Problem 93 - College Algebra with Modeling and Visualization

Question

Exercises \(81-94:\) (Refer to Example \(11 .\) ) Find functions \(f\) and \(g\) so that \(h(x)=(g \circ f)(x) .\) Answers may vary. $$ h(x)=x^{3 / 4}-x^{1 / 4} $$

Step-by-Step Solution

Answer

Functions are \(f(x) = x^{1/4}\) and \(g(x) = x^3 - x\).

Step 1: Identify the Composition Function

Given that the function is expressed as \(h(x) = (g \, \circ \, f)(x)\), which means that \(h(x) = g(f(x))\), our objective is to identify functions \(f(x)\) and \(g(x)\) such that their composition yields \(h(x)\).

Step 2: Choose a Suitable Function for f(x)

Let us select \(f(x)\) to capture the operations that are inside the power functions. A reasonable choice here is to set \(f(x) = x^{1/4}\). This simplifies the structure of the input to the functions involved.

Step 3: Define g(x) Based on f(x)

Now, define \(g(x)\) such that when \(f(x)\) is substituted into it, the result is equivalent to \(h(x)\). Notice that \(h(x) = (x^{1/4})^3 - (x^{1/4})\). Thus, solving this indicates that \(g(x) = x^3 - x\) because it fits the transformation perfectly when \(f(x) = x^{1/4}\).

Step 4: Validate the Functions

Let us verify if the chosen functions \(f(x) = x^{1/4}\) and \(g(x) = x^3 - x\) correctly compose to form \(h(x)\). Compute \(g(f(x))\):\[g(f(x)) = g(x^{1/4}) = (x^{1/4})^3 - (x^{1/4}) = x^{3/4} - x^{1/4},\]which matches \(h(x) = x^{3/4} - x^{1/4}\). Thus, the selection is valid.

Key Concepts

Composite Functions

Composite functions can be thought of as a series of operations performed one after another. By stacking functions, we can simplify complex operations into manageable processes. Consider composite functions like a factory assembly line:

• Each function contributes a specific modification to the input, producing an intermediate result.
• This intermediate output serves as the input for the next function in the sequence.

Let's break down this concept a bit further with our specific example. Suppose we have two functions, \( f(x) \) and \( g(x) \). The composite function \( h(x) = (g \circ f)(x) \) essentially means "apply \( f(x) \), then apply \( g \) to the result from \( f \)."
In a practical sense, if:\[f(x) = x^{1/4}\]and\[g(x) = x^3 - x,\]then the application of these in sequence results in the output \( x^{3/4} - x^{1/4} \), which is the essence of the composite function \((g \circ f)(x)\).
Composite functions are incredibly useful because they allow mathematicians and students to simplify the implementation of complex transformations. Using smaller, simpler steps, it becomes easier to analyze the effects of combined functions on different inputs.

Function Composition

Function composition is a core concept in mathematics, especially useful when dealing with multiple operations. It involves creating a single function from two or more functions by using the output of one function as the input to the next. This step-by-step application can be visualized through equations:

• Given two functions \( f \) and \( g \), function composition is represented as \( g(f(x)) \).
• Here, you first calculate \( f(x) \), and you utilize this result as the input for \( g \).

This sequencing is crucial:

Suppose you have these functions:
\[f(x) = x^{1/4}\]
\[g(x) = x^3 - x\]
The composition \( g(f(x)) \) means putting \( f(x) \) inside \( g(x) \). Computed as:
\[g(f(x)) = g(x^{1/4}) = (x^{1/4})^3 - (x^{1/4})\]
Which simplifies to:
\[x^{3/4} - x^{1/4}\]Thus, function composition provides a systematic way to combine functions, enhancing our ability to tackle complex equations by breaking them into smaller, digestible parts.

Algebraic Expressions

Algebraic expressions are the building blocks of functions and equations in mathematics. They consist of constants, variables, and arithmetic operations that combine to form meaningful mathematical statements. Here's why they are important:

• Algebraic expressions empower us to represent a range of mathematical ideas from the simple (like \( 2x + 3 \)) to the complex (like \( x^{3/4} - x^{1/4} \)).
• They are key in forming equations and functions, allowing us to model real-world problems and perform calculations.

In the case of our example:\[h(x) = x^{3/4} - x^{1/4}\]The task is to express this as a composition of two functions. By viewing it algebraically, we can see it simplifies when split into functions like \( f(x) = x^{1/4} \) and \( g(x) = x^3 - x \).
Algebraic expressions like this help dissect problems into smaller tasks, aligning with smaller functions that can be composed together.
The ability to parse and create such expressions is critical in algebra, as it lays the conceptual foundation for more advanced operations and the study of other mathematical structures.