https://file.aiquickdraw.com/mathbot/file/1726722835713t9m317hp.jpg

To graph the piecewise functions from the image, I'll break down each function and provide instructions on how to graph them. Let's go through them one by one.

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

#### Instructions to Graph:
- **For \( x < 1 \):**
  - The function is \( f(x) = x^2 + 2 \), a parabolic curve shifted upwards by 2 units. Plot points for values like \( x = 0, -1, -2 \), and so on. Do not include \( x = 1 \).
  - Since this is a strict inequality, use an open circle at \( x = 1 \).
  
- **For \( 1 \leq x \leq 3 \):**
  - The function is \( f(x) = 3x - 5 \), a linear function with a slope of 3 and y-intercept at -5.
  - Plot points for \( x = 1, 2, 3 \), and connect them with a straight line. Since the inequality is \( \leq \), use closed circles at \( x = 1 \) and \( x = 3 \).

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

#### Instructions to Graph:
- **For \( x \leq 0 \):**
  - The function is \( f(x) = x^2 \), a parabola opening upwards. Plot points like \( x = 0, -1, -2 \), and use a closed circle at \( x = 0 \).
  
- **For \( x > 0 \):**
  - The function is \( f(x) = x + 1 \), a linear function with slope 1 and y-intercept at 1.
  - Plot points starting from \( x = 0 \) but use an open circle at \( x = 0 \) because \( x > 0 \), and plot more points for \( x > 0 \).

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### 3) \( f(x) = \begin{cases} 2x + 3 & \text{if } x < 1 \\ x - 1 & \text{if } 1 \leq x \leq 6 \end{cases} \)

#### Instructions to Graph:
- **For \( x < 1 \):**
  - The function is \( f(x) = 2x + 3 \), a linear function with slope 2 and y-intercept at 3.
  - Plot points like \( x = 0, -1 \), and use an open circle at \( x = 1 \).
  
- **For \( 1 \leq x \leq 6 \):**
  - The function is \( f(x) = x - 1 \), a linear function with slope 1 and y-intercept at -1.
  - Plot points for \( x = 1, 2, 3, 4, 5, 6 \), and connect them with a straight line. Use closed circles at \( x = 1 \) and \( x = 6 \).

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Let me know if you'd like a detailed breakdown of how to plot these graphs or any clarification!

### Related Questions:
1. How do you determine where to place open and closed circles on piecewise functions?
2. Can a piecewise function have both a quadratic and a linear part?
3. What are the key differences in graphing strict inequalities (\(<\) or \(>\)) vs non-strict inequalities (\(\leq\) or \(\geq\))?
4. How do you evaluate a piecewise function at a point where the definition changes?
5. How would the graph change if the second part of the function had a cubic term?

### Tip:
When graphing piecewise functions, always pay attention to the type of inequality to know whether to use open or closed circles at the boundary points!

Graph the piecewise functions from the image

Learn how to graph the piecewise functions involving quadratic and linear components. The functions include different inequalities, requiring open and closed circles at boundary points. This exercise helps students in Grades 9-11 to graph and understand piecewise functions effectively.

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