Horizontal line test

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In mathematics, the horizontal line test is a test used to determine if a function is injective, surjective or bijective.

Suppose there is a function f : X → Y with a graph., and you have a horizontal line of X x Y : .

• If the function is injective, then it can be visualized as one whose graph is never intersected by any horizontal line more than once.
• If and only if f is surjective, any horizontal line will intersect the graph at least at one point (when the horizontal line is in the codomain).
• If f is bijective, any line horizontal or vertical will intersect the graph at exactly one point.

Passes the test (injective)

Fail the test (not injective)

This test is also used to find whether or not the inverse of the function is indeed a function as well. This is due to the reflective properties of the function over y=x.

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