Posted by ellie on Sunday, November 15, 2009 at 2:04pm.
f: R>R defined by f(x)=x is a function but it is neither surjective nor injective because in order to be injective, each real number y that is an element of the function's Y mapping, which in this case is all reals, can only be gotten to by at MOST one possible real number x in the function's X mapping (NOTE: this allows for the possibility that some values can be in the function's Y mapping that cannot be gotten to by any element of the function's X mapping). For the function f(x)=x, there is exactly two elements x of the function's X mapping that correspond to each element y of the function's Y mapping. Therefore, the function cannot be injective. Whereas to be surjective, the function needs to have for every element y of its Y mapping at LEAST one element in its X mapping that corresponds to y (NOTE: this allows for two elements in the function's X mapping to correspond to the same element in the function's Y mapping). Since the function's Y mapping is all reals, including negative real numbers, no negative element of the function's Y mapping can be gotten to by any element of the function's X mapping.
Note the shape of the graph:
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