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+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
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−1 if x<1−x+2 if x≥1

Answer 1

To determine which piecewise function matches the graph, we would analyze each option based on how the pieces behave in relation to the closed and open endpoints indicated.

1. Stay aware of the endpoints:
   • If a function has a closed endpoint at a certain point, that means the function includes that value.
   • Conversely, an open endpoint means that the function does not include that value.

Let's analyze the piecewise functions provided:

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

   • At \(x = -2\), the first piece is closed and outputs \(-2 + 5 = 3\).
   • The second piece would output \(2(-2) + 3 = -4 + 3 = -1\) for \(x > -2\).

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

   • At \(x = 0\), the first piece is closed and outputs \(2 - 0 = 2\).
   • The second piece \(x - 5\) would go to \(-5\) for \(x > 0\).

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

   • The output is always \(-5\) for \(x < 0\) and is defined starting from \(x=0\) with the closed point \(1\) since at \(x = 0\) we have \(-0 + 1 = 1\).

4. Option 4: (f(x) = \begin{cases} -x - 1 & \text{if } x < 1 \ -x + 2 & \text{if } x \geq 1 \end{cases})

   • The first piece would produce \(-1 - 1 = -2\) at \(x = 1\) and the second would output \(-1 + 2 = 1\) using the closed point.

Based on the characteristics of the graph with the closed and open endpoints, compare to find the correct function. We expect the function to change at specific critical x values, respecting closed/open nature.

• If we notice that we have an open endpoint at \(x = 0\) we can see that Option 3 (the piece that is constant) might fit since the nature of its connection at \(x = 0\) is continuous (the jump does not occur).

Thus, the best likely match to the picture deduced through logic of piecewise behavior would have you verify with the drawn graph to conclude:

Final Answer: (3) f(x)={−5 if x<0; −x+1 if x≥0}