Which graph shows a set of ordered pairs that represents a function?
The graph where every input (x-value) maps to exactly one output (y-value) — so no x-value repeats and no vertical line touches more than one point. If any x appears twice with different y-values, it is NOT a function.
The answer
A set of ordered pairs represents a function when every input (x-value) is paired with exactly one output (y-value). On a graph, that's the set where no vertical line touches more than one point — the vertical line test.
In the figure above, the x-values are −2, −1, 0, 1, and 2 — all different. Slide a vertical line anywhere across it and it never meets two points at once. That's a function.
The one rule to check
Look only at the inputs:
- Repeated x-value with different y-values → NOT a function. Example: {(1, 2), (1, 5)} — the input 1 has two outputs.
- Every x-value distinct (or repeated only with the same y) → function.
Outputs are allowed to repeat. {(1, 4), (3, 4)} is a function — two inputs simply share the output 4. Functions restrict inputs, not outputs, so this trips up a lot of students.
Why the vertical line test works
A vertical line is the set of all points with the same x-value. If that line crosses a graph twice, it means one x is mapped to two different y's — exactly the thing a function forbids. So "passes the vertical line test" and "is a function" are the same statement, one visual and one algebraic.
Spotting the wrong choices
On a multiple-choice version, the distractor graphs usually contain a repeated x: two points stacked vertically (same x, different y), or a curve like a sideways parabola or a circle where a vertical line clearly cuts through twice. Any of those fails. Pick the graph where each input appears once.
Frequently asked
What makes a set of ordered pairs a function?
Each input (x-value) may appear only once. If every x maps to exactly one y, the set is a function. As soon as one x-value is paired with two different y-values, it stops being a function.
What is the vertical line test?
On a graph, imagine sweeping a vertical line across it. If the line ever crosses the graph in more than one point, some x-value has two y-values, so it is not a function. If every vertical line hits at most once, it is a function.
Can two ordered pairs share the same y-value?
Yes. Functions restrict inputs, not outputs. Points like (1, 5) and (3, 5) are fine — different x-values can share a y-value. Only repeated x-values with different y-values break the function rule.
Is {(2, 3), (2, 7)} a function?
No. The input 2 is paired with two different outputs (3 and 7), so a single input has more than one output. That set of ordered pairs is a relation but not a function.