Problem 27 - Elementary and Intermediate Algebra

Question

Let \(f(x)=3 x\) and \(g(x)=4 x .\) Find each function and give its domain. $$ g / f $$

Step-by-Step Solution

Answer

The function is \(\frac{4}{3}\) with domain \(x \in \mathbb{R} \setminus \{0\}\).

Step 1: Understanding Function Division

When dividing two functions, such as \(f(x)\) and \(g(x)\), we want to form the quotient \(\frac{g(x)}{f(x)}\). In this case, this quotient is \(\frac{g(x)}{f(x)} = \frac{4x}{3x}\).

Step 2: Simplify the Expression

To simplify \(\frac{4x}{3x}\), we divide both the numerator and the denominator by \(x\). This gives us \(\frac{4}{3}\). However, this simplification is valid only when \(x eq 0\) to avoid division by zero.

Step 3: Determine the Domain

The domain of a function is the set of all possible input values (\(x\)) for which the function is defined. For \(\frac{4x}{3x}\), the only restriction is that the denominator cannot be zero, so \(xeq 0\). Therefore, the domain is all real numbers except \(x = 0\), often written as \(x \in \mathbb{R} \setminus \{0\}\).

Key Concepts

Domain Calculation

The domain of a function is crucial because it tells us the permissible values of the input variable, typically represented as "\(x\)". When dealing with the quotient of two functions, calculating the domain involves identifying any inputs that would make the function undefined. This often happens when division by zero occurs.
For example, when forming the quotient \(\frac{g(x)}{f(x)}\), the denominator function \(f(x)\) should not equal zero, as division by zero is undefined in mathematics.
In our example, \(f(x) = 3x\), meaning \(f(x) = 0\) when \(x = 0\). Thus, \(x = 0\) is excluded from the domain. We write this domain as all real numbers except zero, denoted as \(x \in \mathbb{R} \setminus \{0\}\). Hence, checking for zero denominators in all function division problems is essential.

Quotient of Functions

Understanding the quotient of functions helps grasp how to divide two functions effectively. Given two functions, say \(f(x)\) and \(g(x)\), the goal is to form the function \(\frac{g(x)}{f(x)}\). This operation involves placing the function \(g(x)\) as the numerator and \(f(x)\) as the denominator.
In our example, this forms the expression \(\frac{4x}{3x}\). The process is straightforward:

• Identify the numerator and denominator based on the functions given.
• Ensure that the denominator does not equal zero to avoid undefined values.

A key point to remember is that forming a quotient creates a new function, which can simplify, transform, or even alter the domain pretty significantly. This illustrates why understanding the relationship between the functions before and after division can be insightful. For instance, the resulting function \(\frac{4}{3}\) offers a different perspective from simply considering \(f(x)\) or \(g(x)\) independently.

Simplifying Expressions

Simplifying expressions is a core step in evaluating the quotient of functions, providing clarity and reducing complexity. Once the expression \(\frac{g(x)}{f(x)}\) is formed, simplification involves reducing terms in both the numerator and the denominator.
In our case, simplifying \(\frac{4x}{3x}\) involves canceling \(x\) because it appears in both the numerator and the denominator. This reduction leads to the simplified constant \(\frac{4}{3}\). The simplification process requires attention to ensure valid operations, particularly ensuring \(x eq 0\), since any simplification involving dividing by zero would lead to erroneous results.

• Check for common factors across the numerator and denominator.
• Ensure conditions for simplification are appropriately set, such as ensuring non-zero values for factors being canceled.

Simplifying expressions not only helps in finding a more manageable result but also aids in understanding the behavior and properties of the function formed by the division.