Defining the Domain of Functions

Defining the Domain

• The domain of a function is the set of all possible input values (often x) for which the function is defined. It essentially describes the input values that produce a valid output.

Types of Restrictions

• Fractions: If a function includes a fraction, the denominator cannot be zero. Exclude any x values that make the denominator zero.

• Square Roots: Functions with square roots require the value inside the radical to be non-negative. Exclude any x that produces a negative value inside the square root.

• Logarithms: The argument of a logarithmic function must be positive. The part inside the log must be greater than zero.

• Even Roots: For functions with even roots (e.g., fourth root, sixth root), the expression inside the root must satisfy the non-negative condition for real outputs.

Determining the Domain in Different Function Forms

• Linear functions: Linear functions (like y = mx + b) generally have no domain restrictions, except for specified restrictions. The domain is typically all real numbers.

• Polynomial functions: Polynomial functions (like y = x² + 2x + 1) generally have no domain restrictions, unless otherwise noted. The domain is usually all real numbers.

• Rational functions: A rational function's denominator cannot equal zero. Find the excluded x-values by solving the denominator equal to zero.

• Radical functions: For square root functions or any functions with even roots, the value underneath the radical must be non-negative. This is found by solving the inequality where the expression underneath the radical is greater than or equal to zero (or zero).

• Logarithmic functions: Logarithms (like y = log₂(x)) require the argument to be strictly positive. The argument must be greater than zero; solve the inequality.

Finding the Domain from a Graph

• Visual Inspection: A graph's domain is the set of x-values the graph covers. Trace horizontally along the graph, and the x-values it spans constitute the domain.

Composite Functions

• Composite function domain: To find the domain of a composite function, consider the restrictions of both component functions. The composite function's domain is limited by inputs that work in both constituent functions.

Interval Notation

• Interval notation expresses the domain.
  • (-∞, ∞) represents all real numbers.
  • [a, b] represents the numbers from a to b, including a and b.
  • (a, b) represents the numbers from a to b, excluding a and b.

Examples

• Example 1: f(x) = 1/(x-2): The denominator cannot be zero, so x ≠ 2. The domain is (-∞, 2) U (2, ∞).

• Example 2: f(x) = √(x+3): The expression inside the square root must be non-negative. Solving x + 3 ≥ 0 gives x ≥ -3. The domain is [-3, ∞).

• Example 3: f(x) = log₂(x-1): The argument must be positive. Solving x - 1 > 0 yields x > 1. The domain is (1, ∞).

Important Considerations

• Extraneous solutions: When solving inequalities, be wary of extraneous solutions that might not be valid in the original problem.

• Context: In real-world applications, functions often have implied domains. For example, if x represents time, then x is typically nonnegative.