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

A. 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

B. 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

C. 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

D. 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

Answer 1

To determine which piecewise function matches the graph described, we can analyze the characteristics of each function and the endpoints given (open and closed).

1. Endpoint characteristics:

   • The open endpoint suggests that the function does not include that point, which would be represented by a < or > symbol.
   • The closed endpoint suggests that the function does include that point, represented by a ≤ or ≥ symbol.

2. Line segments:

   • The functions provided have different slopes and intercepts, which changes how the lines will look.

Now let's evaluate each option based on this information:

• Option A: ( f(x) = \begin{cases} 2 - x & \text{if } x \leq 0 \ x - 5 & \text{if } x > 0 \end{cases} )

  • The first piece is a line with a closed endpoint at \( (0, 2) \), matching with the closed endpoint situation. The second piece is valid for all \( x > 0 \).

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

  • This will not include the characteristics of one open endpoint and one closed endpoint described.

• Option C: ( f(x) = \begin{cases} -x - 1 & \text{if } x < 1 \ -x + 2 & \text{if } x \geq 1 \end{cases} )

  • This involves a change at 1, but doesn’t fit the described endpoint conditions.

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

  • Similar to options B and C, this does not match up with the open/closed endpoints and slopes based on the description.

Based on this analysis, Option A is the only choice that has the characteristics of one closed endpoint and a proper transition at \( x = 0 \) with the correct slope for its segments. Therefore, Option A is the correct answer.