Overview
This lecture covers how to describe the domain and range of functions using interval notation and set builder notation, solve linear inequalities, and interpret piecewise function notation.
Interval and Set Builder Notation
- Interval notation uses parentheses ( ) for values not included and brackets [ ] for values included.
- Set builder notation expresses sets like {x | 2 < x ≤ 5}, meaning all x such that 2 is less than x and x is less than or equal to 5.
- Compound inequalities show ranges (e.g., 2 < x ≤ 5) and can be graphed on a number line with open/closed circles.
Working with Number Lines and Unions
- Closed dots indicate included endpoints; open dots indicate excluded endpoints.
- The union symbol (U) combines separate intervals into one set.
Infinity in Intervals
- Use (−∞, a) or (a, ∞) for unbounded intervals; ∞ and −∞ always use parentheses since infinity is not a number.
- Example: The domain of √x is [0, ∞) since x must be ≥ 0.
Linear Inequality Skills
- Isolate the variable to solve for x (or y, t, etc.).
- Add, subtract, multiply, or divide both sides by a positive without changing the inequality.
- Multiplying or dividing both sides by a negative reverses the inequality sign.
Determining Domain and Range
- Domain: set of all valid input (x) values.
- Range: set of all possible output (y) values.
- To find domain/range on a graph, check starting/ending x-values (domain) and lowest/highest y-values (range).
Domain of Common Functions
- Polynomials: domain is all real numbers (−∞, ∞).
- Rational functions: exclude x-values that make the denominator zero.
- Square root functions: require the inside to be ≥ 0.
Piecewise Functions
- Piecewise functions use different formulas over different x-intervals.
- Use the formula that corresponds to the given value of x only.
- Graph piecewise by plotting each piece over its relevant interval.
Absolute Value Functions
- Absolute value measures distance from zero and always returns a non-negative value.
- Its graph is "V"-shaped and can be represented as a piecewise function.
Key Terms & Definitions
- Interval notation — a way of writing subsets of the real numbers using parentheses and brackets.
- Set builder notation — describes a set by stating the properties its members must satisfy.
- Domain — all valid input (x) values for a function.
- Range — all possible output (y) values for a function.
- Piecewise function — a function defined by different expressions over different intervals.
- Linear inequality — an inequality involving a linear expression.
- Absolute value — distance from zero; always non-negative.
Action Items / Next Steps
- Complete worksheet problems 1-4 on interval notation, inequalities, domain, and piecewise functions.
- Review how to graph intervals and practice translating between notations.
- Prepare for further study on absolute value and piecewise functions.