Overview
This lecture covers describing the domain and range of functions using interval and set-builder notation, solving linear inequalities, and understanding piecewise functions.
Interval and Set-Builder Notation
- Domain is the set of valid inputs for a function; range is the set of possible output values.
- Inequality notation expresses limits, e.g., 2 < x ≤ 5.
- Set-builder notation uses braces: {x | 2 < x ≤ 5} means “the set of all x such that 2 < x ≤ 5.”
- Interval notation uses parentheses for exclusive, brackets for inclusive endpoints: (2, 5], [a, b].
- Use infinity (∞) or negative infinity (−∞) for unbounded intervals; never use brackets with infinity.
Graphing and Linear Inequalities
- A closed dot on a number line means the endpoint is included; an open dot means it’s not.
- Union (U) joins non-contiguous intervals into a combined set.
- To solve linear inequalities, isolate the variable using addition, subtraction, multiplication, or division.
- Multiplying or dividing both sides of an inequality by a negative reverses the inequality sign.
- Express solution sets in interval notation, using brackets for inclusive and parentheses for exclusive endpoints.
Determining Domain and Range
- The domain is typically restricted by denominators (cannot be zero) or even roots (no negatives under the root).
- For polynomial functions, the domain is all real numbers: (−∞, ∞).
- For rational functions, exclude values that make the denominator zero.
- For square root functions, set the radicand ≥ 0.
Piecewise Functions
- Piecewise functions have different formulas for different sections of the domain.
- To evaluate a piecewise function, determine which piece the input belongs to and use only that formula.
- Graph piecewise functions by identifying endpoints (open or closed) and applying each formula to its interval.
Common Real-World Examples
- Reasonable domains and ranges often exclude negative numbers (e.g., weights, ages, costs).
- Use graphs or context (like the Uber fare example) to guide domain and range choices.
Key Terms & Definitions
- Domain — All valid input values (x-values) for a function.
- Range — All possible output values (y-values) of a function.
- Interval Notation — A concise way to represent sets of numbers using parentheses and brackets.
- Set-Builder Notation — A way to describe sets by a property their members satisfy.
- Piecewise Function — A function defined by different expressions over different intervals.
- Union (U) — Combines multiple intervals into one set.
Action Items / Next Steps
- Complete worksheet problems 1–4 to practice interval notation, inequalities, and piecewise functions.
- Review the domain and range rules for polynomials, rationals, and root functions.