Problem 26 - Intermediate Algebra

Question

Let \(f(x)=3 x\) and \(g(x)=4 x .\) Find each function and give its domain. See Example 1. $$ f / g $$

Step-by-Step Solution

Answer

The resulting function is \( \frac{3}{4} \) with domain all real numbers except 0.

Step 1: Understanding the Functions

We have two functions, where \( f(x) = 3x \) and \( g(x) = 4x \). These are simple linear functions given in slope-intercept form. Before proceeding, let's remember their basic forms which will help in subsequent steps.

Step 2: Formulating the Resulting Function

We need to find \( \frac{f}{g} \), which is the division of the function \( f(x) \) by \( g(x) \). This is written as: \[ \frac{f(x)}{g(x)} = \frac{3x}{4x} \]

Step 3: Simplifying the Expression

The expression \( \frac{3x}{4x} \) can be simplified by canceling the common terms in the numerator and the denominator, provided \( x eq 0 \). This simplifies to: \[ \frac{3x}{4x} = \frac{3}{4} \]

Step 4: Determining the Domain

The domain of \( \frac{f}{g} \) is derived from the requirement that \( g(x) eq 0 \) since division by zero is undefined. Given \( g(x) = 4x \), it implies \( 4x eq 0 \), hence \( x eq 0 \). Therefore, the domain of \( \frac{f}{g} \) excludes zero: all real numbers except 0.

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions that you will encounter in mathematics. They are defined by the equation \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form is known as the slope-intercept form. In linear functions, the graph will always be a straight line.
These functions exhibit a constant rate of change, meaning the slope, \( m \), indicates how steep the line is and in which direction it points. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
Let's break down the functions given in the exercise:

• For \( f(x) = 3x \), the slope \( m \) is 3, which means for every unit increase in \( x \), the function value increases by 3. There is no \( b \) term, which means the line passes through the origin \((0,0)\).
• For \( g(x) = 4x \), the slope \( m \) is 4, indicating a steeper ascent than \( f(x) \), and similarly, it also passes through the origin.

Domain of a Function

The domain of a function consists of all possible input values \( x \) that the function can accept without causing any mathematical errors, such as division by zero or taking the square root of a negative number. Essentially, it is the set of all real numbers for which the function is defined.
In the context of the operation \( \frac{f}{g} \), the domain is determined by ensuring \( g(x) \) does not equal zero since dividing by zero is undefined. For the function \( g(x) = 4x \), this implies \( x eq 0 \), as \( g(x) = 4 \cdot 0 = 0 \).
Thus, the domain of \( \frac{f}{g} \) excludes zero. The expression remains valid for all other real numbers, providing us with the final domain: all real numbers except 0.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. The function operation \( \frac{f}{g} \) is a classic example of forming a rational expression. Here, we take two linear functions and create a single rational expression by division.
To simplify a rational expression like \( \frac{3x}{4x} \):

• First, ensure none of the terms result in undefined expressions (like division by zero).
• Next, simplify by cancelling out the common terms. In \( \frac{3x}{4x} \), the \( x \)'s cancel out given \( x eq 0 \), leaving the simplified form of \( \frac{3}{4} \).

This outcome shows that despite the initial complexity of the rational expression, sometimes they can simplify to a much simpler form. Always remember that the simplification is valid as long as you respect the domain of the original rational expression.