Vertical Line Test
Functions & Advanced AlgebraThe 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 shows a function. Draw (or imagine) vertical lines across the graph. If any vertical line crosses the graph in two or more points, it is not a function.
Example
A circle fails the vertical line test: a vertical line through the middle hits the circle twice (once on top, once on bottom). A straight non-vertical line passes the test.
Key Insight
The test works because a function allows only one output per input. If a vertical line (same $x$-value) crosses the graph twice, one $x$ has two $y$-values, breaking the function rule.
Definition
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. This is a graphical restatement of the definition of a function.
Example
$y = x^3$ passes the test (each $x$ gives one $y$). $x = y^2$ fails: at $x = 4$, both $y = 2$ and $y = -2$ are on the graph, so two intersection points occur.
Key Insight
The vertical line test identifies functions "of $x$." A curve that fails can still define a function of $y$, or can be split into pieces that each pass the test (e.g., the upper and lower semicircles of a circle).
Definition
The vertical line test is an informal graphical criterion corresponding to the formal definition: 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
Implicit curves like $x^2 + y^2 = 1$ fail globally but define local functions via the implicit function theorem wherever the partial derivative with respect to $y$ is nonzero.
Key Insight
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.