Overview
This lecture introduces the basic concept of mathematical functions, how to define them, their notation, domains and ranges, and how to represent and compute them.
What is a Function?
- A function is a set of ordered pairs where each first member (argument) is unique.
- Functions can be described by explicitly listing pairs or by using a rule (formula).
- Example: The function f defined by pairs (1,1), (2,1), (3,2) assigns a unique value to each argument.
Function Notation and Terminology
- Functions are named, often as f, g, or h.
- The domain of a function is the set of all possible arguments (first members).
- The range is the set of all possible values (second members).
- Standard notation: f(1) = 1, f(2) = 1, f(3) = 2.
Properties and Uniqueness
- Each argument in the domain is assigned a unique value.
- Functions can assign the same value to different arguments but not different values to the same argument.
Real-Life Example of a Function
- Temperature as a function of time: T assigns a temperature to each time of measurement.
- When a function has an infinite domain, it’s impractical to list all pairs; instead, a formula is used.
Function Formulas and the Identity Function
- Functions are often defined by formulas, e.g., f(x) = x, called the identity function.
- More complex functions: f(x) = 3x, f(x) = x^2, f(x) = x^2 - 1, f(x) = 3/x, etc.
Constructing Functions
- Functions can be made using addition, subtraction, multiplication, and division of numbers and the argument x.
- Spreadsheets or graphing calculators can easily compute and visualize these functions.
Graphing Functions
- The graph of a function is a plot where x is the argument and y is the value.
- Plotting a sample of points can illustrate the function’s behavior, even for infinite domains.
Using Technology with Functions
- Modern technology allows for quick computation and graphing of various functions.
- Built-in functions and numeric computation make it easy to explore function behavior.
Key Terms & Definitions
- Function — A set of ordered pairs where each argument has a unique value.
- Argument — The input or first member of a function’s pair (often called x).
- Domain — The complete set of possible arguments for a function.
- Value — The output or second member of a function’s pair.
- Range — The set of all possible values a function can produce.
- Formula — A rule that assigns a value to each argument in the domain.
- Identity Function — The function f(x) = x, where each value equals its argument.
- Graph — A visual representation of a function using points (x, f(x)).
Action Items / Next Steps
- Complete Exercise 3.1: Determine the domain and specific values for the function g.
- Complete Exercise 3.2: Calculate the value of x^2 + 3 at x = 5 and x = 10.
- Practice writing and graphing functions using formulas and spreadsheets.