Domain

Algebra-2

1. Fundamental Concepts

Definition: The domain of a rational function is the set of all real numbers for which the denominator is not zero.
Exclusion Points: Values that make the denominator zero are excluded from the domain.
Graphical Interpretation: Vertical asymptotes occur at the values that are excluded from the domain.

2. Key Concepts

Basic Rule: $$\text{For } f(x) = \frac{p(x)}{q(x)}, \text{ the domain excludes } x \text{ where } q(x) = 0.$$

Degree Preservation: The highest degree in the result matches input

Application: Used to determine valid inputs for functions in physics and engineering

3. Examples

Example 1 (Basic)

Problem: Determine the domain of $$f(x) = \frac{x + 2}{x - 3}$$

Step-by-Step Solution:

Set the denominator equal to zero: $$x - 3 = 0$$
Solve for \(x\): $$x = 3$$
The domain is all real numbers except \(x = 3\).

Validation: Substitute \(x = 4\) → Original: \(\frac{6}{1}\); Simplified: 6 ✓

Example 2 (Intermediate)

Problem: Determine the domain of $$g(x) = \frac{x^2 - 4}{x^2 - 9}$$

Step-by-Step Solution:

Factor both the numerator and the denominator: $$\frac{(x + 2)(x - 2)}{(x + 3)(x - 3)}$$
Set the denominator equal to zero: $$(x + 3)(x - 3) = 0$$
Solve for \(x\): \(x = -3, 3\)
The domain is all real numbers except \(x = -3, 3\).

Validation: Substitute \(x = 0\) → Original: \(\frac{-4}{-9}\); Simplified: \(\frac{4}{9}\) ✓

4. Problem-Solving Techniques

Visual Strategy: Graph the function to identify vertical asymptotes visually.
Error-Proofing: Always check the denominator for zeros before determining the domain.
Concept Reinforcement: Practice with various rational functions to reinforce understanding of domain exclusions.