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Use the image to answer the question.

A coordinate plane with two lines, one with one open endpoint and one with a closed endpoint. The x-axis ranges from negative 5 to 10 and the y- axis ranges from negative 8 to 10. Both axes are in unit increments and are labeled in increments of 5.

Which of the following piecewise functions matches the graph?

(1 point)
Responses

f(x)={−x−1 if x<1−x+2 if x≥1
f left parenthesis x right parenthesis equals left brace Start Rows 1st Row negative x minus 1 if x less than 1 2nd Row negative x plus 2 if x greater-than-or-equal-to 1 End Rows

f(x)={2−x if x≤0x−5 if x>0
f left parenthesis x right parenthesis equals left brace Start Rows 1st Row 2 minus x if x less-than-or-equal-to 0 2nd Row x minus 5 if x greater than 0 End Rows

f(x)={x+5 if x≤−22x+3 if x>−2
f left parenthesis x right parenthesis equals left brace Start Rows 1st Row x plus 5 if x less-than-or-equal-to negative 2 2nd Row 2 x plus 3 if x greater than negative 2 End Rows

f(x)={−5 if x<0−x+1 if x≥0
f left parenthesis x right parenthesis equals left brace Start Rows 1st Row negative 5 if x less than 0 2nd Row negative x plus 1 if x greater-than-or-equal-to 0 End Rows
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1 answer

To determine which piecewise function corresponds to the given graph, we need to analyze the nature of the lines and their endpoints specified in the question. Since we don’t have the actual image to reference, I'll guide you through the analysis of the piecewise functions based on the structure and characteristics they exhibit.

  1. Function: \( f(x) = { -x - 1 \text{ if } x < 1 \ -x + 2 \text{ if } x \geq 1 } \)

    • This function consists of two lines. The first line is valid for \(x < 1\) and has a negative slope. The second line, valid for \(x \geq 1\), also has a negative slope but could meet the first line at (1, -2).

  2. Function: \( f(x) = { 2 - x \text{ if } x \leq 0 \ x - 5 \text{ if } x > 0 } \)

    • Here, the left segment (for \(x \leq 0\)) has a negative slope, and the right segment (for \(x > 0\)) has a positive slope starting well below zero.

  3. Function: \( f(x) = { x + 5 \text{ if } x \leq -2 \ 2x + 3 \text{ if } x > -2 } \)

    • The left side (for \(x \leq -2\)) is a line with a positive slope, while the right side (for \(x > -2\)) also is a line with a positive slope but steeper than the first.

  4. Function: \( f(x) = { -5 \text{ if } x < 0 \ -x + 1 \text{ if } x \geq 0 } \)

    • The first piece is a horizontal line at \(y = -5\) for \(x < 0\), and the second piece is a line with a negative slope starting from the point (0, 1).

To match the piecewise function to the graph, you will need to evaluate the slopes and the endpoints (open vs closed) depicted in the graph. Look for the following:

  • Open and closed endpoints defining if the point is included in the function or not.
  • Direction of the slope of the line segments.

Once you've done that analysis, you can choose the appropriate function. If you describe specific qualities of the lines in the image (like the slopes or points of intersection), I can help you narrow down which piecewise function fits best.