Problem 21 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The domain is all real numbers except -2, expressed as \((-\infty, -2) \cup (-2, \infty)\).

Step 1: Understand the Rational Function

The given function is \( f(x) = \frac{2x}{x+2} \). This function is a rational function because it is a fraction with polynomials in both its numerator and denominator.

Step 2: Identify the Value that Makes the Denominator Zero

The domain of a rational function is all real numbers except where the denominator is zero. Set the denominator equal to zero to find this value: \( x+2=0 \).

Step 3: Solve for the Excluded Value

Solve the equation \( x+2=0 \) to find the value of \( x \) that would make the denominator zero. Solve for \( x \): \( x = -2 \).

Step 4: Determine the Domain

The domain includes all real numbers except \( x = -2 \) because the function is undefined at this value. In interval notation, this is represented as \( (-\infty, -2) \cup (-2, \infty) \).

Step 5: Express the Solution in Words

The domain of the function is all real numbers except \( -2 \).

Key Concepts

Domain of a Function

Understanding the domain of a function is a foundational skill in mathematics. In simple terms, the domain of a function consists of all the possible input values (usually represented by \( x \)) that can be plugged into the function without causing any mathematical error, such as division by zero.

For rational functions like \( f(x) = \frac{2x}{x+2} \), it's essential to identify the values of \( x \) that make the denominator zero because these values would cause the function to be undefined. Therefore, to find the domain, we need to exclude these problematic values from the set of all real numbers. In other words, the domain is the set of all real numbers except those that lead to division by zero in the function.

Determining the domain systematically involves the following steps:

• Set the denominator equal to zero.
• Solve the equation to find the value(s) of \( x \) that make the denominator zero.
• Exclude these values from the domain.

Interval Notation

Once you've identified the domain of a function, it's helpful to express it in a concise format known as interval notation. Interval notation provides a way to describe the set of values that are included (or excluded) in the domain.

Interval notation uses brackets and parentheses to denote intervals:

• Round brackets \(( )\) are used to indicate that the endpoint is not included in the interval (also known as open interval).
• Square brackets \([ ]\) are used when the endpoint is included (closed interval).

For the function \( f(x) = \frac{2x}{x+2} \), the domain excludes \( x = -2 \). Therefore, the domain in interval notation is written as \((-\infty, -2) \cup (-2, \infty)\).

Here, \((-\infty, -2)\) represents all real numbers less than \(-2\), and \((-2, \infty)\) represents all real numbers greater than \(-2\). The union symbol \(\cup\) indicates that these two intervals together form the complete domain.

Excluded Values

Excluded values are specific points that are removed from the domain of a function due to potential undefined behavior. In rational functions, these are typically the values that cause the denominator to be zero, thus rendering the function undefined.

To determine the excluded values in a rational function, follow these steps:

• Identify the denominator of the rational function.
• Set the denominator equal to zero and solve for \( x \).
• The solutions to this equation are the excluded values.

For the example \( f(x) = \frac{2x}{x+2} \), the denominator \( x + 2 \) equals zero when \( x = -2 \). Therefore, \( -2 \) is an excluded value, meaning the function is undefined at \( x = -2 \).

Understanding excluded values is crucial as it helps in graphing functions accurately and avoiding misunderstandings about where the function exists and where it does not.