Precalc Question

Question

A biochemical experiment involves combining together two protein extracts. Suppose a function φ(t) monitors the amount (nanograms) of extract A remaining at time t (nanoseconds). Assume you know these facts:

1. The function φ is invertible; i.e., it has an inverse function.

2. φ(0) = 6, φ(1) = 5, φ(2) = 3, φ(3) = 1, φ(4) = 0.5, φ(10) = 0.

(a) What is φ −1 (0.5) and what does it tell you?

(b) (True or False) There is exactly one time when the amount of extract A remaining is 4 nanograms.

(c) Calculate φ(φ−1 (1))

(d) Calculate φ−1 (φ(6))

(e) What is the domain and range of φ?

Answer 1

a) φ^-1(0.5) = 4 and that means when the amount remaining is .5, the elapsed time is 4

b) Since you only know the value of the function at specified times, you have to make assumptions about the values at intermediate times. For example, you can assume the function is continuous, but you do NOT know that.

c) 1

d) See answer to b)

e) As stated the domain is the set {0, 1, 2, 3, 4, 10} and the range is the set {6, 5, 3, 1, .5,0}; see answer to b).