https://file.aiquickdraw.com/mathbot/file/1726709636269jepyuhzy.jpg

From the image you uploaded, the graph appears to represent a piecewise function \( f(x) \) where the value of the function changes depending on the value of \( x \). The goal is to fill in the conditions for each value of \( f(x) \).

### Analyzing the Graph
- **For \( f(x) = 5 \):**
  - There is a horizontal line at \( y = 5 \) from \( x = -6 \) to \( x = -2 \). This means \( f(x) = 5 \) when \( -6 \leq x \leq -2 \).

- **For \( f(x) = -1 \):**
  - There is another horizontal line at \( y = -1 \) between \( x = 0 \) and \( x = 2 \), but excluding \( x = 0 \) (open circle at \( x = 0 \)) and including \( x = 2 \) (closed circle at \( x = 2 \)). This means \( f(x) = -1 \) when \( 0 < x \leq 2 \).

- **For \( f(x) = 3 \):**
  - There is a single point at \( (4, 3) \), which means \( f(x) = 3 \) when \( x = 4 \).

### Solution:
The piecewise function \( f(x) \) can be written as:
\[
f(x) = \begin{cases} 
5 & \text{if } -6 \leq x \leq -2 \\
-1 & \text{if } 0 < x \leq 2 \\
3 & \text{if } x = 4 
\end{cases}
\]

Let me know if you'd like more details or explanations!

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Here are 5 related questions for practice:
1. What would \( f(x) \) be if \( x = -3 \)?
2. Is \( f(x) \) defined at \( x = 0 \)? Why or why not?
3. Can you graph the intervals where \( f(x) = 0 \)?
4. How would the graph change if the point at \( x = 4 \) were at \( (4, 5) \)?
5. What happens if the condition \( f(x) = 3 \) is extended to an interval, say \( 3 \leq x \leq 5 \)?

**Tip:** Always check for open and closed circles on graphs to see whether a value is included or excluded at a specific point.

Given a graph representing a piecewise function f(x), identify the conditions for which f(x) takes the values 5, -1, and 3.

Learn how to interpret and define a piecewise function based on a graph. This problem covers key concepts like function intervals and interpreting inequalities in piecewise functions. Suitable for high school students (Grades 9-11).

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