Problem 6 - College Algebra with Corequisite Support

Question

For the following exercises, find the domain of the rational functions. $$ f(x)=\frac{x-1}{x+2} $$

Step-by-Step Solution

Answer

Domain is all real numbers except \( x = -2 \).

Step 1: Understanding the Rational Function

The function given is a rational function. It is expressed as a fraction with a polynomial in the numerator and a polynomial in the denominator. Specifically, the function is \( f(x) = \frac{x-1}{x+2} \).

Step 2: Identifying Potential Restrictions

Rational functions have restrictions on their domain because division by zero is undefined. Therefore, we need to find the values of \( x \) that make the denominator zero and exclude them from the domain.

Step 3: Solving for Restrictions

To find the values that make the denominator zero, solve for \( x \) in the equation \( x + 2 = 0 \). This gives us \( x = -2 \).

Step 4: Defining the Domain

Since \( x = -2 \) makes the denominator zero, \( x = -2 \) is not in the domain. The domain of \( f(x) \) is all real numbers except \( x = -2 \). In interval notation, the domain is \( (-\infty, -2) \cup (-2, \infty) \).

Key Concepts

Domain of a Function

The domain of a function refers to all possible values of the input variable, usually represented as \( x \), that will produce a valid output in the function. For rational functions, which are ratios of two polynomials, the domain can be particularly restricted. These restrictions arise primarily due to the potential for division by zero, which is undefined in mathematics.

To determine the domain, one must identify and exclude any values of \( x \) that make the denominator zero. This process ensures the function remains valid across its domain.

• Consider the rational function \( f(x) = \frac{x-1}{x+2} \).
• The polynomial in the denominator is \( x+2 \).
• To ensure the function remains defined, \( x+2 \) must not equal zero.

Thus, solving \( x+2 = 0 \) helps us find the value to exclude from the domain.

Polynomial

Polynomials form the building blocks of rational functions, appearing in both the numerator and the denominator. In our example, the function \( f(x) = \frac{x-1}{x+2} \) is composed of two simple linear polynomials:

• Numerator: \( x-1 \)
• Denominator: \( x+2 \)

Polynomials are expressions made up of variables raised to non-negative integer powers, and they are combined using operations of addition, subtraction, and multiplication.

Linear polynomials like \( x-1 \) have their highest exponent as one, making them straightforward to handle. However, despite their simplicity, they play a crucial role in determining the behavior and domain of rational functions. Identifying and analyzing these polynomials helps understand how the rational function behaves across its domain.

Division by Zero

Division by zero is undefined in mathematics because it would cause scenarios that do not make sense or are impossible to resolve numerically. When referring to the function \( f(x) = \frac{x-1}{x+2} \), we focus on ensuring the denominator \( x+2 \) never equates to zero, which would lead to a division by zero issue.

Why is division by zero a problem?

• If you attempted to divide a number by zero, the result does not have a clear answer in our current number system.
• This causes the function itself to be undefined for that input value.

To ensure the function is properly defined, you must exclude any values of \( x \) that result in the denominator equaling zero. As identified, solving \( x+2=0 \) outlines these critical points where division by zero must be avoided, specifically at \( x = -2 \).

Interval Notation

Interval notation is a concise way of representing a set of numbers, often used to specify the domain or range of functions. For the function \( f(x) = \frac{x-1}{x+2} \), we determined that \( x = -2 \) would make the denominator zero, so \( x = -2 \) must be excluded from the domain.

To express this domain using interval notation:

• We denote intervals not including \( x = -2 \) as \( (-\infty, -2) \cup (-2, \infty) \).
• \( (-\infty, -2) \) covers all numbers less than \(-2\).
• \( (-2, \infty) \) covers all numbers greater than \(-2\).
• The union symbol \( \cup \) indicates a combination of these two sets.

This notation allows us to clearly communicate which values are permissible within the domain, further ensuring clarity and precision in mathematical descriptions.