Problem 49 - Algebra 1: Concepts and Skills

Question

Find the domain of the function. $$ y=\frac{1}{x+2} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Denominator

In this function, the denominator is \(x + 2\). We need to identify values of 'x' that would make the denominator equal to zero.

Step 2: Find the Value That Makes the Denominator Zero

Set the denominator equal to zero and solve for 'x'. So, \(x + 2 = 0\). Solving for 'x' gives \(x = -2\).

Step 3: Formulate the Domain

The domain of the function is all real numbers except for those that make the denominator equal zero. So, in this case, the domain is all real numbers except \(x = -2\)

Key Concepts

Denominator of a Fraction

The denominator of a fraction plays a pivotal role in determining the functionality and limitations of a mathematical expression. Specifically, in the context of a function, the denominator indicates the values for which the function does not produce a valid result. This is crucial in the study of rational expressions, where the denominator must not be zero, as division by zero is undefined in mathematics.

For example, in the function \(y = \frac{1}{x+2}\), the denominator is \(x + 2\). It is the 'bottom' part of the fraction, and our task is to ensure it is never zero, since dividing by zero would not yield a meaningful result. The exercise, therefore, revolves around avoiding values of 'x' that would render this scenario. Understanding the role of the denominator is fundamental to grasping the broader concept of the domain of rational functions.

Rational Functions

A rational function is any function that can be expressed as the ratio of two polynomials, where the numerator and the denominator are both polynomials and the denominator is not zero. The form of a rational function is usually given by \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) is the numerator polynomial and \(Q(x)\) is the denominator polynomial.

The domain of these functions is all real numbers except where the denominator, \(Q(x)\), equals zero. Since the denominator in our function is \(x + 2\), we exclude the value that turns it to zero from the domain. The intricacies of rational functions require a keen attention to the polynomial forms, as their behavior can greatly differ across their domains, demonstrating unique characteristics such as vertical asymptotes, holes, or horizontal asymptotes.

Undefined Values in Functions

The undefined values in functions are those inputs for which the function does not yield a valid output or result. Typically, these are the values that cause violations in basic mathematical rules, such as division by zero. Identifying these undefined values is critical when determining the domain of a function - the set of all possible inputs.

In the rational function \(y = \frac{1}{x+2}\), the value \(x = -2\) is undefined because it would make the denominator zero, breaking the rule against division by zero. Thus, this value is excluded from the domain. Understanding where a function becomes undefined helps prevent computational errors and informs our interpretation of the function’s graph and characteristics, as these undefined points often correspond to important features, such as breaks in continuity or asymptotic behavior.