DIRECTIONS:

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 since, in economics, q is used for output. Because of this, the inverse notation f 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 output q, while q = q{C) is a function that represents the output q as a function of the cost C. Problems 87–90 illustrate this idea.

PROBLEM:

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(C) = 2.15C – 10.53

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

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

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