Problem 16 - Algebra 1: Concepts and Skills

Question

Find the domain of the function. \begin{equation} \frac{x+1}{x+2} \end{equation}

Step-by-Step Solution

Answer

The domain of the function is \((-∞,-2) ∪ (-2,∞)\).

Step 1: Find the values of x for which the denominator equals zero

Set the denominator equal to zero and solve for x, that is solve the equation \(x+2=0\).

Step 2: Solve the Equation

Solving the equation, we get \(x=-2\).

Step 3: Exclude the Problematic Value from the Domain

This is the only value for which the denominator becomes zero. So, the domain of the given function is all real numbers except -2. In interval notation this is \((-∞,-2) ∪ (-2,∞)\).

Key Concepts

Finding the Domain

Understanding the domain of a function is a fundamental aspect in mathematics. The domain refers to all the possible input values (often represented by x) that a function can accept without causing issues like division by zero or taking the square root of a negative number.

When finding the domain of a function, we look for values that can make the function 'undefined'. For instance, if our function involves a fraction, we must ensure the denominator is never zero. That's because division by zero is undefined in conventional mathematics. If the function includes a square root, we must ensure that the number under the root is non-negative since the square root of a negative number is not a real number.

To determine the domain, start by considering all real numbers and then subtract any values that cause issues. The remaining set of numbers comprises the domain of the function.

Rational Functions

A rational function is any function that can be represented as the quotient of two polynomials. In simpler terms, it's a fraction where both the numerator and the denominator are polynomials. The general form of a rational function is \( \frac{P(x)}{Q(x)} \), where P(x) and Q(x) are polynomials.

The key characteristics of rational functions include their ability to approach asymptotes, both horizontal and vertical, and their potential to have points of discontinuity. It is the points of discontinuity that we are most interested in when determining the function's domain, as these are values for which the function does not produce a real number output.

Excluded Values in Domains

The term excluded values refers to those inputs that we must eliminate from the domain of a function to keep it well-defined. For rational functions, excluded values arise when the denominator is zero, because dividing by zero is undefined.

In the case of the function \( \frac{x+1}{x+2} \), we set the denominator equal to zero and solve for x to find \( x = -2 \). This value is thus excluded from the domain. Understanding that these exclusions are necessary helps prevent mathematical errors and ensures the proper graphing and analysis of functions.

Conveying the full domain, we use interval notation, writing it as \( (-\infty,-2) \cup (-2,\infty) \) for the given function. This notation compactly expresses that all real numbers except -2 are included in the domain of the function, effectively communicating the concept of excluded values.