Overview
This lecture introduces the concepts of domain and range for functions, focusing on interpreting them from a graph and expressing them using interval notation.
Domain and Range Basics
- The domain of a function is the set of all possible x-values.
- The range of a function is the set of all possible y-values.
- Brackets [ ] in interval notation indicate endpoints are included; parentheses ( ) indicate endpoints are not included.
Examples of Domain and Range from Graphs
- For a graph spanning x-values from -4 to 3, the domain is [-4, 3].
- For the same graph, if y-values span from -5 to 4, the range is [-5, 4].
- If a graph has an open circle at x = -6 and a closed circle at x = 6, the domain is (-6, 6].
- If a graph extends to infinity with an arrow, use infinity in interval notation, e.g., [1, ∞) for domain.
- Always use parentheses for infinity, like (–∞, ∞) for a graph that extends endlessly left and right.
Handling Multiple Intervals and Gaps
- When the graph has breaks or jumps:
- Use the union symbol ( ∪ ) to join intervals.
- Example: If y is covered from –4 to –2 (excluding –4), and from 1 to 4 (excluding 1), write (–4, –2] ∪ (1, 4].
- For complex graphs with multiple disjoint sections, write each interval and join with unions.
Strategies for Complex Graphs
- Determine domain by identifying all x-value sections and noting open/closed endpoints.
- Determine range by finding all y-values covered, considering any gaps, and connecting intervals by union.
- To find the overall range for overlapping intervals, find the union of all covered values.
Key Terms & Definitions
- Domain — Set of all x-values for which a function is defined.
- Range — Set of all possible y-values of a function.
- Interval notation — Compact way to describe a range of numbers using brackets and parentheses.
- Union ( ∪ ) — Used to combine multiple intervals in notation.
- Open circle — Point not included in the interval.
- Closed circle — Point included in the interval.
- Infinity (∞) — Used for unbounded intervals; always paired with a parenthesis.
Action Items / Next Steps
- Practice finding the domain and range of various functions using their graphs.
- Review interval notation and practice expressing intervals with brackets, parentheses, and unions.