https://file.aiquickdraw.com/mathbot/file/17282215497156z5jwd9b.jpg

The piecewise function given is:

\[
f(x) = \begin{cases} 
x + 1, & \text{if } x < -5 \\
(x - 1)^2 + 1, & \text{if } -5 \leq x < 5 \\
x + 1, & \text{if } x \geq 5 
\end{cases}
\]

### Let's break this down:

1. **For \( x < -5 \)**:
   - The function is linear: \( f(x) = x + 1 \).
   - This represents a line with slope 1 and y-intercept at \( y = 1 \).
   
2. **For \( -5 \leq x < 5 \)**:
   - The function is quadratic: \( f(x) = (x - 1)^2 + 1 \).
   - This represents a parabola shifted 1 unit to the right and 1 unit up. The vertex of the parabola is at \( (1, 1) \).

3. **For \( x \geq 5 \)**:
   - The function is linear again: \( f(x) = x + 1 \), just like in the first interval.
   - This is the same linear line as for \( x < -5 \).

### Key Points to Graph:
- **First region** \( x < -5 \): Graph the line \( f(x) = x + 1 \).
- **Middle region** \( -5 \leq x < 5 \): Graph the parabola \( f(x) = (x - 1)^2 + 1 \).
- **Last region** \( x \geq 5 \): Graph the same line \( f(x) = x + 1 \).

### Discontinuities:
- There could be potential discontinuities at \( x = -5 \) and \( x = 5 \). Let’s check the values:
  - For \( x = -5 \): 
    - From the first part, \( f(x) = (-5 + 1) = -4 \).
    - From the quadratic part, \( f(-5) = ((-5 - 1)^2) + 1 = 36 + 1 = 37 \).
    - So, there is a **jump discontinuity** at \( x = -5 \).
  
  - For \( x = 5 \):
    - From the quadratic part, \( f(5) = (5 - 1)^2 + 1 = 16 + 1 = 17 \).
    - From the third part, \( f(x) = 5 + 1 = 6 \).
    - There is also a **jump discontinuity** at \( x = 5 \).

Would you like me to provide a detailed plot of this function, or explain any part further? 

### Questions for further exploration:
1. How do we determine whether a function has a discontinuity at a given point?
2. What does the vertex of a parabola represent, and how is it affected by shifts?
3. How would you find the slope of the lines for \( x < -5 \) and \( x \geq 5 \)?
4. How does the quadratic part of the function differ from the linear parts?
5. Can a function have multiple types of discontinuities?

### Tip:
When graphing piecewise functions, always check for discontinuities by calculating the left-hand and right-hand limits at the transition points.

Formulate the graph of f(x) = { (x+1, if x<-5), ((x-1)^2 + 1, if -5 <= x < 5), (x+1, if x >= 5) }.

Learn how to graph a piecewise function that combines linear and quadratic components. Understand discontinuities at transition points and how to break down different sections of the graph. Suitable for high school students in Grades 10-12.

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