Problem 10 - Brief Applied Calculus

Question

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \sqrt[3]{x^{3}+8}-5 $$

Step-by-Step Solution

Answer

The functions are: \( f(u) = u - 5 \) and \( g(x) = \sqrt[3]{x^{3}+8} \).

Step 1: Identify the Outer Function

Examine the given function, which is \[ \sqrt[3]{x^{3}+8}-5 .\] Identify the operation that is applied last, which in this case is the subtraction of 5. This suggests that our outer function could be of the form \( f(u) = u - 5 \).

Step 2: Identify the Inner Function

Look at the expression inside the composition \( \sqrt[3]{x^{3}+8} \). This implies that our inner function \( g(x) \) should simplify to \( x^{3} + 8 \) in order for the cube root to apply before passing it to the function \( f \). Hence, let \( g(x) = \sqrt[3]{x^3 + 8} \).

Step 3: Verify the Composition

Substitute \( g(x) = \sqrt[3]{x^{3}+8} \) into \( f(u) = u - 5 \). We have:\[f(g(x)) = f(\sqrt[3]{x^{3}+8}) = \sqrt[3]{x^{3}+8} - 5 \]This matches the given function, verifying that our choice of \( f \) and \( g \) is correct.

Key Concepts

Outer Function

In mathematics, the concept of function composition involves combining two functions, and understanding how these functions intertwine can be imperative for solving related problems. The outer function is the function applied last in a composition of functions. For the given problem, we are tasked with finding functions such that their composition equals a given expression. The original function is \(\sqrt[3]{x^{3}+8}-5\).
The outer function is denoted by \(f(u)\), where \(u\) stands for the entire expression after all operations by the inner function, except the last one which is the subtraction of 5. Essentially, the outer function completes the operation started by the inner function. Here, the final operation is the subtraction of 5. Therefore, the outer function is \(f(u) = u - 5\). This means, whatever input it takes — in this context, the result of the cube root — it simply subtracts 5 from it. By mastering the art of identifying the outer function, you can confidently break down complex compositions.

Inner Function

Once the outer function is understood, identifying the inner function becomes the next step.
The inner function is applied first, before the outer function. It takes the input \(x\) directly and processes it, setting the stage for the outer function.
In the case of the original expression \(\sqrt[3]{x^{3}+8}-5\), we examine the portion \(x^{3} + 8\) which is within the cube root. This part suggests the inner function, \(g(x)\), that takes the input \(x\) and first transforms it to \(x^3 + 8\). This is because \(g(x)\) prepares the input into a form suitable for further processing by the outer function \(f\). As a result, we set \(g(x) = x^3 + 8\). This decomposition logic also aids in approaching other functional compositions with clarity.

Algebraic Manipulation

Algebraic manipulation is a core skill for breaking down complex expressions into simpler parts. It involves rearranging and simplifying expressions using basic algebra rules. In the context of function composition, algebraic manipulation helps in dissecting an original function into a composition of simpler functions. With the expression \(\sqrt[3]{x^{3}+8}-5\), algebraic manipulation aids in identifying that before any outer operations like subtraction, the inner function has transformed the input \(x\) through operations like power and addition. Understanding these operations lets us redefine the expression part-by-part until it aligns with simple functions \(f\) and \(g\). In practical scenarios, algebraic manipulation can involve:

• Expanding or factoring expressions
• Simplifying complex fractions or roots
• Composing or decomposing functions

Mastering these techniques equips you with the tools to unravel even the most complex of functions by dissecting them into manageable operations.

Cube Root Function

The cube root function is a specific type of root function that is integral in many mathematical applications. It allows you to find a number that, when multiplied by itself three times (cubed), gives the original number. In the display of the function \(\sqrt[3]{x^{3}+8}-5\), the cube root function \(\sqrt[3]{...}\) is applied to the inside expression \(x^3 + 8\). This operation prepares the expression for any further operations, which, in this example, is the subtraction by 5 from its result. Understanding cube root functions goes beyond just computing roots; it includes being able to manipulate them algebraically and integrate them into broader function compositions. The cube root function specifically helps in scaling down cubes back to their base amounts, similar to how square roots work for squares but for one dimension higher. Comprehending this aspect enhances your ability to navigate through various mathematical problems involving root and power functions effectively.