To graph the given piecewise function, follow these steps:

Function:

1. $f(x) = x + 3$ for $-1 < x \leq 2$
2. $f(x) = -x + 9$ for $2 < x \leq 6$

Steps to Graph:

1. For $f(x) = x + 3$ ($-1 < x \leq 2$):

• Start by substituting the boundary points:
  • At $x = -1$: $f(-1) = -1 + 3 = 2$
  • At $x = 2$: $f(2) = 2 + 3 = 5$
• Plot the segment from $(-1, 2)$ (open circle) to $(2, 5)$ (closed circle).

2. For $f(x) = -x + 9$ ($2 < x \leq 6$):

• Substituting the boundary points:
  • At $x = 2$: $f(2) = -(2) + 9 = 7$ (open circle at $(2, 7)$).
  • At $x = 6$: $f(6) = -(6) + 9 = 3$ (closed circle at $(6, 3)$).
• Plot the segment from $(2, 7)$ (open circle) to $(6, 3)$ (closed circle).

Graph:

1. Draw the first line segment for $x + 3$, from $(-1, 2)$ to $(2, 5)$.
2. Draw the second line segment for $-x + 9$, from $(2, 7)$ to $(6, 3)$.

Would you like a detailed illustration or assistance in sketching?

Here are five relevant follow-up questions:

1. Would you like a table of values for each piece of the function?
2. Should I provide step-by-step calculations for each segment?
3. Do you need help identifying open and closed points on the graph?
4. Are you familiar with how to plot piecewise functions?
5. Would a digital or graphical representation of this be more helpful?

Tip: Always check the domain boundaries carefully to ensure you plot open and closed circles correctly!