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 :y_0 \in Y, \{(x,y_0): x \in X\} = (X \times y_0) .

  • If the function is injective, then it can be visualized as one whose graph is never intersected by any horizontal line more than once.
  • Iff f is surjective any horizontal line will intersect the graph at least at one point
  • If f is bijective any horizontal line will intersect the graph at exactly one point.
Image:Horizontal-test-ok.png

Passes the test (injective)

 Image:Horizontal-test-fail.png

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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