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If a function is defined as in the preceding example, the symbol used for the variable is immaterial; that is, expressions such as: and so on, all define the same function.This is true because if a is any number in the domain of f, then the same image a is obtained no matter which expression is employed. Solution As in Example 1, finding images under f is simply a matter of substituting the appropriate number for x in the expression for f(x).Thus to 3 we assign 9, to - 5 we assign 25, and so on. All the examples of correspondences we have given are functions, as defined below.
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As another example, to each human being there corresponds a birth date.
To cite a third example, if the temperature of the air is recorded throughout a day, then at each instant of time there is a corresponding temperature.
The element y is called the image of x under f and is denoted by f(x). The range of the function consists of all images of elements of X.
Earlier, we introduced the notation f(x) for the element of Y which corresponds to x.Example 2 Let X denote the set of nonnegative real numbers and let f be the function from X to R defined by for every x in X. Thus: Many formulas which occur in mathematics and the sciences determine functions.As an illustration, the formula A = pi*r, and we may write A= f(r).For each book x in X there corresponds a positive integer y, namely the number of pages in the book.In the second example, if we let X denote the set of all human beings and Y the set of all possible dates, then to each person x in X there corresponds a birth date y. Our examples indicate that to each x in X there corresponds one and only one y in Y; that is, y is unique for a given x.The examples of correspondences we have given involve two sets X and Y.In our first example, X denotes the set of books in a library and Y the set of positive integers.The curved arrows indicate that the elements f(x), f(w), f(z), and f(a) of Y correspond to the elements x, y, z and a of X.Let us repeat the important fact that to each x in X there is assigned precisely one image f(x) in Y; however, different elements of X such as w and z in Figure 1.18 may have the same image in Y. However, f(x) is an element of Y, namely the element which f assigns to x.We sometimes represent correspondences by diagrams of the type shown in Figure 1.17, where the sets X and Y are represented by points within regions in a plane. However, the same element of Y may correspond to different elements of X.The curved arrow indicates that the element y of Y corresponds to the element x of X. For example, two different books may have the same number of pages, two different people may have the same birthday, and so on.