Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(j(x)=10\left(x^{-2}+2 x^{2}\right)^{3}\). Give two possible decompositions of \(j(x)\) such that \(j(x)=f(g(h(x)))\). None of the functions \(f, g\), and \(h\) should be the identity function.

Step-by-Step Solution

Answer

One decomposition of the function can be \(f(x) = 10x\), \(g(x) = x^{3}\), and \(h(x) = x^{-2}+2x^{2}\). Another decomposition can be \(f(x) = 10x^{3}\), \(g(x) = x^{-2} + 2x\), and \(h(x) = x^{2}\).

Step 1: Identify the innermost function

The innermost operation in the function \(j(x)\) is the \(x^{-2}+2x^{2}\) part. Let's say \(h(x) = x^{-2}+2x^{2}\). So now we have the decomposition \(j(x) = 10(h(x))^{3}\).

Step 2: Identify the intermediate function

The next operation in the function \(j(x)\) is the power of three. Let's make this function \(g\). So now we have \(g(x) = x^{3}\). Substituting \(h(x)\) into \(g(x)\), we get \(g(h(x)) = (x^{-2}+2x^{2})^{3}\), so now we have \(j(x) = 10g(h(x))\).

Step 3: Identify the outermost function

The final operation in the function \(j(x)\) is the multiplication by 10. Let's make this function \(f\). So \(f(x) = 10x\). Substituting \(g(h(x))\), we get \(f(g(h(x))) = 10 (x^{-2}+2x^{2})^{3}\), which equals to our original function \(j(x)\).

Step 4: Provide another decomposition

For another decomposition we can say \(h(x) = x^{2}\), \(g(x) = x^{-2} + 2x\), and \(f(x) = 10x^{3}\). Substituting these values we can again get the original function.

Key Concepts

Decomposition

Decomposition in mathematical functions is akin to breaking down a complex function into simpler parts. When given a function like \( j(x) \), which is already intricate due to multiple operations involved, decomposition helps in isolating its building blocks. This way, we can better understand and manipulate the function.

- In the provided example, the function \( j(x) = 10(x^{-2} + 2x^2)^3 \) is divided into three components: the innermost, intermediate, and outermost functions. - This method allows us to assign each piece to a function: \( h(x) = x^{-2} + 2x^2 \) as the innermost, \( g(x) = x^3 \) for the intermediate, and \( f(x) = 10x \) for the final stretch.

This approach not only clarifies the operations of \( j(x) \) but also enables flexibility in solving or reimagining the problem.

Nested Functions

Nested functions, as their name suggests, are functions working within other functions. Imagine a set of boxes, each one nestled within another; this is similar to how each function in \( f(g(h(x))) \) operates within the others.

- The core principle here is understanding how transformations occur inside out, starting from the innermost part, \( h(x) \). - Once we evaluate \( h(x) \), we pass its result to \( g(x) \). Furthermore, this result is finally processed through \( f(x) \).

This structure is particularly useful in programming and mathematical modeling, where complex transformations are commonplace, and need to be systematically broken down and computed.

Mathematical Functions

Mathematical functions are the backbone of numerous operations in mathematics, acting as rules that assign every input exactly one output.

- In our example, each function \( f, g, \) and \( h \) serves a distinct role. - These functions are algebraic rules that modify inputs in particular ways.

When combined, as in compositional scenarios like \( f(g(h(x))) \), each function interacts in a sequence to produce a result based on operations like polynomials, powers, and arithmetic. Understanding these rules and how they interact is profoundly significant in various fields, from calculus to computer science.