Problem 58 - Applied Calculus

Question

For each pair of functions \(f(x)\) and \(g(x)\), find and fully simplify a. \(f(g(x))\) and b. \(g(f(x))\) $$ f(x)=x^{3}+1 ; g(x)=\sqrt[3]{x-1} $$

Step-by-Step Solution

Answer

a. \(f(g(x)) = x\), b. \(g(f(x)) = x\).

Step 1: Substitute g(x) into f(x) for f(g(x))

To find \(f(g(x))\), substitute \(g(x)\) into \(f(x)\). Therefore, first replace \(x\) in \(f(x) = x^3 + 1\) with \(g(x) = \sqrt[3]{x - 1}\). This gives: \(f(g(x)) = (\sqrt[3]{x - 1})^3 + 1\).

Step 2: Simplify f(g(x))

Simplify the expression from Step 1. The cube and the cube root cancel each other, resulting in: \((\sqrt[3]{x - 1})^3 = x - 1\). Thus, we have: \(f(g(x)) = x - 1 + 1\). Simplifying further: \(f(g(x)) = x\).

Step 3: Substitute f(x) into g(x) for g(f(x))

To find \(g(f(x))\), substitute \(f(x)\) into \(g(x)\). Therefore, replace \(x\) in \(g(x) = \sqrt[3]{x - 1}\) with \(f(x) = x^3 + 1\). This gives: \(g(f(x)) = \sqrt[3]{(x^3 + 1) - 1}\).

Step 4: Simplify g(f(x))

Simplify the expression from Step 3. Subtract \(1\) from \(x^3 + 1\) to get \(x^3\). The expression becomes: \(g(f(x)) = \sqrt[3]{x^3}\). The cube root and the cube cancel each other, resulting in: \(g(f(x)) = x\).

Key Concepts

Step-by-Step Problem Solving

Solving mathematical problems step-by-step helps to break down complex processes into manageable parts. It makes understanding problem-solving clearer and more logical.
Using a step-by-step approach, we can solve exercises involving function composition effectively. For example, given functions like \( f(x) = x^3 + 1 \) and \( g(x) = \sqrt[3]{x-1} \), we are tasked with finding \( f(g(x)) \) and \( g(f(x)) \).

• Start by substituting one function into the other. For \( f(g(x)) \), this means placing the expression for \( g(x) \) into \( f(x) \).
• Next, simplify the resulting expression. This is crucial for getting a clear, final result.
• Repeat these steps for \( g(f(x)) \), substituting \( f(x) \) into \( g(x) \) and simplifying again.

Following each step methodically ensures no details are overlooked, leading to correct simplification.

Simplifying Expressions

Simplifying expressions is an essential skill in mathematics, especially when dealing with function compositions. It's all about reducing expressions to their simplest form to make them easier to work with. Let's explore how to simplify the results of composing functions like \( f(g(x)) \) and \( g(f(x)) \).After substituting, look for operations or terms that can be simplified:

• For \( f(g(x)) = (\sqrt[3]{x-1})^3 + 1 \), notice that the cube and cube root can cancel each other out, simplifying it to \( x - 1 + 1 \), which further simplifies to \( x \).
• For \( g(f(x)) = \sqrt[3]{(x^3 + 1) - 1} \), we simplify the expression by reducing it to \( \sqrt[3]{x^3} \). This again simplifies as the cube root and cube cancel, resulting in \( x \).

Simplification helps reveal the true form of the function compositions, making subsequent operations easier. It involves looking for factors that cancel and knowing fundamental operations like roots and powers.

Mathematical Operations

Mathematical operations are at the core of function composition and simplification. They help manipulate and alter expressions to achieve desired results. In this exercise, the focus is on substitution, root operations, and exponents.When you substitute \( g(x) \) into \( f(x) \), you're performing a composition operation. This involves:

• Replacing the variable \( x \) of \( f(x) \) with an entire function \( g(x) \).
• Then executing basic mathematical operations like cubing an expression \((\sqrt[3]{x-1})^3\).

Recognizing how the cube and cube root interact is important. They cancel each other out, simplifying the expression. Likewise, evaluating \( g(f(x)) \) requires similar awareness. Understanding these interactions paves the way to solving complex expressions through operations like substitution and simplification, ensuring the correct manipulation of terms.