Overview
This lecture explains the vertical line test, which is used to determine if a graph represents a function by analyzing the relationship between inputs and outputs.
The Concept of Functions
- A function gives exactly one output for each input.
- The input axis is horizontal, and the output axis is vertical.
- At every input position, the corresponding output(s) are found vertically above or below.
The Vertical Line Test Explained
- The vertical line test checks if a graph represents a function by using vertical lines.
- For any graph, draw or imagine a vertical line at every input value.
- If a vertical line crosses the graph more than once at any input, the graph is not a function.
- If every vertical line crosses the graph at most once, the graph is a function.
Why the Test Works
- The test is based on the rule that each input must produce only one output.
- Multiple vertical intersections at a single input indicate multiple outputs, breaking the function rule.
Examples & Special Cases
- Diagonal or standard curves typically pass the vertical line test (one output per input).
- Graphs that form loops or vertical sections may fail the test (multiple outputs per input).
- It's acceptable for some inputs to have no outputs (undefined inputs), as in the case of square roots of negative numbers.
- A vertical line itself always fails the vertical line test, representing an infinite number of outputs for one input.
Key Terms & Definitions
- Function — A relation where each input has exactly one output.
- Input Axis (Horizontal Axis) — Axis where input values are located (commonly the x-axis).
- Output Axis (Vertical Axis) — Axis representing output values (commonly the y-axis).
- Vertical Line Test — A method to determine if a graph is a function by checking if any vertical line intersects the graph more than once.
Action Items / Next Steps
- Practice applying the vertical line test to different graphs.
- Use the vertical line test when graphing, especially piecewise functions, to check for function validity.