Horizontal line test

In mathematics, the horizontal line test is a test used to determine if a function is injective and/or surjective. The lines used for the test are parallel to the x axis.

Consider a function f : X → Y with its corresponding graph as a subset of the Cartesian product X x Y. Consider the horizontal lines in X x Y :$\{(x,y_0) \in X \times Y: y_0 \text{ is constant}\} = X \times \{y_0\}$ .

• The function f is injective (i.e., one-to-one) if and only if it can be visualized as one whose graph intersects any horizontal line at MOST once.
• The function f is surjective (i.e., onto) if and only if its graph intersects any horizontal line at LEAST once.
• f is bijective if and only if any horizontal line will intersect the graph EXACTLY once.

Passes the test (injective)

Fail the test (not injective)

This test is also used to determine whether or not the inverse relation of a function is itself a function.

See also

• Vertical line test
• Function (mathematics)
• Inverse (mathematics)