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 $x$-value there can be at most one $y$-value on the graph. A vertical line $x = a$ selects all points with first coordinate $a$. If it hits the curve at two distinct points $(a, b)$ and $(a, c)$ with $b \neq c$, then the relation would require $f(a) = b$ and $f(a) = c$ simultaneously, which is impossible since $b \neq c$.
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 $x^2 + y^2 = 4$: a vertical line $x = 1$ hits it at $(1, \sqrt{3})$ and $(1, -\sqrt{3})$. Fails the vertical line test; not a function.
- The parabola $y = x^2$: every vertical line $x = a$ meets it exactly once at $(a, a^2)$. Passes; it is a function.
- The upper semicircle $y = \sqrt{4 - x^2}$: for each $x \in [-2, 2]$, there is exactly one $y \geq 0$. 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.
