Chapter 5.2, Problem 92AE

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.

Height and Head Circumference The head circumference $C$ of a child is related to the height $H$ of the child (both in inches) through the function

$H\left(C\right)=2.15C-10.53$

(a) Express the head circumference $C$ as a function of height $H$.

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

(c) Predict the head circumference of a child who is 26 inches tall.