Problem 27 - College Algebra

Question

Find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). $$ h(x)=(x-5)^{3} $$

Step-by-Step Solution

Answer

Let \(f(x) = x^3\) and \(g(x) = x - 5\); then \(h(x) = f(g(x))=(x-5)^3\).

Step 1: Identify the Inner Function

To express the function in the form \(h(x) = f(g(x))\), we must first choose an appropriate inner function \(g(x)\).Let's identify \(g(x)\) to be the expression being transformed inside the operation of cube raising in the original function. Here, that expression is \(x-5\).So, we choose \(g(x) = x - 5\).

Step 2: Identify the Outer Function

Now, we need to identify the function \(f(x)\) that takes the result of \(g(x)\) and applies the transformation seen in the original function \(h(x)\).Since \(h(x)\) involves cubing the quantity produced by \(g(x)\), we define \(f(u) = u^3\), where \(u\) is a placeholder for \(g(x)\).

Step 3: Verify the Combination

Finally, let's verify that combining \(f(x)\) and \(g(x)\) gives us the original function \(h(x)\).Substitute \(g(x) = x-5\) into \(f(u) = u^3\):\[ f(g(x)) = f(x-5) = (x-5)^3 \]This matches the original form of \(h(x)\), confirming that our proposed functions are correct.

Key Concepts

Inner Function

In the realm of function composition, the "inner function" plays a crucial role. Essentially, it is the function that undergoes transformation first in a composite function. Identifying the inner function correctly is a fundamental step in understanding how functions interact with each other.

Let's say we start with a composite function given by \( h(x) = f(g(x)) \). Here, \( g(x) \) is the inner function. It is called "inner" because it feeds its output into the next layer, which is the outer function. In the case of the exercise, we have \( g(x) = x - 5 \).

To determine the inner function, look for the expression that is directly transformed within the larger operation. In simple terms, it's the element sitting "inside" any other transformation. Here are some tips on finding the inner function:

• Focus on the values inside parentheses or the arguments of nested functions.
• Understand the order of operations to see which calculations occur first.

Identifying the inner function correctly ensures that the entire composite function can be expressed in terms of its building blocks.

Outer Function

The outer function is the second piece of the composition puzzle and operates on the result of the inner function. In our exercise, once the inner function, \( g(x) = x - 5 \), processes its input, the output flows into the outer function, \( f(x) \), which is responsible for transforming the result further.

In the form \( h(x) = f(g(x)) \), \( f(x) \) is the outer function. It's what changes the output of \( g(x) \) into the final form that we see in \( h(x) \). For our specific example, we define \( f(u) = u^3 \), where \( u \) represents the outcome of the inner function.

Spotting the outer function involves looking at how the final result is shaped:

• Determine the "last" transformation applied to the expression.
• Consider how the function changes the result of the inner function to get the desired output.

By properly identifying and applying the outer function, we create the desired transformation dictated by the composite function, ensuring the output corresponds with the original definition.

Composite Function

The composite function is the combination of the inner and outer functions, creating a new representation of how transformations are applied in sequence. Understanding composite functions is central in many areas of mathematics because they reflect the step-by-step process of combining operations.

The composite function, denoted as \( h(x) = f(g(x)) \), illustrates how the application of one function flows into another. Each function maintains a specific role: the inner function processes the initial input, and the outer function modifies that result into its final form.

Analyzing composite functions involves:

• Identifying both the inner and outer components clearly.
• Understanding the path from the input \( x \), through \( g(x) \), to the final output through \( f(g(x)) \).
• Checking the consistency of applying \( f(x) \) to the outcome of \( g(x) \).

In our example, by substituting the outcome of \( g(x) \) into \( f(u) \), we confirmed that the full composite function \( (x - 5)^3 \) remains true to its initial expression in \( h(x) \). This process underlines the strength and flexibility of using composite functions in mathematical problem-solving.