Problem 27 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The domain is all real numbers except \( x = -2 \); in interval notation: \( (-\infty, -2) \cup (-2, \infty) \).

Step 1: Identify the Rational Function

The given rational function is \( f(x) = \frac{2x}{x+2} \). A rational function is defined as the quotient of two polynomials.

Step 2: Understand the Domain Restrictions

The domain of a rational function is all real numbers except where the denominator is zero. If the denominator is zero, the function is undefined.

Step 3: Set the Denominator to Zero

To find the values that make the denominator zero, set \( x+2 = 0 \).

Step 4: Solve the Denominator Equation

Solve the equation \( x+2 = 0 \). Subtract 2 from both sides to get \( x = -2 \).

Step 5: Determine the Domain in Words

The domain of the function includes all real numbers except \( x = -2 \), because that is where the function is undefined.

Step 6: Express the Domain in Interval Notation

Since \( x = -2 \) is not included in the domain, the domain in interval notation is \( (-\infty, -2) \cup (-2, \infty) \).

Key Concepts

Domain of a Function

The domain of a function is a fundamental concept in understanding how functions behave. It refers to the complete set of all possible input values (usually represented by "x") that a function can accept without leading to any kind of mathematical anomaly. For simple functions like linear ones, the domain might be all real numbers, meaning you can substitute any real number for "x."

But things can get interesting when it comes to rational functions, which are functions defined by the division of two polynomials. For these types of functions, the primary concern with the domain is the denominator - those values of "x" that make the denominator zero must be excluded, because dividing by zero is undefined in mathematics.

So, when determining the domain of a rational function, always start by identifying the values that make the denominator zero, as these will be forbidden in the domain of the function.

Interval Notation

Interval notation is a concise and efficient way to express a set of numbers, often used to describe the domain of functions. This is particularly useful when excluding specific numbers or ranges from the set of possible inputs.

To use interval notation:

• Use parentheses, "()", to denote that an endpoint is not included.
• Use brackets, "[]", to indicate that an endpoint is included.
• Combine ranges with the union symbol "∪" to show a combination of intervals that make up the full domain.

In the original exercise with the rational function \( f(x) = \frac{2x}{x+2} \), the domain in interval notation is expressed as \( (-\infty, -2) \cup (-2, \infty) \). This means all real numbers except \( x = -2 \), neatly representing the exclusion of this undefined point.

Undefined Values in Functions

Undefined values in functions are points where the function cannot operate normally. Often, this occurs in rational functions when the denominator reaches zero, as division by zero is not possible.

When analyzing a function like \( f(x) = \frac{2x}{x+2} \), identifying undefined values is crucial for establishing its domain. The process involves setting the denominator equal to zero and solving for "x." In this case, setting \( x+2 = 0 \) reveals that \( x = -2 \) makes the function undefined.

Understanding undefined values:

• Avoid making calculations that approach these points without adjusting for the exclusion.
• Ensure when graphing that these points are represented as holes or gaps.

Identifying and understanding these points helps clarify where the function and its output remain valid, solidifying the overall comprehension of its behavior.