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The given function \( f(x) \) is defined piecewise with three intervals. Let's analyze and sketch the graph of \( f(x) \) for each part:

1. \( f(x) = -x - 4 \) for \( x \leq -2 \)
   - This is a linear function with a negative slope, so it decreases as \( x \) increases.
   - We can find key points for the graph:
     - At \( x = -2 \), \( f(x) = -(-2) - 4 = -2 \).
     - As \( x \to -\infty \), the function continues to decrease.
     
2. \( f(x) = -5 \) for \( -2 < x \leq 4 \)
   - This is a constant function, meaning the graph is a horizontal line at \( y = -5 \) between \( x = -2 \) and \( x = 4 \).
   - At \( x = -2 \), \( f(x) = -5 \) (the point connects to the first piece), and at \( x = 4 \), \( f(x) = -5 \).

3. \( f(x) = 2x - 4 \) for \( x > 4 \)
   - This is another linear function, with a positive slope (so it increases as \( x \) increases).
   - Key points:
     - At \( x = 4 \), \( f(x) = 2(4) - 4 = 4 \).
     - As \( x \to \infty \), the graph continues to rise.

### Steps to sketch the graph:
- For \( x \leq -2 \), plot the line \( f(x) = -x - 4 \) passing through \( (-2, -2) \) and extending leftward.
- Between \( -2 < x \leq 4 \), plot the horizontal line \( f(x) = -5 \).
- For \( x > 4 \), plot the line \( f(x) = 2x - 4 \) starting from \( (4, 4) \) and extending rightward.

Would you like me to create a detailed plot of the graph or explain any specific part further?

Here are 5 related questions for practice:
1. What is the domain and range of the function \( f(x) \)?
2. How does the slope of each linear piece affect the shape of the graph?
3. Can you find the x-intercepts for the linear segments of the function?
4. What type of discontinuity (if any) does the function have at \( x = -2 \) and \( x = 4 \)?
5. How would the graph change if the interval for \( f(x) = -5 \) were \( -2 < x < 3 \) instead?

**Tip**: Always check the behavior at the boundaries of each piecewise section to determine if the function is continuous or has jumps.

Sketch a graph of f(x) = {-x - 4 if x <= -2, -5 if -2 < x <= 4, 2x - 4 if x > 4}

Sketch the graph of a piecewise function consisting of three sections: a decreasing linear function for x <= -2, a constant function for -2 < x <= 4, and an increasing linear function for x > 4. This problem helps students practice graphing linear equations and understanding piecewise functions, suitable for Grades 10-12.

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