To calculate the average rate of change of the function \( f(x) = 3^x + 2 \) over the interval \([-2, -1]\), we use the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \( a = -2 \) and \( b = -1 \).
From the function table provided:
- \( f(-2) = 1 \)
- \( f(-1) = 3 \)
Now, substituting these values into the formula:
\[ \text{Average Rate of Change} = \frac{f(-1) - f(-2)}{-1 - (-2)} \]
Calculating the numerator:
\[ f(-1) - f(-2) = 3 - 1 = 2 \]
Calculating the denominator:
\[ -1 - (-2) = -1 + 2 = 1 \]
Putting it all together:
\[ \text{Average Rate of Change} = \frac{2}{1} = 2 \]
Thus, the average rate of change over the interval \([-2, -1]\) is:
\[ \boxed{2} \]