https://file.aiquickdraw.com/mathbot/file/1727050424083rgdgve6y.jpg

The image presents a piecewise function \( f(x) \) defined as:

\[
f(x) = 
\begin{cases} 
x - 8 & \text{for } -1 < x < 2 \\
\sqrt{x - 3} + 4 & \text{for } 3 \leq x < 7
\end{cases}
\]

To graph this:

### 1. **Graph for \( -1 < x < 2 \) ( \( f(x) = x - 8 \) ):**
   - This is a linear equation with a slope of 1 and a y-intercept at \( -8 \).
   - For \( x = -1 \), \( f(x) = -1 - 8 = -9 \) (open point).
   - For \( x = 2 \), \( f(x) = 2 - 8 = -6 \) (open point).
   - You need to plot a line from \( (-1, -9) \) to \( (2, -6) \), but remember, both endpoints should be open because of the strict inequality.

### 2. **Graph for \( 3 \leq x < 7 \) ( \( f(x) = \sqrt{x - 3} + 4 \) ):**
   - This is a square root function shifted right by 3 units and up by 4 units.
   - At \( x = 3 \), \( f(x) = \sqrt{3 - 3} + 4 = 4 \) (closed point).
   - At \( x = 7 \), \( f(x) = \sqrt{7 - 3} + 4 = \sqrt{4} + 4 = 6 \) (open point).
   - The curve should follow the shape of the square root function between \( x = 3 \) and \( x = 7 \).

Would you like help with a specific part of the graph or need more details? 

### 5 Related Questions:
1. How do you graph linear and square root functions individually?
2. What are open and closed points, and how do they affect a graph?
3. How does shifting affect the graph of a function?
4. Can you explain how to calculate the domain and range of piecewise functions?
5. How does the slope and y-intercept affect the appearance of a linear function?

### Tip:
Always plot key points first and note whether they are open or closed when graphing piecewise functions.

Graph the piecewise function f(x) = { x - 8 for -1 < x < 2 ; sqrt(x - 3) + 4 for 3 ≤ x < 7 }.

Learn how to graph the piecewise function f(x) = { x - 8 for -1 < x < 2 ; sqrt(x - 3) + 4 for 3 ≤ x < 7 }. This example covers graphing linear and square root functions, highlighting key points and open/closed intervals for students in grades 9-12.

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