How the Vertical Line Test Identifies Functions

Start with the definition of a function

A relation is a function when every input in its domain is paired with exactly one output. On a coordinate graph, the horizontal coordinate is the input $x$ and the vertical coordinate is the output $y$.

The rule does not require different inputs to have different outputs. Several $x$-values may share the same $y$-value. The restriction is only that one particular $x$ cannot point to two different $y$-values.

Why a vertical line tests one input

Every point on a vertical line has the same $x$-coordinate. Sliding a vertical line across a graph therefore asks the same question at every possible input: how many plotted points have this $x$-value?

• At most one intersection for every vertical line means the graph represents $y$ as a function of $x$.
• Two or more intersections for even one vertical line mean the graph is not a function of $x$.

One failing input is enough. There is no need to inspect every coordinate after finding a vertical line that crosses the graph more than once.

Compare two fresh examples

The graph of $y=x^2$ passes the test. For any fixed value of $x$, squaring it produces one value of $y$, so a vertical line meets the parabola at no more than one point.

The sideways parabola $x=y^2$ fails when it is viewed as $y$ in terms of $x$. For a positive input such as $x=9$, both $y=3$ and $y=-3$ satisfy the equation. The vertical line at $x=9$ therefore has two intersections.

Include endpoints and isolated points

Open and closed endpoints matter because only included points count as intersections. The same rule also works for a finite set of plotted points: group the points by their $x$-coordinates and check whether any group contains more than one $y$-coordinate.

Do not confuse vertical and horizontal line tests

The vertical line test asks whether the relation is a function. The horizontal line test asks a later, different question: whether a function is one-to-one. A graph must first pass the vertical test before the horizontal test is useful for deciding whether the function has an inverse that is also a function.