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 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, −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+1, x≤−1x+2, −1<x<1x+3, x≥1
y 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

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
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Answer 1

To identify the correct piecewise function equation based on the provided description of a coordinate plane and the nature of plotted rays and line segments:

1. Examine domain conditions:

   • One ray goes towards negative infinity, indicating a linear function with a defined endpoint.
   • There is another ray with endpoints closed on the sides.
   • A line segment indicates that it has defined endpoints.

2. Identify function pieces based on the given responses:

   • Each piece needs to be matched with the corresponding parts of the graph.
   • Pay attention to open and closed endpoints to establish inequalities.

Looking at the responses provided:

1. First Option:

   • y = { x + 2, x ≤ -1; x + 3, -1 < x < 1; x + 1, x ≥ 1 }
     • Closed at -1 and closed at 1, no closed at 1 contradicts the description.

2. Second Option:

   • y = { x + 2, x < -1; x + 3, -1 ≤ x ≤ 1; x + 1, x > 1 }
     • The middle section has the correct endpoints closed at -1 and 1.

3. Third Option:

   • y = { x + 1, x ≤ -1; x + 2, -1 < x < 1; x + 3, x ≥ 1 }
     • Closed at -1 and closed at 1.

4. Fourth Option:

   • y = { x + 2, x ≤ -1; x + 3, 2 < x < 4; x + 1, x ≥ 2 }
     • Does not match any of the description conditions related to endpoint positions.

Considering the above evaluations, the best match for the described piecewise function based on the information about endpoints and intervals is:

y = { x + 2, x < -1; x + 3, -1 ≤ x ≤ 1; x + 1, x > 1 }

So, the selected response is the second option.