To find the piecewise function that matches a given graph, we would need to examine characteristics of the graph such as slopes, intercepts, and the behavior at the transition points. Since I don't have access to the specific graph you are referring to, I'll guide you through the features of each function based purely on their definitions:
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First Function: ( f(x) = \begin{cases} 2 - x & \text{if } x \leq 0 \ x - 5 & \text{if } x > 0 \end{cases} )
- For \( x \leq 0 \), the graph is a downward sloping line starting from \( (0, 2) \).
- For \( x > 0 \), it is a linear function with a slope of 1 that starts at \( (0, -5) \).
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Second Function: ( f(x) = \begin{cases} -x - 1 & \text{if } x < 1 \ -x + 2 & \text{if } x \geq 1 \end{cases} )
- For \( x < 1 \), it is a downward sloping line starting from \( (1, -2) \).
- For \( x \geq 1 \), it continues as a linear function with a different slope starting from the point \( (1, 1) \).
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Third Function: ( f(x) = \begin{cases} -5 & \text{if } x < 0 \ -x + 1 & \text{if } x \geq 0 \end{cases} )
- For \( x < 0 \), the function is constant at \( -5 \).
- For \( x \geq 0 \), it is a downward sloping line with a y-intercept of 1.
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Fourth Function: ( f(x) = \begin{cases} x + 5 & \text{if } x \leq -2 \ 2x + 3 & \text{if } x > -2 \end{cases} )
- For \( x \leq -2 \), the function is an upward sloping line starting from \( (-2, 3) \).
- For \( x > -2 \), it is a line with a steeper slope starting from \( (-2, 7) \).
To select the correct piecewise function, examine the graph's key features such as values at specific points, slopes, and any constant segments. Compare these characteristics with those given for each function until you find a match. If you can describe the graph or provide any specific points or characteristics, I could help you narrow it down further.