To determine which piecewise function matches the graph described, we need to analyze the information provided about the endpoints of the lines.
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First piecewise function:
( f(x) = \begin{cases} -x - 1 & \text{if } x < 1 \ -x + 2 & \text{if } x \geq 1 \end{cases} )- This function has a line with an open endpoint at \( x = 1 \) (since it uses \( < \) for the first case). It also goes up from \( -1 \) at \( x = 1 \).
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Second piecewise function:
( f(x) = \begin{cases} -5 & \text{if } x < 0 \ -x + 1 & \text{if } x \geq 0 \end{cases} )- This function has a horizontal line at \( y = -5 \) when \( x < 0 \), and it has a slope when \( x \geq 0 \).
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Third piecewise function:
( f(x) = \begin{cases} 2 - x & \text{if } x \leq 0 \ x - 5 & \text{if } x > 0 \end{cases} )- This function starts from a point at \( (0, 2) \) and has a line with a negative slope going downwards.
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Fourth piecewise function:
( f(x) = \begin{cases} x + 5 & \text{if } x \leq -2 \ 2x + 3 & \text{if } x > -2 \end{cases} )- This function has a line that has a positive slope when \( x \leq -2 \) and a steeper line for \( x > -2 \).
Analysis:
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The first function provides a clear open endpoint at \( x = 1 \), matching the description of having a closed endpoint and an open endpoint.
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The third function's endpoint is also inclusive at \( x = 0\), where it transitions from \( 2 - x \) to \( x - 5\).
Given the description of the graph with open and closed endpoints and the behavior around \( x = 0 \) or \( x = 1\), the correct match is:
First piecewise function.
Thus, the response is: f(x) = {−x−1 if x<1, −x+2 if x≥1}