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, 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

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
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Answer 1

To determine the correct piecewise function from the graph described, we need to analyze the endpoints and behavior of each segment:

1. For \(x < -1\), if the ray has a closed endpoint at \((-1, y)\) and a linear behavior observed in the graph, the expression could be something like \(y = x + 2\).

2. For \(-1 \leq x \leq 1\), if this segment is a line from \((-1, y)\) to \((1, y)\), we see that segments could be defined by a function, possibly \(y = x + 3\), since this fits the linear path over that instantiation.

3. For \(x > 1\), if this ray is open on its endpoints after \(x = 1\) and going upwards, it might be \(y = x + 1\).

Given these observations about endpoint inclusivity and exclusivity, the piecewise function can be reconstructed:

The correct choice is: \[ y = \begin{cases} x + 2, & \text{if } x < -1 \ x + 3, & \text{if } -1 \leq x < 1 \ x + 1, & \text{if } x \geq 1 \ \end{cases} \]

This aligns with the fourth option from the provided responses:

y = ⎧⎩⎨⎪⎪x + 2, x < −1; x + 3, −1 ≤ x < 1; x + 1, x ≥ 1.

This is the most likely equation based on your description of the graph characteristics.