Not every curve in the plane is the graph of a function. The vertical line test provides a quick, visual way to check: if any vertical line crosses a curve more than once, the curve cannot represent a function.
Quick Reference
| Test | Rule |
|---|---|
| Vertical line test | A curve is the graph of a function if and only if no vertical line intersects it more than once |
| Passes | Every vertical line meets the curve at most once → it is a function |
| Fails | Some vertical line meets the curve twice or more → it is not a function |
Why the Test Works
A function assigns exactly one output to each input. In graphical terms, for each
A curve in the coordinate plane is the graph of a function if and only if no vertical line intersects it more than once.
Decide which of the following curves represents a function.
- The circle
: a vertical line hits it at and . Fails the vertical line test; not a function. - The parabola
: every vertical line meets it exactly once at . Passes; it is a function. - The upper semicircle
: for each , there is exactly one . Passes; it is a function.
The vertical line test is a quick graphical check, but the ultimate definition of a function is the formal one: each element of the domain is assigned exactly one element of the codomain. The vertical line test and the formal definition are equivalent for curves in the plane.
