To construct a piecewise function based on your description of the image, we need to analyze the segments you provided.
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First Segment:
- A black dot at the point \((-1, 1)\) indicates that this point is included, which means \(f(-1) = 1\).
- The line continues diagonally down to the left. The slope of that line cannot be determined from the provided information, but it suggests a linear function that continues indefinitely in that direction.
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Second Segment:
- An open circle at \((-1, 1)\) suggests this point is not included in this segment.
- The line moves diagonally upwards to the right, indicating a linear function again. The specific slope or endpoint isn't detailed, but we know it starts at point \((-1, 1)\) and goes indefinitely in that direction.
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Third Segment:
- Open circles from the point \((-1, 2)\) to the point \((1, 4)\) suggest that these endpoints are not included in the function, but the line that connects them is part of the function.
- To find the function that connects these points: The slope \(m\) can be calculated as \( m = (y_2 - y_1) / (x_2 - x_1) = (4 - 2) / (1 - (-1)) = 2 / 2 = 1\). This means the equation of the line is \(y = x + 3\) between the points \((-1, 2)\) and \((1, 4)\).
Putting all of this together, the piecewise function can be expressed as:
\[ f(x) = \begin{cases} 1 & \text{for } x = -1 \ \text{y = mx + b, (down to the left) } & \text{for } x < -1 \ x + 3 & \text{for } -1 < x < 1 \ \text{y = mx + b, (up to the right) } & \text{for } x > -1 \end{cases} \]
For a complete answer, we would need more specific details about the slopes or additional points in the "down to the left" and "up to the right" segments. However, from here, you can adapt the function once you have that specific information.
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