Use the image to answer the question.

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)={−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

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
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Answer 1

To determine which piecewise function matches the described graph, we need to analyze the properties of each given function based on their structure and the endpoints specified in the question.

1. Analyzing the first function:

   • f(x) = { -x - 1 if x < 1 -x + 2 if x ≥ 1 }
   • This indicates that for x < 1, the function has a downward slope and for x ≥ 1, the function also has a downward slope but with a different y-intercept.

2. Analyzing the second function:

   • f(x) = { 2 - x if x ≤ 0 x - 5 if x > 0 }
   • Here, for x ≤ 0, the function is decreasing from y = 2, and for x > 0, it increases but starts at -5 for x = 0.

3. Analyzing the third function:

   • f(x) = { -5 if x < 0 -x + 1 if x ≥ 0 }
   • This means a constant value of -5 for x < 0 and a downward linear function starting from 1 (when x = 0).

4. Analyzing the fourth function:

   • f(x) = { x + 5 if x ≤ -2 2x + 3 if x > -2 }
   • This shows a line with a slope of 1 starting from y = 3 when x = -2, changing to another line for x > -2.

If we consider the requirement for open and closed endpoints, we see that the choice needs to match those conditions:

• The first and fourth options appear to have the potential for meeting the open or closed endpoint requirement depending on the graph. Specifically, if the graph is confirmed either to be crossing or meeting the axis.

• Without seeing the graph itself, it becomes challenging to make an accurate choice based solely on this detail. However, based on typical placements and characteristics of these functions,

Assuming the closed endpoints are at specific transition points (like x = 1, x = 0, etc.) and that the other intervals change direction smoothly, the third function aligns with the description of having a set behavior moving from a constant to a variable function at an intersecting axis, likely suggesting movement around those critical intervals.

Thus, the best piecewise function to match the graph in this context is:

f(x) = { -5 if x < 0 -x + 1 if x ≥ 0 }