Overview
This lecture introduced the concept of functions, how to identify and represent them, and how to determine their domain and range. It also covered piecewise functions, function notation, vertical line test, and the difference between removable discontinuities (holes) and vertical asymptotes.
What is a Function?
- A function assigns exactly one output for each input; no input has two outputs.
- Inputs are usually called x (independent variable), and outputs are y or f(x) (dependent variable).
- Functions can be represented by tables, formulas, or graphs.
- If an input repeats with different outputs, it is not a function.
- The vertical line test: if any vertical line intersects a graph more than once, it's not a function.
Function Notation and Representation
- Function notation: f(x) expresses y as a function of x; allows distinguishing between multiple functions.
- To find f(a), substitute a for x in the formula.
- Using function notation helps identify both inputs and outputs clearly.
Examples & Special Cases
- Repeated outputs (e.g., different inputs with the same output) are allowed.
- Circle equation x² + y² = r² is not a function of x (fails vertical line test and gives two y outputs for one x).
- Function can have formula constraints or real-world constraints (e.g., weights, lengths cannot be negative).
Piecewise Functions
- Piecewise functions use different formulas for different intervals of the domain.
- Example: Absolute value function defined as f(x) = x if x ≥ 0 and f(x) = -x if x < 0.
- To graph piecewise functions, graph each piece over its specified domain/range.
Domain and Range
- Domain: all valid input values (usually x-values).
- Range: all possible output values (usually y-values or f(x)).
- Domain may be restricted by real-world context (e.g., side lengths ≥ 0) or by the formula (e.g., division by zero or square roots).
- For square roots: radicand must be ≥ 0; for denominators: cannot equal zero.
Determining Domain (Natural Domain)
- Set denominators ≠0 to find restrictions.
- For roots, set radicand ≥ 0 and solve.
- Use sign analysis for quadratic inequalities in the radicand.
- State domain using interval notation or set notation.
Removable Discontinuities and Asymptotes
- If you can factor and cancel a term causing 0/0, it's a hole (removable discontinuity).
- If you cannot cancel and plugging in gives a nonzero/zero form, it's a vertical asymptote.
- Original domain restriction remains, even after simplification.
Odd and Even Functions
- Even function: f(-x) = f(x); graph is symmetric about the y-axis.
- Odd function: f(-x) = -f(x); graph is symmetric about the origin.
Applying to Word Problems
- Realistic constraints can further restrict domain (e.g., box-making: cut size must be >0 and less than half of length/width).
- Write the volume or other quantities as a function of a variable, considering all restrictions.
Key Terms & Definitions
- Function — Relation assigning one output to each input.
- Domain — All possible input values for a function.
- Range — All possible output values for a function.
- Vertical line test — Graph-based method to determine if a relation is a function.
- Piecewise function — Function defined by multiple sub-functions over different intervals.
- Removable discontinuity (hole) — A point where the function is undefined due to cancelable factors.
- Vertical asymptote — Line where the function grows without bound due to a non-cancelable zero in the denominator.
- Even function — f(-x) = f(x), y-axis symmetry.
- Odd function — f(-x) = -f(x), origin symmetry.
Action Items / Next Steps
- Practice identifying functions from tables, graphs, and formulas.
- Complete assigned exercises on finding domain and range.
- Read textbook sections on piecewise functions and function notation.
- Review homework problems involving domain restrictions and graphing piecewise functions.