In applications, the symbols used for the independent and dependent variables are often ba...

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.

Temperature Conversion The function converts a temperature from C degrees Celsius to F degrees Fahrenheit.

(a) Express the temperature in degrees Celsius C as a function of the temperature in degrees Fahrenheit F.

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(b) Verify that C = C(F) is the inverse of F = F(C) by showing that C(F(C)) = C and F(C(F)) = F.

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(c) What is the temperature in degrees Celsius if it is 70 degrees Fahrenheit?