In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using y = f(x) to represent a function, an applied problem might use C = C(q) to represent the cost C of manufacturing q units of a good since, in economics, q is used for output. Because of this, the inverse notation f–1 used in a pure mathematics problem is not used when finding inverses of applied problems. Rather, the inverse of a function such as C = C(q) will be q = q(C). So C = C(q) is a function that represents the cost C as a function of the output q, while q = q(C) is a function that represents the output q as a function of the cost C. Problems illustrate this idea.
Period of a Pendulum The period T (in seconds) of a simple pendulum as a function of its length f (in feet) is given by
(a) Express the length l as a function of the period T.
(b) How long is a pendulum whose period is 3 seconds?
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