ALGEBRA-FUNCTIONS (MAPPINGS)

May 29th, 2015
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Although the concept of a function plays a central role in nearly every branch of mathematics, the terminology and notation associated with functions vary quite a bit. In this section, we establish ours.

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FUNCTIONS (MAPPINGS) Although the concept of a function plays a central role in nearly every branch of mathematics, the terminology and notation associated with functions vary quite a bit. In this section, we establish ours. We use the shorthand to mean that is a mapping from A to B. We will write or to indicate that carries to .The correspondence from the rational numbers to the integers given by does not define a function since but . To verify that a correspondence is a function, you assume that and prove that There are several kinds of functions that occur often enough to be given names.The term one-to-one is suggestive, since the definition ensures that one element of B can be the image of only one element of A. Alternatively, is one-to-one if implies . That is, different elements of A map to different elements of B.The mapping , with domain R, is not one-to-one, because , although when . The mapping , with domain R, is not one-to-one, because for each R and each integer . With and thought of as mappings from R to R, neither is onto. The image of is the set of nonnegative real numbers. The image of is the set of real numbers between 1 and 1, inclusive. To verify that is one-to-one in the first two cases, notice that if , we may take the cube roots of both sides of the equation to obtain . Clearly, the mapping from Z to Z given by is not onto, since 2 is the cube of no integer. However, defines an onto function from R to R, since

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