Determine If The Following Are Functions Calculator

This calculator helps you determine whether given relations are functions by applying the vertical line test. Learn the definition of a function, understand the vertical line test, and practice with examples.

What is a Function?

A function is a special type of relation between two sets of elements (called the domain and codomain) where each element in the domain is paired with exactly one element in the codomain. In other words, a function assigns exactly one output value to each input value.

Mathematically, a function f from set A to set B is a subset of the Cartesian product A × B such that for every element a in A, there is exactly one element b in B such that (a, b) is in f.

Key properties of functions:

Each input has exactly one output
Not all outputs need to be used
Functions can be represented as equations, tables, or graphs

How to Test if a Relation is a Function

The most common method to determine if a relation is a function is the vertical line test. This test works for functions represented as graphs.

The Vertical Line Test

To perform the vertical line test:

Graph the relation on a coordinate plane
Imagine drawing vertical lines across the graph
If any vertical line intersects the graph more than once, the relation is not a function
If no vertical line intersects the graph more than once, the relation is a function

// Vertical Line Test Algorithm function isFunction(graphPoints) { // Group points by x-coordinate const xGroups = {}; for (const point of graphPoints) { if (!xGroups[point.x]) { xGroups[point.x] = []; } xGroups[point.x].push(point.y); } // Check if any x-coordinate has multiple y-values for (const x in xGroups) { if (xGroups[x].length > 1) { return false; } } return true; }

Examples of Functions and Non-Functions

Example 1: A Function

Consider the relation defined by the equation y = 2x + 3.

For each x-value, there is exactly one y-value
This passes the vertical line test
This is a function

Example 2: Not a Function

Consider the relation defined by the equation y² = x.

For x = 4, y can be 2 or -2
This fails the vertical line test
This is not a function

Example 3: A Function

Consider the relation defined by the table:

x	y
1	2
2	4
3	6

Each x-value has exactly one y-value, so this is a function.

Using the Calculator

Our calculator makes it easy to determine if a given relation is a function. Simply enter the points of the relation and click "Calculate" to see the result.

How to Use the Calculator

Enter the points of the relation in the format "x,y"
Each point should be on a new line
Click "Calculate" to determine if the relation is a function
Review the result and explanation

Tip: For graph-based relations, you can use the vertical line test to verify the result.

FAQ

What is the difference between a relation and a function?: A relation can have multiple outputs for a single input, while a function must have exactly one output for each input.
How do I know if a graph represents a function?: Use the vertical line test - if any vertical line intersects the graph more than once, it's not a function.
Can a function have multiple inputs with the same output?: Yes, a function can have multiple inputs with the same output, as long as each input has exactly one output.
What if my relation has no outputs for some inputs?: This is still a function as long as each input that has an output has exactly one output.
How can I represent a function?: Functions can be represented as equations, tables, or graphs, as long as they satisfy the definition of a function.