To determine whether the graph accurately represents the piecewise function you've described, we need to consider the nature of each segment of the function and how the points are connected.
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For \( f(x) = 0.5 \) if \( 1 \leq x \leq 3 \): This is a constant function for \( x \) values from 1 to 3, inclusive. The function should have closed circles at both \( x = 1 \) and \( x = 3 \) because it includes these endpoints.
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For \( f(x) = 1.0 \) if \( 3 < x \leq 6 \): This segment spans from just above \( x = 3 \) (not including 3, hence an open circle at \( x = 3 \)) to \( x = 6\) (included). Thus, it should have an open circle at \( x = 3 \) and a closed circle at \( x = 6 \).
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For \( f(x) = 0.5x \) if \( x > 6 \): This is a linear function starting from just above \( x = 6 \), and it will have an open circle at \( x = 6 \) (as \( x = 6 \) is not included) and extends infinitely.
Given these observations:
- The first segment includes 1 and 3, indicating closed circles at those points.
- The second segment has an open circle at 3 and a closed circle at 6.
- The third segment starts with an open circle at 6.
Now, let's examine the options:
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Yes. The graph is correct because all three pieces of the function are present. - This is not a complete justification since we need to consider the continuity and endpoints.
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Yes, because all of the points at the ends of the pieces of the function should have closed circles - This statement is partially true (for 1 and 6), but not for 3 (which should have an open circle).
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No. The graph should be continuous, with all the pieces connected. - This is false because piecewise functions can have discontinuities.
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No. There should be an open circle on the first point of two parts of the function. - This is the most accurate option, specifically referring to the requirement of an open circle at \( x = 3 \).
Therefore, the best answer is: No. There should be an open circle on the first point of two parts of the function.
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