To determine which option represents the piecewise function of the graph based on the description you provided, we need to analyze the provided options and their respective intervals.
Option #1: \[ f(x)= \begin{cases} 2x & \text{if } -2 < x < -1 \ 2 & \text{if } -1 \leq x \leq 1 \ -2x & \text{if } 1 < x < 2 \end{cases} \]
Option #2: \[ f(x)= \begin{cases} -2x & \text{if } 2 < x < 4 \ 2 & \text{if } x = 2 \ 2x & \text{if } 2 < x < 4 \end{cases} \]
Option #3: \[ f(x)= \begin{cases} -2x & \text{if } -2 < x < -1 \ 2 & \text{if } -1 \leq x \leq 1 \ 2x & \text{if } 1 < x < 2 \end{cases} \]
Comparing these options, we can see that:
- Option #1 has a constant value of 2 from \(-1\) to \(1\), which might not match a typical graph whose continuous behavior is broken at those points unless specified.
- Option #2 seems to define the function for intervals starting from \(2\) and extending to \(4\), which doesn’t align with the \(x\) ranges provided in the question.
- Option #3 shows behaviors on ranges from \(-2 < x < -1\), at \(-1 \leq x \leq 1\), and \(1 < x < 2\) with a mix of linear functions.
Without actually seeing the graph, Option #3 seems most coherent with a typical piecewise function of three segments across three parts of the horizontal range.
Thus, based on the provided descriptions and the analysis, Option #3 is the correct choice:
Option #3 is the correct equation for the piecewise graph.