To solve this problem, let's break it down into a few steps.
1. Understanding the function:
Given the function [tex]\( f(x) = \sqrt{x + 3} - 2 \)[/tex].
2. Calculate the values of the function at the given points:
- For [tex]\( x = 6 \)[/tex]:
[tex]\[
f(6) = \sqrt{6 + 3} - 2 = \sqrt{9} - 2 = 3 - 2 = 1
\][/tex]
- For [tex]\( x = 13 \)[/tex]:
[tex]\[
f(13) = \sqrt{13 + 3} - 2 = \sqrt{16} - 2 = 4 - 2 = 2
\][/tex]
3. Plug in the function values into the average rate of change formula:
The average rate of change of a function [tex]\( f(x) \)[/tex] between [tex]\( x = a \)[/tex] and [tex]\( x = b \)[/tex] is given by:
[tex]\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\][/tex]
Here, [tex]\( a = 6 \)[/tex] and [tex]\( b = 13 \)[/tex].
Substitute [tex]\( f(6) = 1 \)[/tex] and [tex]\( f(13) = 2 \)[/tex] into the formula:
[tex]\[
\text{Average rate of change} = \frac{f(13) - f(6)}{13 - 6} = \frac{2 - 1}{13 - 6} = \frac{1}{7}
\][/tex]
4. Match the result with the given options:
The average rate of change is [tex]\(\frac{1}{7}\)[/tex], which matches choice:
[tex]\[
\frac{1}{7}
\][/tex]
Thus, the answer is:
[tex]\[
\boxed{\frac{1}{7}}
\][/tex]
