Problem 20 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find functions \(f, g\), and \(h\) such that \(k(x)=f(g(h(x)))\) and \(f(x) \neq x, g(x) \neq x\), and \(h(x) \neq x\). \(k(x)=\sqrt{\left(x^{2}+1\right)^{3}+5}\)

Step-by-Step Solution

Answer

The functions \(f\), \(g\), and \(h\) which satisfy the equation \(k(x)=f(g(h(x)))\) are \(f(x)=\sqrt{x}\), \(g(x) = x^{3}+5\), and \(h(x) = x^{2} + 1\) respectively.

Step 1: Identify function h

From the definition \(k(x)=\sqrt{\left(x^{2}+1\right)^{3}+5}\), we identify the innermost operation as \(x^{2}+1\). Therefore, \(h(x) = x^{2} + 1\).

Step 2: Identify function g

Once we applied function \(h\), we are left with the function \(\sqrt{(x)^{3}+5}\), where \(x = h(x)\). The next operation is raising x to the power of 3 and adding 5. Thus, \(g(x) = x^{3}+5\).

Step 3: Identify function f

Now that we applied functions \(h\) and \(g\), we are left with the function \(\sqrt{x}\), where \(x = g(h(x))\). The square root is the operation handled by function \(f\), which gives \(f(x) = \sqrt{x}\).

Key Concepts

Function Composition

Before leaping into the complexities of function composition, it's crucial to understand the simplicity behind the concept. Imagine a relay race, where a baton is passed from one runner to the next. Function composition operates in a similar manner; the output of one function becomes the input of another. In calculus, this seamless handoff is denoted as \( f(g(x)) \), which reads as 'the function \( f \) of the function \( g \) of \( x \)'. This leads to a new function that combines both the actions of \( f \) and \( g \) into a single operation.

To transform this from an abstract concept into something more tangible, consider this analogy: If \( g \) represents a process that packages a product and \( f \) represents a delivery system, then \( f(g(x)) \) is the complete service of packaging and delivering the product. This concept is not only cornerstone in calculus but it's also a vital tool across various branches of mathematics and applied sciences.

Inner Functions and Operations

Zooming into the anatomy of composite functions reveals an important element - the 'inner functions'. These are the functions that are nested within other functions, and they're the first to act on the input. Think of them as the foundation of a building; they need to be sturdy and well-understood because everything else is built upon them.

Returning to our previous example, \( h(x) = x^{2} + 1 \) is an inner function for the given composite function \( k(x) \). It's akin to prepping the product before it's packaged – squaring the input and adding one. After \( h \) has transformed \( x \) into \( h(x) \), only then can \( g \) and \( f \) proceed with their operations. For a student tackling calculus problems, identifying and understanding these inner workings is crucial for successfully decomposing and composing functions.

Identifying Component Functions

Dissecting a composite function into its component functions is like reverse-engineering a complex gadget to understand its parts. The composite function \( k(x) \) can be seen as an intricate machinery made up of individual operations, each performed by a component function. The trick is to peel the layers, starting from the outermost operation and moving inward.

In our case with the function \( k(x) = \sqrt{(x^{2}+1)^{3}+5} \) the square root is the final operation – handled by \( f \), the raising to the power of three and addition of five is done by \( g \), and the initial squaring and increment by one is executed by \( h \). Learning to identify these components requires practice, but once mastered, it provides a powerful framework for understanding more complex mathematical relationships.

• \bf{h(x)}: Start with identifying the most nested function, typically involving the simplest arithmetic operation on \( x \).
• \bf{g(x)}: Uncover the next function which takes the output of \( h(x) \) as its input.
• \bf{f(x)}: Finally, reveal the outermost function applying its operation to the result of \( g(h(x)) \).