Summary
Highlights
A relation is any set of input and output values (ordered pairs). There are no special rules for a relation. Relations can be represented as ordered pairs, graphs, tables, or mappings. A function is a specific type of relation where a special rule applies, ensuring one output for each input.
A function can be compared to a vending machine: for a given input (code), you always get the same output (item). The input is called the domain, and the output is called the range. Functions are predictable; if an input gives multiple outputs, it's not a function. Inputs cannot repeat, but outputs can.
To determine if a graph represents a function, use the vertical line test. If a vertical line intersects the graph at more than one point, it is not a function. This is because a single input (x-value) would have multiple outputs (y-values).
For tables, check if any input (x-value) repeats with different outputs. If an input has more than one corresponding output, it's not a function. For mappings, ensure only one arrow originates from each input to an output. If an input points to multiple outputs, it's not a function.
The video provides several examples of tables and mappings, demonstrating how to identify if they are functions and how to determine their domain (all unique inputs) and range (all unique outputs). The domain lists all x-values, and the range lists all y-values.
For graphs, the domain is how far left to how far right the graph extends (x-values), and the range is how far down to how far up the graph extends (y-values). Pay attention to whether endpoints are included (brackets) or excluded (parentheses), and use infinity for graphs that extend indefinitely. Special attention is given to asymptotes for specific functions like exponential functions.