Problem 2 - Beginning and Intermediate Algebra

Question

Find the domain of each rational expression. See Example 1. $$ g(x)=\frac{4-3 x}{2} $$

Step-by-Step Solution

Answer

The domain of \(g(x)\) is all real numbers.

Step 1: Identify the Rational Expression

The given expression is \( g(x) = \frac{4 - 3x}{2} \). A rational expression is one that can be expressed as a ratio of two polynomials.

Step 2: Analyze the Denominator

The denominator of the expression is 2. We need to determine the values of \(x\) for which the denominator becomes zero, because division by zero is undefined.

Step 3: Set Denominator Equal to Zero

Since the denominator is a constant \(2\), solving \(2 = 0\) would result in no solutions. This indicates that the denominator never becomes zero for any value of \(x\).

Step 4: Conclude the Domain

Because the denominator is never zero, there are no restrictions on the values of \(x\). Therefore, the domain of the function \(g(x)\) is all real numbers.

Key Concepts

Rational Expressions

Rational expressions are an important concept in algebra. They are expressions formed by dividing one polynomial by another. In a more straightforward way, you can think of them as fractions where both the numerator and the denominator are polynomials.For example, in the expression \( g(x) = \frac{4 - 3x}{2} \), the numerator is \(4 - 3x\), and the denominator is \(2\). The rule to remember here is that the denominator should not be zero, as this makes the expression undefined. Rational expressions can often look complex, but they're just like the fractions you're used to working with, involving variable terms. They allow us to represent relationships that can change depending on the values chosen for the variables.

Denominator Analysis

Analyzing the denominator is a crucial step when working with rational expressions. The denominator tells us the values that make the expression undefined. Because division by zero is not possible, we must ensure our denominator is never zero.For the expression \( g(x) = \frac{4 - 3x}{2} \), the denominator is \(2\), a constant. This shows us that it will never be zero, regardless of what value \(x\) takes. The analysis of the denominator becomes even more essential when the denominator includes variables. In such cases, setting the denominator equal to zero helps identify any restricted values for the domain. But for constant denominators like \(2\), no further action is needed, as the expression is valid for all real numbers.

Division by Zero

Division by zero is a concept that often comes up when dealing with rational expressions. It refers to dividing a number by zero, which is undefined in mathematics. This is why it is critical to examine the denominator of a rational expression.Whenever a possibility of division by zero is spotted, we must exclude those values of the variable from the domain. Failing to do so results in expressions that do not have valid numeric values. In our specific case, \( g(x) = \frac{4 - 3x}{2} \), we see the denominator is \(2\), which will not become zero for any real number \(x\). Therefore, division by zero is not an issue here, and \(g(x)\) is defined for all values of \(x\). Understanding division by zero helps avoid errors in evaluations or simplifications of expressions.