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. 

If a rock falls from a height of 100 meters above Earth, the height H (in meters) after t seconds is approximately H(t) = 100 − 4.9t²

(a) In general, quadratic functions are not one-to-one. However, the function H is one-to-one. Why?

(b) Find the inverse of H and verify your result.

(c) How long will it take a rock to fall 80 meters?