Step-by-step explanation:
A)
the domain of a function is the interval or set of all valid values for the input variable (the parameter of the function) - in our case here "n".
as the description says, "n" is the number of days observing the growth of the plant.
so, what are useful values for the number of observation days ?
negative numbers ? no, for sure not.
0 ? yes, as the starting size of the plant, when the study began.
and greater than 0 ? yes, up to the n, where the functional value reaches 11.04.
beyond that ? no, because it does not make any sense, as nobody was checking, if the curve still fits. and a plant for sure will not grow to infinity.
so, the interval is [0, x].
and x is the solution for n of
11.04 = 10(1.02)^n
1.104 = (1.02)^n
[tex] log_{1.02}(1.104) = n[/tex]
and to calculate that with our calculators we know that
[tex] log_{b}(a) = \frac{ log_{c}(a) }{ log_{c}(b) } [/tex]
so, let's simply use log to the base of c=10 :
[tex] \frac{ log(1.104) }{ log(1.02) } = n[/tex]
n = 0.042969073... / 0.008600172... = 5
the study ended after day 5.
therefore, the domain interval is [0, 5]
B)
I answered this already under A.
the y-intercept is the point of n=0, and it is the starting size of the plant, when the study began (day 0).
C)
the average rate of change of a function in an interval [begin, end] is simply
(f(end) - f(begin)) / (end - begin)
in our case
begin = 1
end = 5
(f(5) - f(1))/(5-1) = (10(1.02)⁵ - 10(1.02)¹)/4 = (11.04 - 10.2)/4 =
= 0.84/4 = 0.21
that means the plant grew as average 0.21 cm per day during that observation period between day 1 and day 5.
