Problem 37 - Calculus

Question

Determine functions \(f\) and \(g\) such that \(h(x)=f(g(x)) .\) [Note: There is more than one correct answer. Do not choose \(f(x)=x \text { or } g(x)=x\).] $$h(x)=x^{3}+1$$

Step-by-Step Solution

Answer

Functions can be \(f(u) = u + 1\) and \(g(x) = x^3\).

Step 1: Initial Understanding

We know that the function \(h(x) = x^3 + 1\) needs to be decomposed into two functions \(f\) and \(g\) such that \(h(x) = f(g(x))\). Our task is to find such functions.

Step 2: Identify Function g(x)

Consider \(g(x) = x^3\). This is part of \(h(x)\) where the variable \(x\) is raised to the power of 3.

Step 3: Determine Function f(g(x))

Since \(h(x) = x^3 + 1\) and \(g(x) = x^3\), we can write \(h(x) = f(g(x)) = f(x^3)\). Hence, \(f(u)\) must be a function such that \(f(u) = u + 1\), where \(u = g(x)\).

Step 4: Verify the Decomposition

By substituting \(g(x) = x^3\) into \(f(u) = u + 1\), we have \(f(g(x)) = (x^3) + 1 = x^3 + 1 = h(x)\), confirming the functions satisfy the original equation.

Key Concepts

Composite Functions

A composite function is a powerful mathematical tool that combines two functions to form a new one. In simpler terms, it's like a function within a function. Imagine you have two functions, say \(f(x)\) and \(g(x)\). A composite function \(h(x)\) is formed by replacing the input of one function with the output of another. Formally, this is written as \(f(g(x))\), which means you're first applying \(g(x)\) and then using its output as the input for \(f(x)\).
Composite functions allow us to solve complex problems by breaking them down into simpler, more manageable parts. This is particularly useful in calculus, where composing and decomposing functions helps to analyze and understand curves and behaviors of functions.
In the context of our exercise, the composite function \(h(x) = x^3 + 1\) can be seen as being constructed from the functions \(g(x) = x^3\) and \(f(u) = u + 1\), where \(h(x)\) is a combination of these simpler functions.

Function Composition

Function composition refers to the process of applying one function to the results of another. This is where the beauty of mathematics shines, allowing for flexibility and creativity in solving equations. When we have \(f(g(x))\), this means we evaluate \(g(x)\) first and then apply \(f\) to the outcome.
Why do we use this? Consider situations where direct calculation is complicated or where breaking down a function into simpler parts makes it easier to manage. With function composition, complex expressions can become less intimidating.
Using our exercise, the function composition \(h(x) = f(g(x)) = f(x^3) = (x^3) + 1\) demonstrates how composing functions can simplify the original function \(h(x)\). It helps to focus on understanding each part separately and then combining them effectively.

Decomposition Techniques

Decomposition techniques are methods used to break down a composite function into its constituent parts. When a function is expressed as \(h(x) = f(g(x))\), decomposition involves identifying \(g(x)\) and \(f(u)\) such that when composed, they reproduce \(h(x)\).
Decomposing functions is like solving a puzzle. It allows us to understand complicated functions by looking for patterns or features within the function. For instance, if you notice a part of the function \(h(x) = x^3 + 1\) can stand alone (like \(x^3\)), it might be \(g(x)\). The remaining part then assists in forming \(f(u)\).
In our given problem, we decomposed \(h(x) = x^3 + 1\) by identifying \(g(x) = x^3\) and \(f(u) = u + 1\). These structured steps show how decomposition can be used to simplify and make sense of the function \(h(x)\). It's a technique that empowers us to handle functions with an analytical approach, offering insights into their construction and behavior.