To calculate the average rate of change of the function \( f(x) = \sqrt{x + 2} \) over the interval \([2, 7]\), we use the formula for the average rate of change:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \( a = 2 \) and \( b = 7 \).
- First, evaluate \( f(a) \) and \( f(b) \):
\[ f(2) = \sqrt{2 + 2} = \sqrt{4} = 2 \]
\[ f(7) = \sqrt{7 + 2} = \sqrt{9} = 3 \]
- Now compute the average rate of change:
\[ \text{Average Rate of Change} = \frac{f(7) - f(2)}{7 - 2} = \frac{3 - 2}{7 - 2} = \frac{1}{5} \]
Thus, the average rate of change over the interval \([2, 7]\) is
\[ \boxed{\frac{1}{5}} \]