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.