# Download e-book for kindle: Abstract convex analysis by Singer I. By Singer I.

ISBN-10: 0471160156

ISBN-13: 9780471160151

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Additional resources for Abstract convex analysis

Example text

8696. In each case we replaced the variable x in f 1x 2 ϭ x2 with the indicated value, x ϭ 3, x ϭ 13 , and so on, and then evaluated the expression 32, A 13 B 2, and so on. Similarly, the function g that takes the square root of any nonnegative real number x can be written as y ϭ g 1x 2 ϭ 2x . 2; g A 14 B ϭ 3 14 ϭ 12 . But g 1Ϫ25 2 ϭ 1Ϫ25 does not make sense because it isn’t possible to take the square root of a negative number in the real number system. That is, the function g is not defined for x ϭ Ϫ25.

17(b). 17 x2 y x3 x4 x5 x6 x7 x8 x9 x x1 (a) x2 x3 x4 x5 x6 x7 x8 x9 x (b) Next, we look for particular points on the curve. e. The turning points for this function are at x ϭ x3 , at x ϭ x5 , and at x ϭ x7 . f. The function has a local maximum at x ϭ x1 (when compared to other nearby points); the function also has a local maximum at x ϭ x5 and again at x ϭ x9 . g. Similarly, the function reaches a local minimum at x ϭ x3 (when compared to other nearby points) and again at x ϭ x7 . h. The points of inflection occur where the concavity changes, which happens at x ϭ x2 , at x ϭ x4 , at x ϭ x6 , and at x ϭ x8 .

Similarly, x2 is carried into y2 and x3 is mapped into y3 . Note that x4 and x5 are both transformed into y4 , which is perfectly legitimate for a function. Each x-value must be mapped into a single y-value, although it is certainly possible for several different x’s to be mapped into the same y. Think about the function y ϭ f 1x 2 ϭ x2, where both x ϭ 2 and x ϭ Ϫ2 are transformed into y ϭ 4. 21 Domain Range We now summarize the preceding ideas in a formal definition of a function. Definition of a Function A function f is a rule that assigns to each permissible value of the independent variable x one and only one value of the dependent variable y.