In order to determine which piecewise function matches a graph, it's important to analyze the behavior of the function at the specific regions defined by the conditions. Let's briefly break down each of the provided functions:
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Function 1: \[ f(x) = \begin{cases} 2 - x & \text{if } x \leq 0 \ x - 5 & \text{if } x > 0 \end{cases} \]
For \( x \leq 0 \), this line has a y-intercept of 2 and a slope of -1. For \( x > 0 \), it is a line that starts from (0, -5) and has a slope of +1.
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Function 2: \[ f(x) = \begin{cases} -5 & \text{if } x < 0 \ -x + 1 & \text{if } x \geq 0 \end{cases} \]
This function is a horizontal line at y = -5 for \( x < 0 \) and a decreasing line starting from (0, 1) for \( x \geq 0 \) with a slope of -1.
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Function 3: \[ f(x) = \begin{cases} -x - 1 & \text{if } x < 1 \ -x + 2 & \text{if } x \geq 1 \end{cases} \]
For \( x < 1 \), the slope is -1 with y-intercept at -1. For \( x \geq 1 \), the line has a slope of -1 starting from (1, 1).
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Function 4: \[ f(x) = \begin{cases} x + 5 & \text{if } x \leq -2 \ 2x + 3 & \text{if } x > -2 \end{cases} \]
This function is a line with a slope of +1 and a y-intercept of +5 for \( x \leq -2 \), and for \( x > -2 \) it has a slope of +2 starting from ( -2, 1).
To match the function with the graph, consider the characteristics at the breakpoints (where the piecewise definition changes) and the overall slope and intercepts to determine which function corresponds to the visual representation shown in the graph.
If you can provide details about the graph's characteristics (like intercepts, whether it has constant sections, slopes etc.), I could help narrow down the comparison!
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