To find the average rate of change of the function \( f(x) = 3x + 2 \) over the interval \( [-2, -1] \), we can use the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
Here, \( a = -2 \) and \( b = -1 \). We need to find \( f(-2) \) and \( f(-1) \).
From the given function, we have:
\[ f(-2) = 3(-2) + 2 = -6 + 2 = -4 \] \[ f(-1) = 3(-1) + 2 = -3 + 2 = -1 \]
Now, we can substitute these values into the average rate of change formula:
\[ \text{Average Rate of Change} = \frac{f(-1) - f(-2)}{-1 - (-2)} = \frac{-1 - (-4)}{-1 + 2} \]
Calculating the numerator and denominator:
\[ = \frac{-1 + 4}{1} = \frac{3}{1} = 3 \]
Thus, the average rate of change over the interval \([-2, -1]\) is:
\[ \boxed{3} \]