Overview
This lecture covers the definition of functions, ways to represent them (tables, graphs, equations), function notation, and how to evaluate and operate with functions.
Definition and Characteristics of Functions
- A function assigns exactly one output value to each input value.
- The input (independent) variable determines the output (dependent) variable.
- Not all relationships between variables are functions; each input must have only one output.
Ways to Define Functions
- Functions can be defined using tables, graphs, or equations.
- In tables, if any input has more than one output, the relation is not a function.
- In graphs, the input is plotted on the horizontal axis and the output on the vertical axis.
Examples and Non-Examples
- Distance as a function of speed and time is a function since speed uniquely determines distance.
- Cholesterol count is NOT a function of age if multiple people of the same age have different cholesterol counts.
- Several inputs can have the same output in a function, but each input must have only one output.
Function Notation and Evaluation
- Function notation uses symbols like f(x), where x is the input and f(x) is the output.
- To evaluate a function, substitute the input value into the function and simplify.
- Example: For f(x) = 5x + 3, f(2) = 13.
Operations with Functions
- You can evaluate functions at algebraic expressions (e.g., f(a + b)), but f(a + b) ≠f(a) + f(b) in general.
- Parentheses in function notation do NOT represent multiplication.
Using Graphing Calculators for Functions
- Graphing calculators can generate tables of function values given an equation.
- Table settings allow for different starting points and increments for input values.
Key Terms & Definitions
- Function — a rule assigning one output to each input value.
- Input variable (independent variable) — the variable you choose values for.
- Output variable (dependent variable) — the variable whose value depends on the input.
- Function value — the output for a specific input, e.g., f(3).
- Function notation — notation like f(x) meaning "the function f at input x".
Action Items / Next Steps
- Practice identifying and evaluating functions using tables, graphs, and equations.
- Complete Homework 1.2 exercises on distinguishing and evaluating functions.
- Review the rules and notation for functions to prepare for problems involving function operations.