Problem 18 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

In Problems 18 through 20, find functions \(f, g\), and \(h\) such that \(k(x)=f(g(h(x)))\) and \(f(x) \neq x, g(x) \neq x\), and \(h(x) \neq x\). \(k(x)=\frac{3}{\sqrt{x^{2}+4}}\)

Step-by-Step Solution

Answer

The functions \(f, g, h\) are: \(f(x) = \frac{3}{x}\), \(g(x) = \sqrt{x}\), \(h(x) = x^2 + 4\).

Step 1: Recognize the structure

Look at the structure of the given function and observe if you can locate places where you could 'insert' other functions. Observe function \(k(x) = \frac{3}{\sqrt{x^2+4}}\). This can be seen as a composition of 3 functions where \(k(x) = f(g(h(x)))\) with \(h(x) = x^2 + 4\), \(g(x) = \sqrt{x}\), and \(f(x) = \frac{3}{x}\).

Step 2: Verification

Verify if the specified function delivers the same result as the original when composed. Substitute the result of each step into the next one. First substitute \(x\) in the function \(h(x)\) to get \(h(x)=x^2 +4\). The substitute \(h(x)\) into \(g(x)\) and get \(g(h(x))= \sqrt{x^2 + 4}\). Finally substitute \(g(h(x))\) into \(f(x)\) to get \(f(g(h(x)))= \frac{3}{\sqrt{x^2 + 4}}\). Since this matches with the original function, thus functions \(f\), \(g\), and \(h\) work as desired.

Step 3: Checking non-identity

Check if these functions agree with the other conditions, especially if none of the functions is an identity function. Here, \(f(x) = \frac{3}{x}\), \(g(x) = \sqrt{x}\), and \(h(x) = x^2 + 4\). None of them is identical to \(x\), therefore, they satisfy the conditions given in the problem.

Key Concepts

Composite Functions

Composite functions are akin to combining simple ingredients to create more complex dishes in a kitchen. Imagine you have different processes represented by functions, and you want to perform them one after another on your input.

With the mathematics of function composition, this is precisely what we're doing. Let us say you have functions named 'Peel', 'Slice', and 'Fry'. If you wanted to prepare fried potatoes, you would perform these functions in a sequence on your potato: Fry(Slice(Peel(potato))). In mathematical terms, if you have functions such as 'f', 'g', and 'h', the composite function allows you to perform them successively; you would calculate 'h' first, then apply 'g' to the result of 'h', and finally apply 'f' to the result of 'g'. The notation for this is written as 'f(g(h(x)))'.

The exercise provided is a beautiful illustration of this concept. The function 'k' is the equivalent of our 'Fry' operation, and we decompose it into simpler operations (functions 'h', 'g', and 'f'). The function decomposition is intuitive once you recognize the 'layers' at which the composition happens.

Function Operations

Function operations are the basic arithmetic manipulations applied to functions, including addition, subtraction, multiplication, division, and composition. When we talk about operations with functions, we imagine each function as a machine that takes an input and spits out an output.

Operations combine these machines' outputs in different ways. For example, if we add the outputs of two functions, 'f' and 'g', for the same input 'x', we get a new function '(f + g)(x) = f(x) + g(x)'. Similarly, with composition, we feed the output of one function directly into the input of another, creating a new function altogether.

In the exercise, we are not just adding or multiplying the functions. Instead, we are inserting the output of one directly into the input of the other. Hence, we find that 'g(h(x))' becomes the new 'x' for the function 'f'. This ability to build new functions from old ones is one of the reasons why studying function operations is incredibly powerful in mathematics.

Higher-level Functions

Higher-level functions refer to those that are created by combining simpler functions, which means they can perform more complex tasks. In computer science, this concept is similar to higher-order functions, which are functions that take other functions as arguments or return them as results.

In mathematics, higher-level functions often come from compositions, like in our lesson's exercise. They exhibit more intricate behavior than their components since each 'layer' could modify the output in ways that simplify or enhance certain characteristics of the input.

For instance, in our example 'k(x)', we constructed it by composing non-identity functions 'f', 'g', and 'h'. This is a higher-level function because it is the result of layering three simpler functions, each contributing to the overall complexity of 'k'. The output of 'h' alters the domain for 'g', and the result of 'g(h(x))' further adjusts the domain for 'f', leading to a nested relationship that ultimately defines the behavior of 'k'.