Overview
This lecture introduces the concept of functions in algebra, explaining how they relate sets of inputs (domain) to unique outputs (range), and covers notation and graphical representations.
Sets and Functions
- A set is a collection of items, often numbers or objects, enclosed in curly brackets.
- Sets can be finite (like alphabet letters) or infinite (like all integers).
- A function connects one set (input) to another set (output) using a specific rule.
- The input set is called the domain; the output set is called the range.
Function Tables and Examples
- A function table displays input values and their corresponding output values side by side.
- Example: For inputs {triangle, square, pentagon}, a function could output their number of sides.
- Algebraic functions often relate variables, e.g., y = 2x, with x as input and y as output.
Function Properties and Limitations
- Each input in the domain must correspond to exactly one output in the range.
- Equations like y² = x are not functions because one input can give multiple outputs.
- Functions must avoid "one-to-many" relationships.
Graphing Functions and the Vertical Line Test
- Ordered pairs (input, output) from a function can be plotted on a coordinate plane.
- A graph is a function if every vertical line crosses it at most once (Vertical Line Test).
- Linear functions (y = x + 1) and others (quadratic, cubic, trigonometric) can be checked this way.
Function Notation
- Function notation: f(x) represents a function named f with input x, outputting a value.
- f(x) = y and y = ... are interchangeable; f(x) emphasizes the function's structure.
- Evaluating functions: Substitute a value for x, e.g., f(4) for f(x) = 3x + 2 yields 14.
Key Terms & Definitions
- Set — A collection of objects or numbers.
- Function — A rule linking each input in the domain to exactly one output in the range.
- Domain — The set of all possible input values for a function.
- Range — The set of all possible output values for a function.
- Function Table — A table showing input-output pairs for a function.
- Vertical Line Test — A method to determine if a graph represents a function.
- Function Notation (f(x)) — A way to write functions indicating the function's name and input.
Action Items / Next Steps
- Practice creating function tables and evaluating functions for given values.
- Try drawing graphs of simple functions and use the Vertical Line Test.
- Complete exercises on identifying and representing functions with proper notation.