Let f(x) = cos(x). Recall that we define the inverse of cosine by restricting the domain of cosine to the interval [0,].

(a) The cosine function is periodic with period 2, so why don't we restrict the domain of f to the interval [0,2]? (Explain by means of an example.)

(b) When we restrict the sine function in order to define its inverse, we use the interval [/2 , /2 ]. Explain (through the use of an example) why we cannot use the same interval for defining the inverse of cosine.

(c) Explain why we don't use the interval [0, /2 ] for the restricted domain of cosine when defining the inverse of cosine.

(d) Find an interval which does not contain 0 that we can use as a restricted domain for cosine in order to define its inverse. If g(x) = cos^1(x) is the inverse of cosine over this new restricted domain, then what is the value of g(1)?

2. Suppose that the number P(t) of bacteria in a culture after t days is given by the formula P(t) = 1500 2^t/6.

(a) What is the initial population of the bacteria?

(b) How many days does it take for the population to triple in size? Give the exact answer. (That means do not write it as a decimal approximation.) Then give the answer in decimal form, using three decimal places.

(c) Find constants k and r for which P(t) = ke^rt.