Master the fundamental rules of functions and relations. Learn to apply the Vertical Line Test, analyze
domain and range mappings, and distinguish injective, surjective, and non-function relations.
Core Concept
What is a Function?
A function is a mathematical relation that maps every element in a domain set X to
exactly one unique element in a range set Y. If an input branches to multiple outputs, it is a
general relation, not a function.
Function Machine Rule
f(x) = x² + 1
Deterministic Single-Output Rule
Think of a function as an automated black-box machine. When you drop an input value x into the
machine, it executes a set mathematical transformation and outputs a single, consistent value
f(x). Dropping the same input into a function will never yield two different results.
Graphical Test
Vertical Line Test Simulator
The Vertical Line Test states that a curve in the Cartesian plane represents a function if and only if
no vertical line intersects the curve at more than one point.
Select Equation Curve
y = 0.5x + 1
Move
mouse over graph to test
It is a Function (Maximum of 1 intersection).
Move your cursor across the graph canvas. If any vertical line crosses the curve twice,
the relation fails the test and is not a function.
Set Theory
Mapping Diagrams
Domain and Range mapping diagrams connect input set X to output set Y. Inspect how one-to-one,
many-to-one, and one-to-many relations behave.
Select Mapping Type
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.