To determine whether the endpoints of the piecewise function are opened or closed, we need to examine the definitions of each piece of the function and their respective endpoints:
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For the first piece: \( f(x) = 2x \) for \( x < 2 \)
- The endpoint is \( x = 2 \). Since the inequality is strict (\(<\)), this endpoint is not included in the function. Therefore, it is an open endpoint.
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For the second piece: \( f(x) = 3x \) for \( 2 \leq x < 4 \)
- The endpoints are \( x = 2 \) and \( x = 4 \). Here, \( x = 2 \) is included in the function (due to the \(\leq\)), making it a closed endpoint. For \( x = 4 \), since the inequality is strict (\(<\)), this endpoint is not included, making it an open endpoint.
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For the third piece: \( f(x) = 4x \) for \( x > 5 \)
- The endpoint is \( x = 5 \). The inequality is strict (\(>\)), meaning this endpoint is not included in the function, making it an open endpoint.
Summary of the endpoints:
- The endpoint at \( x = 2 \) is closed.
- The endpoint at \( x = 4 \) is open.
- The endpoint at \( x = 5 \) is open.