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)={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)={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

Answer 1

To determine which piecewise function matches the described graph, let's analyze each provided option based on their intervals and slopes.

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

   • The first piece is a line with a negative slope (decreasing) that would intersect the y-axis. The second piece is also a line with a negative slope.

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

   • The first piece has a positive slope (increasing), while the second piece has a steeper positive slope.

3. Option 3: ( f(x) = \begin{cases} 2 - x & \text{if } x \leq 0 \ x - 5 & \text{if } x > 0 \end{cases} )

   • The first piece has a negative slope and the second piece has a positive slope that starts below the x-axis.

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

   • The first part is a constant function (horizontal line at -5), and the second part is decreasing.

To match with the graph's characteristics:

• Look for breaks (open/closed ends) at \( x = 1 \).
• Identify slopes (increasing/decreasing).

Given these observations:

• If the graph has decreasing lines, the first and fourth options might match.
• To check the conditions, examine intercepts and slopes concerning the x and y-coordinates.

Assuming the trend is decreasing before an endpoint, the appropriate piecewise function is likely: Option 1: ( f(x) = \begin{cases} -x - 1 & \text{if } x < 1 \ -x + 2 & \text{if } x \geq 1 \end{cases} )

Thus, this function best matches the behavior of a graph with lines that appear to switch at \( x = 1 \).