Vertical Line Test
The vertical line test determines whether a graph represents a function by checking that no vertical line crosses the graph more than once.
Definition
The vertical line test is a visual way to tell whether a graph represents a function of $x$: draw or imagine vertical lines across the graph, and if any vertical line crosses the graph in two or more points, it is not a function. Formally, a curve in the $xy$-plane represents a function of $x$ if and only if every vertical line $x = c$ intersects the curve at most once, which restates the requirement that a relation $R$ on $\mathbb{R} \times \mathbb{R}$ is a function of $x$ if and only if for all $c \in \mathbb{R}$, the set $\{y : (c, y) \in R\}$ has at most one element.
Example
A circle fails the test: a vertical line through the middle hits it twice, once on top and once on bottom, while a straight non-vertical line passes. $y = x^3$ passes (each $x$ gives one $y$), but $x = y^2$ fails since at $x = 4$ both $y = 2$ and $y = -2$ lie on the graph. Implicit curves like $x^2 + y^2 = 1$ fail the test globally but define local functions via the implicit function theorem wherever the partial derivative with respect to $y$ is nonzero.
Key Insight
The test works because a function allows only one output per input: if a vertical line crosses the graph twice, one $x$-value has two $y$-values, breaking the function rule. A curve that fails can still define a function of $y$, or can be split into pieces that each pass the test, like the upper and lower semicircles of a circle. The implicit function theorem generalizes the test to higher dimensions, guaranteeing that near a point where the Jacobian is nonsingular, an implicit equation defines a locally unique function.