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Variations of the horizontal line test can be used to determine whether a function is surjective or bijective: 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.
A graphical approach for a real-valued function of a real variable is the horizontal line test. If every horizontal line intersects the curve of f ( x ) {\displaystyle f(x)} in at most one point, then f {\displaystyle f} is injective or one-to-one.
A function is bijective if it is both injective and surjective. A bijective function is also called a bijection or a one-to-one correspondence (not to be confused with one-to-one function, which refers to injection). A function is bijective if and only if every possible image is mapped to by exactly one argument. [1]
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5 potentially merging Horizontal line test and Vertical line test into monotonicity?
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The farthest this end of the needle can move away from this line horizontally in its region is t. The probability that the farthest end of the needle is located no more than a distance l cos θ away from the line (and thus that the needle crosses the line) out of the total distance t it can move in its region for 0 ≤ θ ≤ π / 2 is ...