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.
Income Taxes The function
T(g) = 4386.25 + 0.25(g - 31,850)
represents the 2007 federal income tax T (in dollars) due for a “single” filer whose modified adjusted gross income is g dollars, where 31,850 ≤ g ≤ 77,100.
(a) What is the domain of the function T?
(b) Given that the tax due T is an increasing linear function of modified adjusted gross income g, find the range of the function T.
(c) Find adjusted gross income g as a function of federal income tax T. What are the domain and the range of this function?
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