Vertical Line Test

A function is a relationship in which each input corresponds to exactly one output. But how can we tell from a graph whether a relationship is a function?

We can check this graphically using the vertical line test. A graph represents a function if every vertical line intersects it at most once.

For example, a parabola that opens upward passes the vertical line test because each x-value corresponds to only one y-value, so the graph shows a function.

Now imagine turning the parabola sideways. Some vertical lines cross the graph twice, giving two y-values for the same x-value. As a result, the sideways parabola fails the vertical line test and does not show a function.

But if we keep only one branch of the sideways parabola, each x-value again has only one output. This restriction turns the relation into a function.

We can see the same idea with a car moving along a road. If time is the input and distance traveled is the output, each moment can have only one distance value. So, a valid distance-time graph must pass the Vertical Line Test.