Answer:
Part A: To determine a reasonable domain to plot the growth function, we need to consider the context of the problem. The equation given is f(m) = 4.61 - 3(1.09)^m, where f(m) represents the length of the fish in cm after m months.
Since the length of the fish cannot be negative, we can conclude that the domain of the function is all non-negative values of m. In other words, m must be greater than or equal to zero.
Therefore, a reasonable domain to plot the growth function would be m ≥ 0.
Part B: The y-intercept of the graph of the function f(m) represents the value of the function when m equals zero. In this case, the equation is f(m) = 4.61 - 3(1.09)^m.
Substituting m = 0 into the equation, we get f(0) = 4.61 - 3(1.09)^0. Since any number raised to the power of zero is 1, we can simplify the equation to f(0) = 4.61 - 3(1).
Therefore, the y-intercept of the graph of the function f(m) is f(0) = 4.61 - 3 = 1.61 cm. This represents the initial length of the fish when the study began, or the length of the fish at the start of the observation period.
Part C: The average rate of change of the function f(m) from m = 2 to m = 5 represents the average rate at which the length of the fish changes over that time interval. To calculate the average rate of change, we need to find the difference in the function values divided by the difference in the input values.
First, we find f(2) and f(5) by substituting the values of m into the equation f(m) = 4.61 - 3(1.09)^m:
f(2) = 4.61 - 3(1.09)^2
f(5) = 4.61 - 3(1.09)^5
Next, we calculate the difference in the function values: f(5) - f(2).
Finally, we divide the difference in the function values by the difference in the input values: (f(5) - f(2))/(5 - 2).
The result of this calculation will give us the average rate of change of the function f(m) from m = 2 to m = 5, and it represents the average rate at which the length of the fish changed over that time interval.
Explanation:
<3