To determine which piecewise function matches a given graph, you must analyze characteristics such as the slopes, intercepts, and breakpoints of each function. However, since the graph is not available, I'll describe the nature of each function based on the provided options:
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Option 1:
- ( f(x) = \begin{cases} -5 & \text{if } x < 0 \ -x + 1 & \text{if } x \geq 0 \end{cases} )
- This function is constant at -5 for \( x < 0 \) and has a line with a negative slope (\(-1\)) for \( x \geq 0 \), starting at \( (0, 1) \).
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Option 2:
- ( f(x) = \begin{cases} -x - 1 & \text{if } x < 1 \ -x + 2 & \text{if } x \geq 1 \end{cases} )
- This function has a negative slope for both parts but changes behavior at \( x = 1 \).
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Option 3:
- ( f(x) = \begin{cases} 2 - x & \text{if } x \leq 0 \ x - 5 & \text{if } x > 0 \end{cases} )
- This piece shows a linear function with a positive slope starting at \( (0, 2) \) for \( x < 0 \) and then changing to a different slope for \( x > 0\).
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Option 4:
- ( f(x) = \begin{cases} x + 5 & \text{if } x \leq -2 \ 2x + 3 & \text{if } x > -2 \end{cases} )
- For \( x \leq -2\), there is a line with a slope of \(1\), while for \( x > -2\), there is a steeper line with a slope of \( 2\).
To find the function that matches a specific graph, you would look for the function that has the correct constant values, slopes, and any transitions at the specified points (breaks) revealed by the graph.
If you have a specific graph in mind, you could identify slope changes, y-intercepts or constant values to match it to one of these functions. Since I don’t have the graph, I can’t determine which piecewise function corresponds to it. If you could describe the graph or key features, I would be able to provide a more precise answer!