[3]{x - 1} \), which matches the original function \( h(x) = \sqrt[3]{x-1} \). This confirms our functions are correctly defined.

Key Concepts

Inner Function

When working with function composition, it's important to first identify the inner function. This is the function that will be "nested" inside another function. Think of it as the starting point; in the problem, it is the expression within a more complex function. In this exercise, we have the function \( h(x) = \sqrt[3]{x-1} \). The task is to represent it as the composition of two functions: \( f(x) \) and \( g(x) \).

For \( h(x) \), the part \( x-1 \) is referred to as the inner function, because it is the component that needs to be addressed first before anything else can be solved or operated on. Thus, we define:

• Inner Function: \( g(x) = x - 1 \)

The inner function \( g(x) \) is simply taking our input \( x \) and subtracting 1 from it. It's crucial to clearly identify this since it will be our input for the outer function.

Outer Function

Once you know the inner function, it's time to look at the rest of the function to identify the outer function. The outer function operates on the result of the inner function. In this exercise, after identifying \( g(x) = x - 1 \) as our inner function, the process of taking the cube root of \( x-1 \) is the next major operation. Thus, the cube root itself becomes our outer function.

• Outer Function: \( f(x) = \sqrt[3]{x} \)

The outer function \( f(x) \) takes the input from \( g(x) \) and applies the cube root to it. This function effectively completes the composition, transforming the input from the inner function into the intended output \( h(x) = \sqrt[3]{x-1} \). Understanding the role of the outer function helps to break down complex functions into manageable parts.

Mathematical Verification

Mathematical verification in the context of function composition is about ensuring that the breakdown of a complex function into its inner and outer components is correct. It's a crucial step to confirm that the identified functions truly combine to form the original given function.To verify the composition \( h(x) = f(g(x)) \), substitute \( g(x) = x - 1 \) into \( f(x) \). According to our functions:

• \( f(g(x)) = f(x - 1) \)
• \( = \sqrt[3]{x - 1} \)

Comparing this with the original function, \( h(x) = \sqrt[3]{x-1} \), the verification step shows that these expressions match exactly. This confirms that the functions \( f(x) = \sqrt[3]{x} \) and \( g(x) = x - 1 \) are correctly defined.Performing such verification offers not just clarity, but confidence, that your composition is accurate. It is always advised to follow through with this check to avoid mistakes, especially when dealing with more complex functions.