a) A reasonable domain to plot the growth function would be m ∈ [0, 7]. This means that the number of months (m) is between 0 and 7, inclusive.
b) The y-intercept is 4, which means the fish was 4 cm long at the start of the study.
c) The average rate of change of the function f(m) from m = 3 to m = 7 is 4.12 cm per month. This means that on average, the length of the fish decreased by approximately 4.12 cm per month over this period.
How to solve the exponential?
Part A: The domain of a function is the set of all possible input values (in this case, the number of months).
Since the length of the fish was approximately 9.19 cm at the end of the study, we need to find the number of months it took to reach this length.
We can do this by solving the equation 4(1.08)^{m} = 6.86.
Solving this equation gives:
(1.08)^{m} = 6.86/4
m In 1.08 = In 1.715
m = 7
Therefore, a reasonable domain to plot the growth function would be m ∈ [0, 7]. This means that the number of months (m) is between 0 and 7, inclusive.
Part B: The y-intercept of the graph of the function f(m) represents the length of the fish at the start of the study (when m = 0).
In this case, the y-intercept is 4, which means the fish was 4 cm long at the start of the study.
Part C: The average rate of change of the function f(m) from m = 3 to m = 7 is given by the formula [f(7) - f(3)] / (7 - 3).
Substituting the given function into this formula gives:
[4(1.08)^{7} - 4(1.08)^{3}] / (7 - 3)] ≈ -4.12
This means that, on average, the length of the fish decreased by approximately 4.12 cm per month over this period.