Overview
This lesson covers the definition of functions, how to identify them using various methods, function notation, evaluating functions, and interpreting function values from tables, equations, and graphs.
Functions: Definition and Identification
- A function relates each value in the domain (x-values) to only one value in the range (y-values).
- Domain values must be unique in a function; range values may repeat.
- Relations can be represented as ordered pairs, mappings, tables, graphs, or equations.
- The vertical line test (VLT) determines if a graph represents a function: if a vertical line touches more than one point at once, it's not a function.
Examples of Functions and Non-Functions
- A list of ordered pairs is a function if all x-values are unique.
- If an x-value repeats in the set of ordered pairs or in a mapping diagram, it's not a function.
- Circles (e.g., x² + y² = 36) are not functions; quadratics (parabolas) are functions.
- If a graph fails the VLT, it is not a function; if it passes, it is.
Function Notation
- Function notation uses f(x) instead of y, where f is the function's name.
- f(x) reads as "f of x" or "f at x" and represents the output for a given input x.
- The variable inside the parentheses is the input (can change to t, a, etc. if needed).
Evaluating Functions
- To evaluate f(a), substitute a for x in the function.
- For expressions like f(2) - f(1), evaluate each value separately and subtract.
- With more complex inputs (e.g., a + 1), replace x with the entire input expression.
- When given function tables, match the input with its corresponding output.
Solving for Inputs Given Outputs
- To solve for x when given f(x) = value, set the function equal to that value and solve for x (e.g., by factoring a quadratic).
Working with Graphs and Tables
- To find f(a) from a graph, locate a on the x-axis and read off the corresponding y-value.
- From a table, find the row where x equals the given input, and read the output.
Key Terms & Definitions
- Function — A relation where each input in the domain maps to only one output in the range.
- Domain — The set of all possible input (x) values.
- Range — The set of all possible output (y) values.
- Vertical Line Test (VLT) — A method to test if a graph represents a function by checking if any vertical line crosses more than one point.
- Function Notation — A way to name and evaluate functions, usually written as f(x).
Action Items / Next Steps
- Complete the assigned homework problems on functions and function notation.