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≥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
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Answer 1

To accurately select the piecewise function based on the description provided, let's analyze the characteristics of each segment:

1. For x < -1: The description mentions that there is a ray with a closed endpoint, indicating that this line segment is included at x = -1. Therefore, if it has a slope that leads to a value on the y-axis, we can conclude it could be of the form \(x + 2\).

2. For -1 ≤ x ≤ 1: Since there is a closed endpoint at both -1 and 1, it indicates that the function includes values in this range, taking a form like \(x + 3\).

3. For x > 1: The ray for this section has a closed endpoint at 1 and extends indefinitely. Thus, it should take the form of \(x + 1\) since it starts from 1.

Based on this information, the correct piecewise function is:

\[ y = \begin{cases} x + 2 & \text{for } x < -1 \ x + 3 & \text{for } -1 \leq x \leq 1 \ x + 1 & \text{for } x > 1 \end{cases} \]

This matches the first response option:

y=⎧⎩⎨⎪⎪x+2, x<−1 x+3, −1≤x≤1 x+1, x>1

So, the correct choice is the first option.