By George F. Simmons

This article is a spin-off of Appendices A ("A number of extra Topics") and B ("Biographical Notes") of Simmons' profitable CALCULUS WITH ANALYTIC GEOMETRY. The textual content is acceptable as a complement for a calculus direction and/or background of arithmetic direction. The textual content can also be applicable for a liberal arts arithmetic direction for college kids with minimum arithmetic historical past.

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**Example text**

Note that A Â B is not deﬁned as a set of ordered pairs if either A or B is empty. However, A Â {} ¼ {(1, ), (2, ), (3, )}. But this set of ordered pairs is of no use to us. 1 Domain of a Relation In any relation (in the form of a set of ordered pairs), the set consisting of the “ﬁrst” element of each pair constitutes the domain of the relation. Let us consider the above relations. . The domain of R1 ¼ {1, 2, 3, 4} ¼ A. But there are two ordered pairs (1, 2) and (1, 5) in which the ﬁrst element is same.

To study functions in details, it is useful to ﬁx certain terms, which will be needed frequently. (3) Note that there are two reasons due to which R2 is not a function. In fact, any one reason is sufﬁcient for this conclusion. 5 DOMAIN, CODOMAIN, IMAGE, AND RANGE OF A FUNCTION Let f be a function from set A to set B ( f: A ! B), then . . The (entire) set A is called the domain of f. The (entire) set B is called the codomain of f. An element of B that corresponds to some element x of A is denoted by f(x), and it is called the image of x under f.

2 > a a > 1 1 > > 2 ¼ a þ 2À2> aÀ > ; a a . . . ða þ b þ cÞ2 ¼ a2 þ b2 þ c2 þ 2ðab þ bc þ caÞ a3 þ b3 þ c3 À 3abc ¼ ða þ b þ cÞða2 þ b2 þ c2 À ab À bc À caÞ 3 3 3 If a þ b þ c ¼ 0; then a þ b þ c ¼ 3abc: 1 1 1 1 ¼ À aÁb bÀa a b If n is a natural number, then the expansion ðx þ yÞn ¼ n C 0 xn þ n C1 xnÀ1 Á y þ n C 2 xnÀ2 Á y2 þ . . þ n Cn yn is called the binomial expansion, where x and y can be any real numbers. – This expansion has (n þ 1) terms. – The general term is of the form n Cr xnÀr yr and it is the (r þ 1)th term in the expansion.