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Use the image to answer the question.

Three lines, a solid line, a dotted line, and a dashed line, are drawn on a coordinate plane. The -axis ranges from negative 3 to 3 in 1-unit increment, and the y-axis ranges from 0 to 5 in 1-unit increments.

Which option represents the piecewise function of the graph?

Option #1: f(x)=⎧⎩⎨⎪⎪2x,2,−2x, −2<x<−1−1≤x≤11<x<2


Option #2: f(x)=⎧⎩⎨⎪⎪−2x,2,2x, 2<x<4x=2 2<x<4


Option #3: f(x)=⎧⎩⎨⎪⎪−2x,2,2x, −2<x<−1−1≤x≤11<x<2


(1 point)
Option #
is the correct equation for the piecewise graph.

1 answer

To accurately determine the correct piecewise function, we need to assess the information provided about the lines drawn on the coordinate plane.

Based on the structure of the options:

  • Option #1 states:

    • ( f(x) = \begin{cases} 2x & \text{for } -2 < x < -1 \ 2 & \text{for } -1 \leq x \leq 1 \ -2x & \text{for } 1 < x < 2 \end{cases} )

  • Option #2 states:

    • ( f(x) = \begin{cases} -2x & \text{for } 2 < x < 4 \ 2 & \text{for } x = 2 \ 2x & \text{for } 2 < x < 4 \end{cases} )

  • Option #3 states:

    • ( f(x) = \begin{cases} -2x & \text{for } -2 < x < -1 \ 2 & \text{for } -1 \leq x \leq 1 \ 2x & \text{for } 1 < x < 2 \end{cases} )

Given the options, without viewing the graph directly, we can use the structure of the piecewise functions and typical behavior of linear functions to eliminate options:

  • Option #2 is unlikely to apply, as it refers to domains that range above 2 and introduces an equation for \( x = 2 \), which suggests a function value holds at that point but lacks continuity in terms of intervals.

Both Options #1 and #3 show piecewise definitions spanning different intervals commonly represented in piecewise graphs. Without loss of generality, if we consider the typical mappings, the behavior for values \( -2 < x < 2 \) is represented in both cases but the signs of the slopes in these intervals differ.

Given:

  • Option #1 has a linear function increase in value for \( -2 < x < -1\).
  • Option #3 has a linear function decrease in value in a similar range.

Thus, without the context of the graph, I would suggest reviewing the behavior depicted to finalize on either options based upon whether the segments reflect increasing or decreasing trends.

If the function sections mirror, for example, a rise or decrease correctly matching linear intersections or limits on respective end values, then indicating key evaluation points on either interval will help confirm the match.

Conclusively, if you can provide a descriptive overview or graph context, I can help derive the precise picking! Based on general structuring, Option #3 seems appropriate if the slope aligns with typical decreases in interval sections, whereas Option #1 would suit a more floral peak rising to a given height in region overlap.

Please choose either representation or verify with the graph presented for definitive recognition.