Chapter 5.2, Problem 93AYU

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.

Ideal Body Weight One model for the ideal body weight $W$ for men (in kilograms) as a function of height $h$ (in inches) is given by the function

$W\left(h\right)=50+2.3\left(h-60\right)$

(a) What is the ideal weight of a 6-foot male?

(b) Express the height $h$ as a function of weight $W$.

(c) Verify that $h=h\left(W\right)$ is the inverse of $W=W\left(h\right)$ by showing that $h\left(W\left(h\right)\right)=h$ and $W\left(h\left(W\right)\right)=W$.

(d) What is the height of a male who is at his ideal weight of 80 kilograms?

[Note: The ideal body weight $W$ for women (in kilograms) as a function of height $h$ (in inches) is given by $W\left(h\right)=45.5+2.3\left(h-60\right)$.