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Use the image to answer the 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, 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+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+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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1 answer

To determine the correct piecewise function from the given options, we need to analyze the behavior of the function over different intervals on the coordinate plane based on the description provided.

  1. Identify endpoints and intervals:

    • A ray with a closed endpoint means the function is defined at that point.
    • A ray with open endpoints means the function values are not defined at those points.
    • A line segment means the function is defined throughout that segment.

  2. Examine each option:

    • The equations need to be checked against the specified conditions about the intervals for x (whether they are open or closed).

Based on your descriptions, the most likely piecewise function that matches with the description where rays have closed endpoints and a line segment has open endpoints would be:

y=⎧⎩⎨⎪⎪ x + 2, x ≤ -1
y= x + 3, -1 < x < 1
y= x + 1, x ≥ 1

This piece corresponds to:

  • \(y = x + 2\) for \(x \leq -1\) (closed ray starting from \((-1, 1)\) and going left, includes \(-1\))
  • \(y = x + 3\) for \(-1 < x < 1\) (line segment between \((-1, 2)\) and \( (1, 4)\), excluded at \(-1\) and \(1\))
  • \(y = x + 1\) for \(x \geq 1\) (open ray starting from \( (1, 2) \) including \((1)\))

Therefore, the correct option looks like this:

y=⎧⎩⎨⎪⎪x+2, x≤−1
x+3, −1<x<1
x+1, x≥1

Matching one of the given choices:

y=⎧⎩⎨⎪⎪x+2, x≤−1
x+3, −1<x<1
x+1, x≥1

Choose the above option.