Problem 7 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The domain is all real numbers except \( x = \frac{1}{3} \).

Step 1: Identify the restriction on the denominator

In a rational expression, the domain is all real numbers except those that make the denominator zero. Thus, we need to find values of \( x \) such that \( 3x-1=0 \).

Step 2: Solve for the restriction value

Set the equation \( 3x - 1 = 0 \). Solve for \( x \) by adding 1 to both sides to get \( 3x = 1 \), then divide both sides by 3 to get \( x = \frac{1}{3} \).

Step 3: Write the domain excluding the restriction

Since the expression \( 3x-1 \) cannot be zero, the domain of \( f(x) \) is all real numbers except \( x = \frac{1}{3} \). Thus, the domain is \( \{ x \in \mathbb{R} : x eq \frac{1}{3} \} \).

Key Concepts

Domain of a Function

When dealing with functions, particularly rational expressions, it's important to understand the "domain of a function." The domain of a function consists of all input values (usually represented by the variable \( x \)) for which the function is defined. For a function to be valid, its denominator must never be zero, as division by zero is undefined in mathematics.
In rational expressions, this means we need to identify any values of \( x \) that make the denominator zero, because these are precisely the values excluded from the domain. Once you determine these values, you can express the domain as all real numbers except these specific values. This is often represented in function notation or set builder notation, such as \( \{ x \in \mathbb{R} : x eq a \} \), where \( a \) represents any value making the denominator zero.

Denominator Restrictions

Rational expressions are fractions where both the numerator and the denominator are polynomials. When considering the domain, the denominator holds specific importance because it must not equal zero.
For the expression \( f(x) = \frac{x}{3x - 1} \), the denominator is \( 3x - 1 \). To find the restriction, we solve for when \( 3x - 1 = 0 \). This occurs when \( x = \frac{1}{3} \).
Understanding denominator restrictions lets us pinpoint the "problematic" values which would make the function undefined, helping us clearly determine the function's domain. In practical terms, this means we exclude \( x = \frac{1}{3} \) from the set of possible \( x \) values the function can accept.

Solving Equations with Denominators

Determining values that invalidate a function involves solving equations that stem from the denominator equaling zero. This process involves basic algebraic steps to eliminate unknowns.
For the function \( f(x) = \frac{x}{3x - 1} \), set the denominator equation \( 3x - 1 = 0 \) and solve it. Start by adding 1 to both sides, yielding \( 3x = 1 \).
Next, divide both sides by 3 to isolate \( x \):

• Adding 1 gives \( 3x = 1 \).
• Dividing by 3 provides \( x = \frac{1}{3} \).

By following these simpler steps, you can efficiently find any restrictions necessary to establish the domain of a rational function, safeguarding against undefined operations.