The function f: R !R given by f(x) = x2 is not injective as, e.g., ( 21) = 12 = 1. Only one-to-one functions have inverses, so if your line hits the graph multiple times then don’t bother to calculate an inverse—because you won’t find one. The second graph and the third graph are results of functions because the imaginary vertical line does not cross the graphs more than once. 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. You can also use a Horizontal Line Test to check if a function is surjective. In general, you can tell if functions like this are one-to-one by using the horizontal line test; if a horizontal line ever intersects the graph in two di er-ent places, the real-valued function is not … In the example shown, =+2 is surjective as the horizontal line crosses the function … "Line Tests": The \vertical line test" is a (simplistic) tool used to determine if a relation f: R !R is function. With this test, you can see if any horizontal line drawn through the graph cuts through the function more than one time. 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. See the horizontal and vertical test below (9). from increasing to decreasing), so it isn’t injective. Injective = one-to-one = monic : we say f:A –> B is one-to-one if “f passes a horizontal line test”. If f(a1) = f(a2) then a1=a2. You can find out if a function is injective by graphing it.An injective function must be continually increasing, or continually decreasing. The \horizontal line test" is a (simplistic) tool used to determine if a function f: R !R is injective. If the horizontal line crosses the function AT LEAST once then the function is surjective. \$\begingroup\$ See Horizontal line test: "we can decide if it is injective by looking at horizontal lines that intersect the function's graph." A few quick rules for identifying injective functions: 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. Examples: An example of a relation that is not a function ... An example of a surjective function … 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. \$\endgroup\$ – Mauro ALLEGRANZA May 3 '18 at 12:46 1 If a horizontal line can intersect the graph of the function only a single time, then the function … If a horizontal line intersects the graph of the function, more than one time, then the function is not mapped as one-to-one. Example. This means that every output has only one corresponding input. 2. The first is not a function because if we imagine that it is traversed by a vertical line, it will cut the graph in two points. The horizontal line test lets you know if a certain function has an inverse function, and if that inverse is also a function. Horizontal Line Testing for Surjectivity. All functions pass the vertical line test, but only one-to-one functions pass the horizontal line test. If no horizontal line intersects the function in more than one point, the function is one-to-one (or injective). An injective function can be determined by the horizontal line test or geometric test.