Example. A few quick rules for identifying injective functions: If a horizontal line can intersect the graph of the function only a single time, then the function … Horizontal Line Testing for Surjectivity. Examples: An example of a relation that is not a function ... An example of a surjective function … "Line Tests": The \vertical line test" is a (simplistic) tool used to determine if a relation f: R !R is function. The horizontal line test lets you know if a certain function has an inverse function, and if that inverse is also a function. The \horizontal line test" is a (simplistic) tool used to determine if a function f: R !R is injective. If f(a1) = f(a2) then a1=a2. The second graph and the third graph are results of functions because the imaginary vertical line does not cross the graphs more than once. If the horizontal line crosses the function AT LEAST once then the function is surjective. Look for areas where the function crosses a horizontal line in at least two places; If this happens, then the function changes direction (e.g. This means that every output has only one corresponding input. All functions pass the vertical line test, but only one-to-one functions pass the horizontal line test. See the horizontal and vertical test below (9). 2. An injective function can be determined by the horizontal line test or geometric test. You can also use a Horizontal Line Test to check if a function is surjective. The horizontal line test, which tests if any horizontal line intersects a graph at more than one point, can have three different results when applied to functions: 1. $\begingroup$ See Horizontal line test: "we can decide if it is injective by looking at horizontal lines that intersect the function's graph." Injective = one-to-one = monic : we say f:A –> B is one-to-one if “f passes a horizontal line test”. If a horizontal line intersects the graph of the function, more than one time, then the function is not mapped as one-to-one. You can find out if a function is injective by graphing it.An injective function must be continually increasing, or continually decreasing. from increasing to decreasing), so it isn’t injective. If no horizontal line intersects the function in more than one point, the function is one-to-one (or injective). With this test, you can see if any horizontal line drawn through the graph cuts through the function more than one time. ex: f:R –> R. y = e^x This function passes the vertical line test, but B ≠ R, so this function is injective but not surjective. The function f: R !R given by f(x) = x2 is not injective as, e.g., ( 21) = 12 = 1. Example picture (not a function): (8) Note: When defining a function it is important to limit the function (set x border values) because borders depend on the surjectivness, injectivness, bijectivness. 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