https://file.aiquickdraw.com/mathbot/file/1725060825815oitx7yxc.jpg

Let's evaluate the piecewise function \( f(x) \) for the given values of \( x \).

The function is defined as:

\[
f(x) = \begin{cases} 
3 + x & \text{if } -3 \leq x < 0 \\
3 & \text{if } x = 0 \\
\sqrt{x} & \text{if } x > 0 
\end{cases}
\]

Now, let's find \( f(10) \), \( f(-1) \), \( f(16) \), \( f(-3) \), and \( f(0) \):

1. **\( f(10) \):**

   Since \( 10 > 0 \), we use \( f(x) = \sqrt{x} \).
   
   \[
   f(10) = \sqrt{10} \approx 3.16
   \]

2. **\( f(-1) \):**

   Since \( -1 \) falls in the interval \( -3 \leq x < 0 \), we use \( f(x) = 3 + x \).
   
   \[
   f(-1) = 3 + (-1) = 2
   \]

3. **\( f(16) \):**

   Since \( 16 > 0 \), we use \( f(x) = \sqrt{x} \).
   
   \[
   f(16) = \sqrt{16} = 4
   \]

4. **\( f(-3) \):**

   Since \( -3 \) is exactly at the lower boundary of the interval \( -3 \leq x < 0 \), we use \( f(x) = 3 + x \).
   
   \[
   f(-3) = 3 + (-3) = 0
   \]

5. **\( f(0) \):**

   Since \( x = 0 \), we use the corresponding rule, \( f(x) = 3 \).
   
   \[
   f(0) = 3
   \]

### Final Answers:
- \( f(10) = \sqrt{10} \approx 3.16 \)
- \( f(-1) = 2 \)
- \( f(16) = 4 \)
- \( f(-3) = 0 \)
- \( f(0) = 3 \)

Would you like more details or have any other questions?

Here are some related questions:

1. What is the general form of a piecewise function?
2. How do you determine which part of a piecewise function to use?
3. Can a piecewise function be continuous?
4. What are the implications of the square root function for \( x < 0 \)?
5. How do you graph a piecewise function?

**Tip:** When evaluating piecewise functions, always carefully check which interval your input value belongs to before applying the corresponding function rule.

Explore how to evaluate the piecewise function f(x) = { 3 + x if -3 ≤ x < 0, 3 if x = 0, √x if x > 0 } for specific values such as f(10), f(-1), f(16), f(-3), and f(0). Understand the rules governing piecewise functions and their application in mathematical contexts. Suitable for Grades 9-12.

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