https://file.aiquickdraw.com/mathbot/file/17259066866514eynj5jf.jpg

The question asks for the range of the function \( f(x) \), which is defined piecewise, as seen in the graph. Let's analyze the function and the corresponding graph step by step.

The function \( f(x) \) is given as a piecewise function:

\[
f(x) = 
\begin{cases} 
4 & \text{if } -6 \leq x \leq -2 \\
|x| & \text{if } -2 \leq x < 8 \\
\sqrt{x} & \text{if } 8 \leq x \leq 13
\end{cases}
\]

### Step-by-step analysis:
1. **For the interval** \( -6 \leq x \leq -2 \), \( f(x) = 4 \).
   - This is a constant function, so the value of \( f(x) \) is 4 over this entire interval.

2. **For the interval** \( -2 \leq x < 8 \), \( f(x) = |x| \).
   - Here, \( f(x) \) takes values from \( |x| = |-2| = 2 \) to \( |x| = 8 \) (as the absolute value increases from 2 to 8 over the range).

3. **For the interval** \( 8 \leq x \leq 13 \), \( f(x) = \sqrt{x} \).
   - In this case, \( f(x) \) takes values from \( \sqrt{8} \approx 2.83 \) to \( \sqrt{13} \approx 3.61 \).

### Finding the range of \( f(x) \):
From the graph and the piecewise function, we can now determine the range:

- The function takes the constant value 4 in the interval \( -6 \leq x \leq -2 \).
- Then, from \( -2 \leq x < 8 \), the function takes values from \( 2 \) to \( 8 \) (due to \( |x| \)).
- Finally, in the interval \( 8 \leq x \leq 13 \), the function's values range from \( \sqrt{8} \approx 2.83 \) to \( \sqrt{13} \approx 3.61 \).

Thus, the complete range of \( f(x) \) is \( [0, 8] \), as the function covers all values from 0 (though it never actually reaches 0 but approaches it) to 8 across the various intervals.

### Correct Answer: \( [0, 8] \)

Let me know if you need further details or clarifications! 

Here are five follow-up questions related to piecewise functions and range concepts:
1. What is the definition of a piecewise function?
2. How do you find the domain of a piecewise function?
3. Why is the square root function only defined for non-negative values of \( x \)?
4. How can absolute value functions affect the range of a function?
5. What are some common strategies for graphing piecewise functions?

**Tip:** When analyzing a piecewise function, always look at each segment individually to determine its effect on the overall range and domain!

Based on the graph, find the range of y = f(x).

Learn how to determine the range of a piecewise function by analyzing its graph. This problem involves understanding absolute value and square root functions, suitable for high school students in Grades 10-12.

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