Overview
This lecture covers how to describe the domain and range of functions using interval and set builder notation, as well as how to solve linear inequalities and interpret piecewise function notation.
Interval and Set Builder Notation
- Interval notation uses parentheses ( ) for non-inclusive endpoints and brackets [ ] for inclusive endpoints.
- Set builder notation uses braces { } to describe sets, e.g., {x | 2 < x ≤ 5}.
- Compound inequalities combine two inequalities, e.g., 2 < x ≤ 5, showing a range of valid values.
- Negative infinity (−∞) and infinity (∞) denote unbounded intervals in interval notation.
Graphing Inequalities on a Number Line
- Closed dots indicate included endpoints; open dots indicate excluded endpoints.
- The union symbol "∪" combines separate intervals.
Linear Inequalities and Solving Techniques
- Isolate the variable using addition, subtraction, multiplication, or division.
- Multiplying or dividing both sides by a negative number reverses the inequality sign.
- Example: If −6y + 18 > 48, subtract 18 and then divide by −6, flipping the inequality.
Domain and Range: Definitions and Examples
- Domain: set of all valid inputs (x-values) for a function.
- Range: possible outputs (y-values) the function can produce.
- For square root functions, the radicand must be ≥ 0.
- For rational functions, denominators cannot equal zero.
- Polynomials have domains of all real numbers.
Determining Domain and Range from Graphs
- Domain is the set of x-values covered by the graph; check for open and closed endpoints.
- Range is the set of y-values from the lowest to highest point of the graph.
Piecewise Functions
- Piecewise functions have different expressions for different intervals of the domain.
- Use the correct rule based on the input's interval; do not plug into both pieces.
- Graphs of piecewise functions may include open and closed circles depending on endpoint inclusion.
Application: Absolute Value and Real-World Scenarios
- Absolute value functions are a special type of piecewise function, always returning non-negative distance from zero.
- Real-world examples like Uber pricing use piecewise functions to model costs.
Key Terms & Definitions
- Domain — Set of all valid input values for a function.
- Range — Set of possible output values for a function.
- Interval Notation — Compact way to describe intervals using parentheses and brackets.
- Set Builder Notation — Uses braces and conditions to describe a set.
- Piecewise Function — A function defined by different expressions over different intervals.
Action Items / Next Steps
- Complete worksheet problems 1–4 on interval notation, inequalities, and domains.
- Practice identifying domain and range from graphs and function rules.
- Review piecewise function evaluation and try related problems on the worksheet.