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Understanding Domain and Range

Jul 15, 2025

Overview

This lecture explains how to find the domain and range of relations and functions using ordered pairs, mapping diagrams, tables, graphs, and equations, including restrictions and notation.

Domain and Range Basics

  • The domain is the set of first coordinates (x-values) in a relation or function.
  • The range is the set of second coordinates (y-values).
  • List domains and ranges in ascending order, using curly brackets for sets.

Restrictions in Finding Domain and Range

  • For radicals with even indices, the expression under the radical must be non-negative.
  • For fractions, the denominator must not be zero (division by zero is undefined).

Examples Using Ordered Pairs, Mapping Diagrams, and Tables

  • Identify the domain from the set of all distinct x-values in the relation.
  • Identify the range from the set of all distinct y-values in the relation.
  • Organize results in ascending order.

Finding Domain and Range from Graphs

  • The domain consists of all x-values covered by the graph (from left to right).
  • The range consists of all y-values covered by the graph (from bottom to top).
  • Use brackets [ ] for included endpoints (solid dots); use parentheses ( ) for excluded endpoints (open dots).
  • Use interval notation (e.g., [-10, -2]) or set notation (e.g., {x | -10 ≤ x ≤ -2}).

Infinite Domains and Ranges

  • If the graph extends infinitely left/right or up/down, domain and/or range is all real numbers.
  • Use interval notation: (-∞, ∞) or set notation: {x | x ∈ ℝ}.

Domain and Range from Equations

  • For polynomials (e.g., y = 2x - 5), domain and range are typically all real numbers.
  • For equations with radicals, restrict domain/range so the expression under the radical is non-negative.
  • For rational functions, exclude values that make the denominator zero.

Sample Problems Analyzed

  • For y = x² - 9: Domain is all real numbers; range is y ≥ -9.
  • For x² + y - 4 = 0: Domain is all real numbers; range is y ≤ 4.
  • For rational equations, exclude values that make denominators zero.

Key Terms & Definitions

  • Domain — The set of all possible input values (x-values).
  • Range — The set of all possible output values (y-values).
  • Interval notation — A way to describe all numbers between two endpoints.
  • Set notation — Expresses domain/range by specifying conditions (e.g., {x | x ≥ 0}).
  • Polynomial — An algebraic expression with variables and coefficients, involving only non-negative integer powers of variables.
  • Radical — An expression that includes a root (e.g., √x).
  • Rational function — A function expressed as a fraction with polynomials in the numerator and denominator.

Action Items / Next Steps

  • Practice finding domain and range from various representations (ordered pairs, tables, graphs, equations).
  • Review restrictions for radicals and rational functions.
  • Prepare for exercises on representing domains and ranges in both set and interval notation.

Full transcript