Overview
This lecture introduces key characteristics of functions, including definitions, types, tests for function properties, notation for domain and range, linearity, intercepts, and symmetry.
Functions: Definitions and Types
- A function is a relation where every input (x-value) has exactly one output (y-value).
- One-to-one function: every output has exactly one input.
- Many-to-one function: some outputs have multiple inputs.
- A relation fails to be a function if an input maps to more than one output.
Graphical Tests for Functions
- The vertical line test: a graph is a function if any vertical line crosses it at most once.
- One-to-one functions pass the horizontal line test: any horizontal line crosses the graph at most once.
Continuity and Discreteness
- Continuous functions: can be drawn without lifting your pencil (unbroken line/curve).
- Discrete functions: consist of unconnected points.
- Some functions are neither continuous nor discrete (e.g., piecewise with jumps).
Domain and Range: Definitions and Notation
- Domain: set of all possible x-values (inputs) for a function.
- Range: set of all possible y-values (outputs) for a function.
- Notation for domain/range:
- Interval notation: (a, b) or [a, b], where ( ) means exclusive, [ ] means inclusive.
- Algebraic notation: uses inequalities (e.g., x > -2).
- Set builder notation: {x | x > -2}.
Determining Domain and Range from Graphs
- Arrows indicate extension to infinity.
- Open circles mean the endpoint is not included; closed circles mean it is included.
- Piecewise functions: include a value in domain/range if it exists somewhere on the graph.
Linearity
- A linear function has no variable with power other than one.
- Standard form: y = mx + b, where m and b are real numbers.
- Graph of a linear function is a straight line (constant slope).
- Constant functions (e.g., y = 9) are linear with zero slope.
Intercepts
- x-intercept: point(s) where the graph crosses the x-axis (set y=0 and solve for x).
- y-intercept: point where the graph crosses the y-axis (set x=0 and solve for y).
- Functions can have multiple x-intercepts but only one y-intercept.
Symmetry
- Line symmetry: graph is reflected over a vertical line so both halves match.
- Point symmetry: graph maps onto itself after a 180° rotation about a point.
- Not all functions have symmetry; graphs may be neither.
Even and Odd 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.
- If neither condition is met, the function is neither even nor odd.
- Shortcut: all even exponents/none = even; all odd exponents = odd; mixture = neither.
Key Terms & Definitions
- Function — A relation where each input has one output.
- Domain — Set of all possible inputs (x-values).
- Range — Set of all possible outputs (y-values).
- One-to-one function — Each output relates to only one input.
- Many-to-one function — Some outputs relate to multiple inputs.
- Continuous function — Can be drawn without lifting pencil; unbroken.
- Discrete function — Made up of unconnected points.
- Linear function — No variable raised to power other than one; straight line.
- x-intercept — Where graph crosses x-axis (y=0).
- y-intercept — Where graph crosses y-axis (x=0).
- Line symmetry — Reflection over a vertical line.
- Point symmetry — 180° rotational symmetry about a point.
- Even function — Satisfies f(-x)=f(x); y-axis symmetry.
- Odd function — Satisfies f(-x)=-f(x); origin symmetry.
Action Items / Next Steps
- Practice identifying domain and range in interval, algebraic, and set builder notations.
- Solve for x- and y-intercepts in given equations.
- Use algebraic tests to check if functions are even, odd, or neither.
- Review and memorize key terminology and notations.