Master mathematical mapping criteria. Visualize domain-to-range connectivity, run the vertical line test dynamically, and identify function constraints.
Algebra Core
What is a Function?
A **function** is a specific type of relation where each element in the **Domain** (set of inputs) maps to *exactly one* element in the **Range** (set of outputs). If an input maps to multiple outputs, it is a **Relation**, not a function.
Definitions & Rules
Domain: The set of all possible input values (typically x).
Range: The set of all actual output values (typically f(x) or y).
Vertical Line Test: If any vertical line intersects a graph more than once, the graph represents a relation, not a function.
Visual Mapping Machine
Think of a function as a machine:
- You drop an input x in.
- The machine performs a rule: f(x) = x² + 1.
- You get exactly one output y.
If a machine could output both +3 and -3 for the single input 9 (like y² = x), it is not a function.
Interactive Graph
Vertical Line Test Simulator
Select a relation preset, then drag the mouse across the canvas to draw a vertical line. If the line intersects the curve more than once, it fails the test.
It is a Function (Maximum of 1 intersection).
Relation Audit
Equation
y = 0.5x + 1
Intersections Count
1 intersection
If a vertical line crosses at multiple points, a single x maps to multiple y values, violating the mapping criteria.
Set Mapping
Mapping Diagrams
Examine mapping connections between domain inputs and range outputs. Toggles different relation types.
Classification details
One-to-One Mapping
Each input in the Domain points to a unique output in the Range. It represents a strictly Injective Function.
Notice how in the 'One-to-Many' mode, a single node branches out to multiple outputs, rendering it an invalid function.
Evaluation
Conceptual Quiz
Solve the questions below to test your understanding of domain, range, and vertical line tests.
Question 1 of 4Score: 0/0
Does the equation x² + y² = 9 define y as a function of x?
Correct Answer!
No, it defines a circle of radius 3. Moving a vertical line across x in (-3, 3) gives two y-intersections, failing the vertical line test.