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. 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 number q of units manufactured, and q = q(C) is a function that represents the number q as a function of the cost C. Problem illustrate this idea.
Vehicle Stopping Distance Taking into account reaction time, the distance d (in feet) that a car requires to come to a complete stop while traveling r miles per hour is given by the function D(r) = 6.97r − 90.39
(a) Express the speed r at which the car is traveling as a function of the distance d required to come to a complete stop.
(b) Verify that r = r(d) is the inverse of d = d(r) by showing that r(d(r))= r and d(r(d))= d.
(c) Predict the speed that a car was traveling if the distance required to stop was 300 feet.
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