For the first piece f(x)=x−1*f*(*x*)=*x*−1, start at the point (1,0)(1,0) with an open circle, and draw the line going towards the left (negative direction) with a slope of 1.

For the second piece f(x)=−2x+4*f*(*x*)=−2*x*+4, start at the point (1,2)(1,2) with a closed circle, and draw the line going towards the right (positive direction) with a slope of -2.

draw 

Here's how you can draw the graph based on the instructions:

### 1. First Piece: \( f(x) = x - 1 \) for \( x < 1 \)
- Start at the point \( (1, 0) \) with an **open circle** because the inequality is \( x < 1 \).
- From the point \( (1, 0) \), draw a line with a slope of 1 towards the left (decreasing x values).
  - For example, when \( x = 0 \), \( f(x) = 0 - 1 = -1 \), so the point is \( (0, -1) \).
  - Continue this to extend the line to the left.

### 2. Second Piece: \( f(x) = -2x + 4 \) for \( x \geq 1 \)
- Start at the point \( (1, 2) \) with a **closed circle** because the inequality is \( x \geq 1 \).
- From the point \( (1, 2) \), draw a line with a slope of -2 towards the right (increasing x values).
  - For example, when \( x = 2 \), \( f(x) = -2(2) + 4 = 0 \), so the point is \( (2, 0) \).
  - Continue this to extend the line to the right.

### Final Graph Description
- **Line 1**: Starts at \( (1, 0) \) with an open circle and extends to the left with a positive slope.
- **Line 2**: Starts at \( (1, 2) \) with a closed circle and extends to the right with a negative slope.

Here's a rough sketch you can use:

1. Plot an open circle at \( (1, 0) \).
2. Draw a line from \( (1, 0) \) down and to the left.
3. Plot a closed circle at \( (1, 2) \).
4. Draw a line from \( (1, 2) \) down and to the right.

Since I can't draw directly on the image here, you'd follow these steps on your graph. If you have any more questions, or if you'd like further help, feel free to ask!

Here are 5 related questions:
1. What is the significance of open and closed circles in piecewise functions?
2. How do you find the slope of a line given two points?
3. How can you determine the range of a piecewise function from its graph?
4. What are the key differences between continuous and discontinuous functions?
5. How would you determine the intercepts of a piecewise function?

**Tip:** When graphing piecewise functions, always pay close attention to the inequality signs, as they dictate whether you use open or closed circles at the boundary points.

Learn how to graph piecewise functions step-by-step with this detailed guide. Includes examples and explanations for graphing functions with open and closed circles, and understanding slopes in piecewise functions. Suitable for high school students studying algebra and graphing functions.

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