https://file.aiquickdraw.com/mathbot/file/1726722340015bm8p33cx.jpg

To solve the problem, I’ll provide instructions on how to graph the two piecewise functions shown in your image. Let's take them one by one:

### 1) Function 1: 
\[ 
f(x) = 
\begin{cases} 
x^2 + 2 & \text{if } x < 1 \\
3x - 5 & \text{if } 1 \leq x \leq 3 
\end{cases}
\]

#### Graphing Instructions:
- **For the first piece**: \( f(x) = x^2 + 2 \) when \( x < 1 \).
  - This is a parabola shifted upwards by 2. 
  - For values \( x < 1 \), plot the points according to \( f(x) = x^2 + 2 \). You could plot points such as \( (-1, 3), (0, 2) \), and stop the curve just before \( x = 1 \).
  
- **For the second piece**: \( f(x) = 3x - 5 \) when \( 1 \leq x \leq 3 \).
  - This is a linear function with a slope of 3 and intercept at -5.
  - Plot points like \( (1, -2), (2, 1), (3, 4) \), and draw a straight line between them.

- **Check continuity at \( x = 1 \)**: 
  - For the first piece, at \( x = 1 \), the value of \( f(x) \) would be \( 1^2 + 2 = 3 \), but we do not include this point since \( x < 1 \).
  - For the second piece, \( f(1) = 3(1) - 5 = -2 \). Since the values don't match, there is a jump discontinuity at \( x = 1 \). Plot an open circle at \( (1, 3) \) and a solid point at \( (1, -2) \).

### 2) Function 2: 
\[ 
f(x) = 
\begin{cases} 
x^2 & \text{if } x \leq 0 \\
x + 1 & \text{if } x > 0 
\end{cases}
\]

#### Graphing Instructions:
- **For the first piece**: \( f(x) = x^2 \) when \( x \leq 0 \).
  - This is a simple parabola for non-positive values of \( x \).
  - Plot points like \( (-2, 4), (-1, 1), (0, 0) \), and continue plotting for \( x \leq 0 \).

- **For the second piece**: \( f(x) = x + 1 \) when \( x > 0 \).
  - This is a straight line with slope 1 and intercept 1.
  - Plot points like \( (1, 2), (2, 3) \), and draw a straight line for \( x > 0 \).

- **Check continuity at \( x = 0 \)**: 
  - For the first piece, at \( x = 0 \), the value is \( f(0) = 0^2 = 0 \), and for the second piece, \( f(0) = 0 + 1 = 1 \).
  - There is a jump discontinuity at \( x = 0 \), so plot a solid point at \( (0, 0) \) for the first piece and an open circle at \( (0, 1) \) for the second piece.

### Additional Practice:
If you need further clarification or assistance in graphing, let me know!

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Would you like a detailed breakdown of how to calculate more specific points for these graphs? Also, here are some related questions to deepen your understanding:

1. How do you determine whether a piecewise function is continuous or discontinuous?
2. What role do open and closed circles play when graphing piecewise functions?
3. How can you find the derivative of a piecewise function?
4. How would you describe the behavior of the function as \( x \to 1 \) in the first function?
5. How can you express the domain and range for these piecewise functions?

**Tip**: When graphing piecewise functions, always check if the endpoints are included by looking at the inequality symbols (open or closed circles).

Graph the following piecewise functions:
1) f(x) = { x^2 + 2  if x < 1
          { 3x - 5   if 1 <= x <= 3
2) f(x) = { x^2  if x <= 0
          { x + 1  if x > 0

Learn how to graph piecewise functions with step-by-step instructions for functions defined in different intervals. This example covers f(x) = { x^2 + 2  if x < 1, 3x - 5  if 1 <= x <= 3 } and f(x) = { x^2  if x <= 0, x + 1  if x > 0 }. Understand continuity, discontinuity, and how to handle different behaviors in each piece. Suitable for high school students.

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