The first step is to evaluate our function at the points of interest - the endpoints of our interval.
For x = -6 (the lower limit of our interval), we substitute x into our function. This gives us:
f(-6) = (-6)^3 + 8*(-6) + 9 = -216 - 48 + 9 = -255.
Then, for x = 0 (the upper limit of our interval), the function becomes:
f(0) = (0)^3 + 8*(0) + 9 = 0 + 0 + 9 = 9.
Now that we have the function's values at the lower and the upper limits of our interval, we can calculate the average rate of change.
The average rate of change of a function over the interval [a, b] is given by [f(b) - f(a)]/(b - a). Substituting our calculated function values and the interval limits, we get:
The average rate of change of f(x) over the interval -6<=x<=0 is [f(0) - f(-6)] / (0 - -6) = (9 - (-255)) / (0 - -6) = 264 / 6 = 44.
In conclusion, the average rate of change of the function f(z) = z^3+8x+9 over the interval -6<=x<=0 is 44.