Use the image to answer the question.

Question

A coordinate plane's axes range from negative 5 to 5 by 1-unit increments. Two rays with closed endpoints and one line segment with two open endpoints are plotted.

Select the equation of the piecewise function shown in the graph.

(1 point)
Responses

y=⎧⎩⎨⎪⎪x+2, x<−1x+3, −1≤x≤1x+1, x>1
y equals left brace line 1 x plus 2, x is less than negative 1 line 2 x plus 3 comma negative 1 less than or equal to x less than or equal to 1 line 3 x plus 1 comma x greater than 1

y=⎧⎩⎨⎪⎪x+2, x≤−1x+3, −1<x<1x+1, x≥1
y equals left brace line 1 x plus 2, x is less than or equal to negative 1 line 2 x plus 3 comma negative 1 less than x less than 1 line 3 x plus 1 comma x greater than or equal to 1

y=⎧⎩⎨⎪⎪x+2, x≤1x+3, 2<x<4x+1, x≥2
y equals left brace line 1 x plus 2, x is less than or equal to negative 1 line 2 x plus 3 comma 2 less than x less than 4 line 3 x plus 1 comma x greater than or equal to 2

y=⎧⎩⎨⎪⎪x+1, x≤−1x+2, −1<x<1x+3, x≥1y equals left brace line 1 x plus 1, x is less than or equal to negative 1 line 2 x plus 2 comma negative 1 less than x less than 1 line 3 x plus 3 comma x greater than or equal to 1
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Answer 1

To determine the correct piecewise function from the descriptions given, we need to consider the following:

1. The behavior of the function in different intervals of \(x\).
2. The closed and open endpoints as indicated by the rays and line segments in the graph.

Analyzing the Function Components:

According to the equations:

1. For \(x < -1\):

   • First option: \(x + 2\)
   • Second option: \(x + 2\)
   • Third option: \(x + 2\)
   • Fourth option: \(x + 1\)

2. For \(-1 \leq x \leq 1\):

   • The options vary in terms of stability and continuity.
   • First option: \(x + 3\) (closed on \(-1\)), \(x + 3\) (both closed)
   • Second option: \(x + 3\) (open endpoints)

3. For \(x > 1\):

   • Again, multiple responses have \(x + 1\) but check their closed and open endpoints.

Choosing the Correct Function:

From our analysis:

• You need to pay attention to which segments of the graph have open and closed endpoints.
• Check correct intervals, as indicated by the graph features.

Given that the function has closed endpoints at points like \(x = -1\) and \(x = 1\), we would lean towards:

Correct response:

• y = { x + 2, x ≤ -1 { x + 3, -1 < x < 1 { x + 1, x ≥ 1}

This captures the behavior correctly with the closed and open endpoints as described in your question.