Problem 30 - College Algebra

Question

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(\mathrm{g}(x))\) $$h(x)=4+\sqrt[3]{x}$$

Step-by-Step Solution

Answer

Choose \(f(u) = 4 + u\) and \(g(x) = \sqrt[3]{x}\).

Step 1: Identify the Outer Function

To express the function as a composition, we first identify the outer function. Looking at the given function \(h(x) = 4 + \sqrt[3]{x}\), we can see that it consists of adding 4 to a cubic root expression. This suggests that the outer function \(f(u)\) can be defined as adding 4 to any input \(u\). Thus, define \(f(u) = 4 + u\).

Step 2: Identify the Inner Function

The inner function should account for the expression under the control of the outer function. The only non-linear component present inside the expression \(4 + \sqrt[3]{x}\) is the cube root of \(x\). Therefore, we define the inner function \(g(x)\) as the cube root of \(x\): \(g(x) = \sqrt[3]{x}\).

Step 3: Verification

Verify that the composition of \(f(x)\) and \(g(x)\) equals the given function \(h(x)\). Substituting \(g(x)\) into \(f(x)\), we have \(f(g(x)) = f(\sqrt[3]{x}) = 4 + \sqrt[3]{x}\), which matches the original function \(h(x)\). Therefore, the functions are correctly identified.

Key Concepts

Outer Function

In mathematics, especially when dealing with function composition, the **outer function** plays a crucial organizational role. It's the function that wraps around the result of the **inner function**. For a clear understanding, let's dive deeper into what this means with our example.

We begin with the function: \( h(x) = 4 + \sqrt[3]{x} \). Here, the **outer function** is the one where another function's output becomes its input. Specifically, in this case, when we compose functions to express \( h(x) \) with composition notation as \( f(g(x)) \), the outer function \( f(u) \) is operating on the output of \( g(x) \), where:

• \( f(u) = 4 + u \)

The key point is that \( f(u) \) simply takes whatever \( g(x) \) calculates and adds 4. This highlights the outer function's role in interpreting and manipulating results from the inner function before presenting a final output. It's like putting a final touch on a work that's almost complete.

Inner Function

Function composition relies heavily on understanding the **inner function**. This function performs an initial computation on the variable \( x \), setting the stage for any additional processing by the outer function.

In the expression \( h(x) = 4 + \sqrt[3]{x} \), the inner function is what we compute before applying the outer function. Here, we define the **inner function** as:

• \( g(x) = \sqrt[3]{x} \)

This function extracts the cube root of \( x \). It essentially breaks down the calculation of \( h(x) \) into simpler parts, handling the non-linear aspect of the overall function. By re-expressing the problem in steps— first calculating \( g(x) \), then applying \( f(u) \) to it— understanding and solving complex problems become much more manageable. The inner function can be seen as the foundation or the preliminary step in the composition that prepares the input for the outer function.

Cube Root Function

The **cube root function** is a specific type of mathematical function that is crucial to nonlinear modeling, particularly when simplifying expressions involving cubes.

The cube root of a number \( x \) is the number which, when multiplied by itself three times, equals \( x \). It is denoted mathematically as \( \sqrt[3]{x} \). In the context of the composition problem at hand, the cube root function represents the inner function \( g(x) \).

• \( g(x) = \sqrt[3]{x} \)

This procedural step simplifies the input \( x \) for the outer function by extracting the cube root. Here's a quick example: if \( x = 8 \), then \( g(x) = \sqrt[3]{8} = 2 \), because \( 2 \times 2 \times 2 = 8 \).

Understanding cube root functions is crucial as they frequently appear in mathematical models, engineering problems, and various real-world applications. By mastering the concept of the cube root function, you're well-equipped to tackle more complex compositions and transformations in mathematical contexts.