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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)={−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)={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

1 answer

To determine which piecewise function corresponds to the graph, you will need to analyze the graph in terms of slopes, intercepts, and the nature of the endpoints (open or closed).

  1. Identify the lines in the graph:

    • If the endpoint at x = 1 is closed, this means the function at x = 1 includes a specific value (hence the equal in the piecewise function).
    • If the endpoint at x = 1 is open, then the value at that point is not included.

  2. Analyze the slopes:

    • The function will switch from one equation to another at the point where x equals the threshold defined in the piecewise function.

  3. Check each option against these characteristics:

    • Option 1: ( f(x) = \begin{cases} -x - 1 & \text{if } x < 1 \ -x + 2 & \text{if } x \geq 1 \end{cases} )
    • Option 2: ( f(x) = \begin{cases} -5 & \text{if } x < 0 \ -x + 1 & \text{if } x \geq 0 \end{cases} )
    • Option 3: ( f(x) = \begin{cases} 2 - x & \text{if } x \leq 0 \ x - 5 & \text{if } x > 0 \end{cases} )
    • Option 4: ( f(x) = \begin{cases} x + 5 & \text{if } x \leq -2 \ 2x + 3 & \text{if } x > -2 \end{cases} )

  4. Compare the lines:

    • For each option, calculate points where they intersect with the x-axis and their slopes to visually check against the graph.

  5. Cross-verify open/closed endpoints:

    • Ensure that if a function has an open endpoint at a particular x-value, its defined equality should not include that endpoint.

Without having the graph in front of me, I suggest following this analysis for each piecewise function based on the behavior described, specifically focusing on the value at x = 1 and slopes of each line to determine which is correct based on the visual characteristics you observe.

If you confirm slopes and endpoints from the graph correspond to one of the options, that option is the correct piecewise function.