Chapter 5.2, Problem 94AE

In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using $y=f\left(x\right)$ to represent a function, an applied problem might use $C=C\left(q\right)$ 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 $C=C\left(q\right)$ will be $q=q\left(C\right)$. So $C=C\left(q\right)$ is a function that represents the cost $C$ as a function of the number $q$ of units manufactured, and $q=q\left(C\right)$ is a function that represents the number $q$ as a function of the cost $C$. Problems 91-94 illustrate this idea.

Temperature Conversion The function $F\left(C\right)=\frac{9}{5}C+32$ 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 $F$.

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

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