Lesson 5-1: Identify Functions

Property

A relation is a set of ordered pairs. The set of the first components (inputs) is called the domain, and the set of the second components (outputs) is called the range.

A function is a special relation where each possible input value leads to exactly one output value.

Examples

• The relation {(1, 3), (2, 5), (3, 7)} is a function because each input (1, 2, 3) has exactly one output.
• The relation {(A, 1), (B, 2), (A, 3)} is not a function because the input 'A' is paired with two different outputs, 1 and 3.
• In a school, if each student's name is an input and their assigned homeroom number is the output, this is a function because each student is assigned to only one homeroom.

Property

Relations can be represented in three equivalent forms: ordered pairs (x, y), tables, and mapping diagrams with arrows.

To be a function, every input must have exactly one arrow pointing away from it (in a diagram) or correspond to exactly one output (in a table). It is completely acceptable for different inputs to share the same output.

Examples

• Valid Function: A table shows the months of the year (input) and the number of days in that month (output). Multiple months like January and March have the exact same output (31 days), which is perfectly allowed.

Property

The vertical line test determines if a graph represents a function by checking whether any perfectly vertical line intersects the graph at more than one point. If every vertical line intersects the graph at most once, then the graph represents a function.

Examples

• An upward-opening parabola passes the vertical line test because each vertical line intersects it at most once, meaning it is a function.
• A horizontal straight line passes the vertical line test because each vertical line intersects it exactly once.
• A circle fails the vertical line test because a vertical line drawn through its center will intersect the circle at two different points (a top point and a bottom point).

Explanation

The vertical line test works because functions must have exactly one output (y-value) for each input (x-value). When a vertical line hits a graph at multiple points, it proves that a single x-value is producing multiple y-values, breaking the ultimate rule of a function. This visual trick gives you a split-second answer!