You already know that a function is a rule that assigns each input to exactly one output. For graphs, the Vertical Line Test (VLT) is a quick visual check for this rule. If you can slide a vertical line across the entire graph and it never hits the curve at more than one point, you're looking at a function.

While this is straightforward for simple lines, the same principle holds for more complex curves. The core idea doesn't change: the vertical line represents a single input value, $x$. Any point where that line intersects the graph represents an output, $y$. If there's more than one intersection, that single $x$ has multiple $y$ values, violating the definition of a function.

Consider the cubic function graphed above. No matter where you draw a vertical line, it will only ever cross the blue curve once. This visual test confirms that for any given $x$-value you choose, there is only one corresponding $y$-value. The graph successfully represents a function.

Not every equation produces a graph that is a function. Some graphs represent what we call relations, where an input can be linked to multiple outputs. A common example is a circle. While a circle is defined by a single equation, its graph fails the Vertical Line Test.

A vertical line drawn through most of the circle will intersect it in two places. This immediately tells you it's not a function. The single $x$-value corresponds to two different $y$-values: one on the top half of the circle and one on the bottom.

This graph illustrates a relation, not a function. For the input $x=1$, there are two outputs: $y = \sqrt{3}$ and $y = -\sqrt{3}$. Because a single input maps to more than one output, the relationship shown by the circle does not meet the definition of a function.

The Vertical Line Test is especially useful for graphs that are composed of multiple pieces, like piecewise functions or relations defined in separate parts. The test must be applied to the entire graph. Even if each individual piece would pass the test on its own, the overall graph fails if any two pieces are stacked vertically.

The key is to look for any $x$-value where the graph has points from more than one piece. A common place for this to occur is at the endpoints of the different sections of the graph.

On the left, we see a valid piecewise function. At $x=0$, the open circle indicates a point that is not included, so there is only one defined output (the solid dot). The graph passes the VLT everywhere.

On the right, the two pieces overlap for $x$-values between 0 and 1. A vertical line drawn in this interval will intersect both the line and the curve. Since these inputs have multiple outputs, the graph as a whole is not a function.

Ultimately, the Vertical Line Test is a simple, powerful tool for applying the fundamental definition of a function to a visual representation.