We say a function f: a -> b is injective (one-to-one) iff for all xe a, r2a, f(x1) = f(x2) implies r = r2. what does it mean for a function not to be injective?

By definition a function is injective if and only if implies . Notice that this means that different elements of the domain have different images. That is why injective functions are also called one-to-one, because each element from A has only one image in B.

Now, if a function is not injective means that there are, at least, two elements of the domain with the same image. A very good example is the function . Recall that .

For real functions of a real variable we have a geometrical interpretation of injective property. The function is injective if for each line we draw parallel to the X-axis, it has, at most, one intersect with the graph of . Then, if one line has more than one intercept, then the function is not injective.

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