18 Jan 2009

Domains and Intervals in Edwards & Penney

presentation of Edwards & Penney's work, by

presentation of Edwards & Penney's work, by Corry Shores
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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

[the following will not stray from Edwards & Penney's procedure, so it is largely quotation.]

Domains and Intervals:

The function f and the value or expression f (x) are different in the same sense that a machine and its output are different. Nevertheless, it is common to use an expression like

to define a function merely by writing its formula. In this situation the domain of the function is not specified. Then, by convention, the domain of the function f is the set of all real numbers x for which the expression f (x) makes sense and produces a real number y. For instance, the domain of the function h (x) = 1/x is the set of all nonzero real numbers (because 1/x is defined precisely when

Domains of functions frequently are described in terms of intervals of real numbers.

Recall that a closed interval [a, b] contains both its endpoints x = a and x = b, whereas the open interval (a, b) contains neither endpoint. Each of the half-open intervals [a, b) and (a, b] contains exactly one of its two endpoints. The unbounded interval

contains its endpoint x = a, whereas

does not. The previously mentioned domain of h (x) = 1/x is the union of the unbounded intervals

Example 6:
Find the domain of the function

Division by zero is not allowed, so the value g (x) is defined precisely when

This is true when

and thus when

Hence the domain of g is the set

which is the union of the two unbounded open intervals

shown below

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, p.4c-5b.

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