Problem 30 - 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 \(h(x)=f(g(x))\). \(h(x)=4+\sqrt[3]{x}\)

Step-by-Step Solution

Answer

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

Step 1: Analyze the Problem

The given function is \( h(x) = 4 + \sqrt[3]{x} \). Our task is to express it as a composition of two functions \( h(x) = f(g(x)) \).

Step 2: Identify the Inner Function

Observe that \(\sqrt[3]{x}\) can be considered as an inner function. Let's define \( g(x) = \sqrt[3]{x} \).

Step 3: Define the Outer Function

Now that \( g(x) = \sqrt[3]{x} \), recognize that \( h(x) = 4 + g(x) \). Thus, \( f(x) = 4 + x \).

Step 4: Verify the Composition

Substitute \( g(x) \) in \( f(x) \): \( f(g(x)) = f(\sqrt[3]{x}) = 4 + \sqrt[3]{x} \), which matches \( h(x) \).

Key Concepts

Inner Function

The concept of the "inner function" plays a crucial role when we decompose a complex function into a composition of simpler functions. In the context of function composition, the inner function is the part that is applied first. It is essentially the input to the outer function when two functions are composed together.

For example, in our problem with the function \( h(x) = 4 + \sqrt[3]{x} \), the inner function is \( g(x) = \sqrt[3]{x} \). The job of the inner function here is to take the input \( x \) and transform it by computing the cubed root of \( x \).

• This step simplifies the input before the outer function is applied.
• The choice of the inner function frequently depends on the operations being performed within the composite function.

Always remember that the inner function's output becomes the input for the subsequent function in the chain of composition.

Outer Function

After identifying the inner function, the next step is to determine the "outer function". The outer function is applied to the result of the inner function. It completes the transformation by carrying out the final operations needed to achieve the desired outcome described by the original function.

In our workaround for \( h(x) = 4 + \sqrt[3]{x} \), once the cubed root is computed (\( g(x) = \sqrt[3]{x} \)), we add 4 to it. Thus, the outer function is defined as \( f(x) = 4 + x \).

• This addition can be seen as further processing the result from the inner function.
• The outer function directly determines the final shape and behavior of the composite function.

Think of the outer function as the final touch that tailors our overall function \( h(x) \) back to its original form.

Composition of Functions

The "composition of functions" is a way to combine two or more functions into a single, new function. When we talk about function composition, we denote it with the notation \( f(g(x)) \), which means the function \( g(x) \) is applied first and then its output becomes the input to the function \( f(x) \).

Using our function \( h(x) = 4 + \sqrt[3]{x} \), we identified \( g(x) = \sqrt[3]{x} \) (inner function) and \( f(x) = 4 + x \) (outer function). The composition is then \( f(g(x)) = 4 + \sqrt[3]{x} \).

• This demonstrates how to break down complex expressions into simpler parts.
• The final composed function still encapsulates the same logic as the original function \( h(x) \).

This method makes complicated functions easier to work with and analyze, offering a step-by-step procedure to manage and understand them.

Cubed Root Function

The "cubed root function" is an essential mathematical concept, especially when dealing with functions involving roots or exponents. The cubed root of a number \( x \) is a value that, when multiplied by itself three times, gives \( x \). In mathematical notation, it is represented as \( \sqrt[3]{x} \).

In our function setup, the cubed root function is expressed through the inner function \( g(x) = \sqrt[3]{x} \).

• This function transforms an input \( x \) to make it easier to manage in subsequent operations.
• It is especially useful when simplifying expressions or solving equations involving powers of three.

Understanding how to work with the cubed root function, as well as its properties, equips you with the tools needed to handle similar functions efficiently.