https://file.aiquickdraw.com/mathbot/file/17268157815871j5w72i2.jpg

To graph the piecewise function \( f(x) \) from the image, let's break down the conditions:

The function is given as:
\[
f(x) = \begin{cases} 
-5 & \text{if } x \leq -2 \\
2x - 1 & \text{if } -2 < x \leq 1 \\
0 & \text{if } x > 1 
\end{cases}
\]

### Steps to graph:

1. **For \( x \leq -2 \):**
   The function is constant at \( f(x) = -5 \). This is a horizontal line from \( x = -2 \) to the left, so plot the point \( (-2, -5) \) and draw a horizontal line extending to the left (including a closed circle at \( x = -2 \)).

2. **For \( -2 < x \leq 1 \):**
   The function follows the equation \( f(x) = 2x - 1 \). 
   - When \( x = -2 \), \( f(-2) = 2(-2) - 1 = -5 \) (though this point is already included in the first case, so it will be an open circle here).
   - When \( x = 1 \), \( f(1) = 2(1) - 1 = 1 \).
   
   Plot the points from \( x = -2 \) to \( x = 1 \), and since the function is linear, it will be a straight line between these points. Use an open circle at \( x = -2 \) and a closed circle at \( x = 1 \).

3. **For \( x > 1 \):**
   The function is constant at \( f(x) = 0 \), which is a horizontal line along the x-axis. This line starts at \( x = 1 \) and extends to the right. Use an open circle at \( x = 1 \) and extend the horizontal line for \( x > 1 \).

Would you like me to generate a visual graph based on this explanation? Let me know if you need further details or have any questions! 

### Relative Questions:
1. What is a piecewise function, and how do you graph different sections?
2. Why do some points have open and closed circles when graphing piecewise functions?
3. How do you calculate the slope of the linear section \( 2x - 1 \) for \( -2 < x \leq 1 \)?
4. What role do inequalities play in defining the behavior of piecewise functions?
5. How can we check continuity in this piecewise function?

### Tip:
Always check for open and closed endpoints in piecewise functions to ensure that you correctly represent the function at boundary points.

Sketch a graph of the piecewise function: f(x) = { -5 if x ≤ -2, 2x - 1 if -2 < x ≤ 1, 0 if x > 1 }.

Learn how to graph the piecewise function f(x) = { -5 if x ≤ -2, 2x - 1 if -2 < x ≤ 1, 0 if x > 1 } using basic graphing techniques. Explore the behavior of this function over different intervals with a step-by-step solution. Suitable for Grades 9-12.

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