Stitz-Zeager_College_Algebra_e-book

Since the outputs from f are the inputs to f 1 and

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Unformatted text preview: d refer the interested reader to a solid course in Calculus for a more rigorous explanation. √ The number 3 = 1.73205 . . . is an irrational number3 and as√ such, its decimal representation neither repeats nor terminates. We can, however, approximate 3 by terminating decimals, and √ √ it stands to reason4 we can use these to approximate√ 3 . For example, if we approximate 3 2 √ 173 100 173 by 1.73, we can approximate 2 3 ≈ 21.73 = 2 100 = 2 . It is not, by any means, a pleasant number, but it is at least a number that we understand in terms of powers and roots. It also stands √ to reason that better √ and better approximations of 3 yield better and better approximations of √ 2 3 , so the value of 2 3 should be the result of this sequence of approximations.5 Suppose we wish to study the family of functions f (x) = bx . Which bases b make sense to study? We find that we run into difficulty if b < 0. For example, if b = −2, then the function f (x) = (−2)x √ 1 has trouble, for instance, at x = 2 since (−2)1/2 =...
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