Problem 32 - Calculus with Concepts in Calculus

Question

Write \(h\) as the composite \(g \circ f\) of two functions \(f\) and \(g\) (neither of which is equal to \(h\) ). $$ h(x)=\frac{1}{\left(x^{3}-2 x^{2}\right)^{5}} $$

Step-by-Step Solution

Answer

Define \( f(x) = x^3 - 2x^2 \) and \( g(x) = \frac{1}{x^5} \), so \( h(x) = g(f(x)) \).

Step 1: Identify the innermost function

Examine the expression \( h(x) = \frac{1}{(x^3 - 2x^2)^5} \). Identify the innermost operation within the function. Here, \( x^3 - 2x^2 \) is the innermost operation, which we will define as the function \( f(x) \).

Step 2: Define function f

Define \( f(x) = x^3 - 2x^2 \). This represents the core element which transforms \( x \) before applying further operations.

Step 3: Identify the remaining outer operation

From the expression \( \frac{1}{(x^3 - 2x^2)^5} \), identify the operation applied to \( f(x) = x^3 - 2x^2 \), which is raising it to the 5th power and then taking the reciprocal.

Step 4: Define function g

Define \( g(x) = \frac{1}{x^5} \). This function takes an input, raises it to the 5th power, and then takes the reciprocal.

Step 5: Compose functions g and f

Express \( h(x) \) as the composition of \( g(f(x)) \). Therefore, \( g(f(x)) = \frac{1}{(x^3 - 2x^2)^5} \) exactly as \( h(x) \) is defined.

Key Concepts

Function Composition

Function composition is a fundamental concept in mathematics, especially when dealing with complex expressions. It allows you to combine two or more functions into one by letting the output of one function become the input of another. This is represented by the symbol \( \circ \), and the expression \( g \circ f \) means that you apply function \( f \) first, and then apply function \( g \) to the result of \( f \).

• Let's consider an example: if \( f(x) = 2x + 3 \) and \( g(x) = x^2 \), then to find \( g \circ f(x) \), we first compute \( f(x) \), and then apply \( g \) to this result.
• In symbols, \( g \circ f(x) = g(f(x)) = (2x + 3)^2 \).

The key to mastering function composition is understanding that the order of functions matters, and careful step-by-step evaluation is crucial.
Function composition helps simplify complex operations by breaking them down into manageable steps, making it easier to analyze and solve problems.

Inner and Outer Functions

When working with composite functions, you often encounter terms like 'inner' and 'outer' functions. These terms help you identify which function to apply first and which one to apply second. In the given example of composing \( h(x) = \frac{1}{(x^3 - 2x^2)^5} \) using \( g \) and \( f \), you can break it down as follows:

• Inner Function: The function \( f(x) = x^3 - 2x^2 \) is used first. It transforms the input \( x \) before any other operations.
• Outer Function: The function \( g(x) = \frac{1}{x^5} \) is then applied to the result of the inner function, which further transforms the intermediate result.

Identifying which function acts as the inner and outer is crucial because it guides you through the function composition process. This structure highlights that any function inside another function's expression is typically handled before proceeding to the `outer` operations.
Understanding inner and outer functions is vital for systematically approaching and resolving composite function problems.

Function Notation

Function notation is a way to denote functions in mathematics clearly and concisely. It typically uses letters and parentheses to indicate that something is a function, such as \( f(x) \), where \( f \) is the name of the function, and \( x \) is the variable or input.

• Each function has its unique notation, which helps us understand which function we are referring to at any given time. For instance, \( h(x) \) refers to a specific function \( h \) applied to \( x \).
• In the context of composite functions, \( g(f(x)) \) denotes that \( f(x) \) is evaluated first, and the resulting output serves as the input for \( g \).

The notation helps avoid ambiguity and clarifies the steps involved in function evaluation and composition.
Function notation is vital not only in solving mathematical problems but also in communicating methods and processes effectively between students and educators.