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Write the piecewise function for the graph 

The graph provided appears to depict a piecewise function \( g(x) \), composed of three distinct linear segments. Let's analyze each section of the graph:

1. **First segment (leftmost):**
   - This is a horizontal line on the interval \( -9 \leq x \leq -4 \) with an open circle at \( x = -9 \) and a closed circle at \( x = -4 \).
   - The function has a constant value of \( y = -9 \) over this interval.
   - Equation: \( g(x) = -9 \) for \( -9 < x \leq -4 \).

2. **Second segment (middle):**
   - This segment is an increasing line from \( (1, 4) \) to \( (3, 8) \), with closed circles at both endpoints.
   - The slope of this line can be calculated as \( m = \frac{8 - 4}{3 - 1} = 2 \), and the equation of the line in point-slope form is:
     \[
     g(x) = 2(x - 1) + 4 = 2x + 2
     \]
   - Equation: \( g(x) = 2x + 2 \) for \( 1 \leq x \leq 3 \).

3. **Third segment (rightmost):**
   - This is another increasing line from \( (6, 2) \) to \( (8, 6) \), with an open circle at \( x = 6 \) and a closed circle at \( x = 8 \).
   - The slope of this line is \( m = \frac{6 - 2}{8 - 6} = 2 \), and the equation of the line is:
     \[
     g(x) = 2(x - 6) + 2 = 2x - 10
     \]
   - Equation: \( g(x) = 2x - 10 \) for \( 6 < x \leq 8 \).

### The Piecewise Function:

\[
g(x) =
\begin{cases}
-9 & \text{for } -9 < x \leq -4 \\
2x + 2 & \text{for } 1 \leq x \leq 3 \\
2x - 10 & \text{for } 6 < x \leq 8
\end{cases}
\]

Would you like a deeper explanation of any part, or do you have any specific questions?

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Here are 5 related questions:
1. How do you calculate the slope of a line between two points?
2. What is the difference between an open and a closed circle on a graph?
3. How can you derive the equation of a line given two points?
4. What is a piecewise function and how is it used in mathematics?
5. What are the conditions for a function to be continuous?

**Tip:** When writing piecewise functions, always check the behavior at the endpoints (open or closed circles) to define the correct inequalities for each segment.

Write the piecewise function for the graph.

This problem involves interpreting a graph with three distinct linear segments to write a piecewise function. Students will calculate slopes, identify line equations, and learn how to express piecewise functions. Suitable for Grades 9-11.

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