Problem 19 - Intermediate Algebra

Question

Find the domain of each rational function. Express your answer in words and using interval notation. $$ f(x)=\frac{2}{x} $$

Step-by-Step Solution

Answer

The domain of \( f(x) = \frac{2}{x} \) is all real numbers except zero, or \((-\infty, 0) \cup (0, \infty)\) in interval notation.

Step 1: Understanding the Rational Function

A rational function is a fraction of two polynomials. In the given function \( f(x) = \frac{2}{x} \), the denominator is \( x \). For the function to be defined, the denominator cannot be zero.

Step 2: Identify the Restriction on the Denominator

Set the denominator equal to zero to find the restricted values of \( x \). \( x = 0 \) means the denominator becomes zero, which is undefined. Therefore, \( x eq 0 \).

Step 3: Determine the Domain in Words

Since \( x \) cannot be zero, the domain of the function includes all real numbers except zero. In words, the domain is all real numbers except zero.

Step 4: Express the Domain in Interval Notation

To express the domain in interval notation, exclude zero from the set of all real numbers. In interval notation, the domain is \( (-\infty, 0) \cup (0, \infty) \).

Key Concepts

Domain of a Function

The domain of a function is essentially the set of all possible input values (usually represented by \( x \)) that will not lead to any undefined or non-real value outputs. For rational functions like \( f(x) = \frac{2}{x} \), the domain is determined by identifying the values that make the denominator zero. This is because division by zero is undefined in mathematics. By setting the denominator equal to zero and solving \( x = 0 \), we can locate the values that need to be excluded from the domain.

Therefore, for \( f(x) = \frac{2}{x} \), \( x \) cannot be zero, meaning the domain includes all real numbers except zero. This systematic approach of identifying prohibited values is crucial for correctly determining the domain of rational functions. You can picture the domain as the boundary where the function exists without any mathematical errors.

Interval Notation

Interval notation is a simplified way to represent sets of numbers using parentheses and brackets. It provides a clear visual format for stating the domain of functions.

For rational functions, where certain values might be excluded due to undefined behavior, interval notation is particularly handy. For instance, the domain of \( f(x) = \frac{2}{x} \) excludes zero, represented as \((-\infty, 0) \cup (0, \infty)\). This indicates the set of all numbers from negative infinity up to, but not including, zero, combined with all numbers greater than zero extending to positive infinity.

Here's how the notation works:

• Parentheses \(( , )\) are used to indicate that an endpoint is not included (also known as an open interval).
• Brackets \([ , ]\) would indicate that an endpoint is included (a closed interval), but in our current example, zero is excluded, hence parentheses are used.

These symbols help in concisely communicating the range without confusion.

Undefined Values

Undefined values occur when there is a mathematical operation that does not produce a valid result. In rational functions, this most commonly happens when division by zero occurs. For \( f(x) = \frac{2}{x} \), dividing by zero would mean attempting to calculate \( \frac{2}{0} \), which is undefined because you cannot divide any number by zero and get a meaningful result.

When determining undefined values, the process involves identifying numbers that, when used in the function, do not produce real or existent outputs. For rational expressions, this means setting the denominator equal to zero, as done in our example where \( x = 0 \). Recognizing and excluding undefined values is essential for defining the correct domain and ensuring the function behaves in a mathematically valid manner.

By pinpointing these values, you ensure that the function formula only corresponds to real numbers, preventing errors in calculations or subsequent operations.