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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≥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

y=⎧⎩⎨⎪⎪x+2, x≤−1x+3, −1<x<1x+1, x≥1

1 answer

To determine the correct piecewise function from the provided options, we need to analyze the characteristics of each segment of the function according to the given conditions.

1. Identify Each Piece of the Function:

  • For \( x < -1 \), does the function appear to be \( x + 2 \)?
  • For \( -1 \leq x \leq 1 \), does the function appear to be \( x + 3 \)?
  • For \( x > 1 \), does the function appear to be \( x + 1 \)?

2. Analyzing the Options:

  1. First Option: \[ y = \begin{cases} x + 2 & \text{if } x < -1 \ x + 3 & \text{if } -1 \leq x \leq 1 \ x + 1 & \text{if } x > 1 \end{cases} \]

    • This structure matches the described segments.

  2. Second Option: \[ y = \begin{cases} x + 2 & \text{if } x \leq -1 \ x + 3 & \text{if } 2 < x < 4 \ x + 1 & \text{if } x \geq 2 \end{cases} \]

    • This option does not match the segments since it misplaces conditions.

  3. Third Option: \[ y = \begin{cases} x + 1 & \text{if } x \leq -1 \ x + 2 & \text{if } -1 < x < 1 \ x + 3 & \text{if } x \geq 1 \end{cases} \]

    • The order of segments here does not correspond to the description.

  4. Fourth Option: \[ y = \begin{cases} x + 2 & \text{if } x \leq -1 \ x + 3 & \text{if } -1 < x < 1 \ x + 1 & \text{if } x \geq 1 \end{cases} \]

    • This also does not match the conditions set earlier.

Conclusion: The first option accurately represents the piecewise function described. Therefore, the correct answer is:

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