How can I tell whether a graph represents a function?

Imagine placing a vertical line—one that goes straight up and down—over the graph, then sliding it from left to right. Check how many points of the graph the line meets at each position. If any vertical line meets the graph more than once, it fails the vertical line test and does not represent a function. That crossing means the same x-coordinate, or input, is associated with more than one y-coordinate, or output. [S1]

If every vertical line meets the graph at most once, it passes: each input shown on the graph has at most one output. A quick sketch of the rule is: one intersection or fewer: passes; two or more: fails.

Try an upright U-shaped parabola and a circle. A vertical line can meet the upright parabola at most once, so it passes. A vertical line through the middle of a circle meets it twice—once above and once below—so the circle fails. The lower and upper points have the same x-coordinate but different y-coordinates. That is precisely the multiple-output problem. [S1]

The shape itself is not the decision rule: follow a vertical line and count its intersections. For a worked check, suppose your sketch has a vertical line at x = 2. If that line crosses the graph at two points, record “fails at x = 2”; the graph is not a function. If it crosses once, that position is okay, but you still need to check the rest of the graph for a line that crosses more than once.

Failing means at least one input has more than one output: a vertical line at the input’s x-coordinate hits multiple points on the graph. That graph therefore does not define a function. [S1]

It is a different question from whether a function is one-to-one. The vertical line test checks whether each input has at most one output; it does not check whether different inputs can share an output. [S1] For instance, an upright parabola passes the vertical line test even though some horizontal lines can meet it twice. Do not use a horizontal-line test to decide whether a graph is a function.

Use this routine:

1. Imagine a vertical line moving across the entire graph.
2. At each position, count its intersections with the graph.
3. If any position gives two or more intersections, the graph fails.
4. Name the x-coordinate where it fails, if you can read it.
5. Explain that this input has multiple y-values.

Copyable answer: “This graph is not a function. A vertical line at x = [input] intersects the graph at [number] points, so the input [input] has more than one output.” If no vertical line intersects more than once, write: “This graph represents a function because every vertical line intersects it at most once.” The test is based on the graph you are checking; it does not tell you whether a graph is one-to-one. [S1]