Final answer:
The average rate of change for the function f(x) = 1/2(3^x) - 4 over the interval (-1, 4] is calculated to be approximately 8.067, but this does not match any of the provided options, indicating a possible error in the question or the options.
Step-by-step explanation:
The question asks for the rate of change of the function f(x) = 1/2(3^x) - 4 over the interval (-1, 4]. To calculate this, we need to find the average rate of change which is the change in function values over the change in x-values.
We start by evaluating the function at the given x-values:
- f(-1) = 1/2(3^-1) - 4 = 1/6 - 4 = -23/6
- f(4) = 1/2(3^4) - 4 = 1/2(81) - 4 = 40.5 - 4 = 36.5
Then, we use these values to calculate the average rate of change:
Average rate of change = (f(4) - f(-1)) / (4 - (-1))
= (36.5 - (-23/6)) / 5
= (219/6 + 23/6) / 5
= (242/6) / 5
= 40.333... / 5
= 8.0666...
However, none of the given options (A) 36, (B) -26, (C) 12, (D) 16 match our calculated value. It seems there may be an error in the provided options or a misunderstanding in the calculation of the rate of change.