Question
Children’s cough syrup has dosage guidelines based on a child’s age. In the following piecewise function, x represents the child’s age in years and f(x)
represents the dosage in fluid ounces.
f(x)=⎧⎩⎨⎪⎪0.5 if 1≤x≤31.0 if 3<x≤60.5x if x>6
Does the graph accurately represent the piecewise function? Why or why not?
(1 point)
Responses
No. The graph should be continuous, with all the pieces connected.
No. The graph should be continuous, with all the pieces connected.
No. There should be an open circle on the first point of two parts of the function.
No. There should be an open circle on the first point of two parts of the function.
Yes. The graph is correct because all three pieces of the function are present.
Yes. The graph is correct because all three pieces of the function are present.
Yes, because all of the points at the ends of the pieces of the function should have closed circles.
represents the dosage in fluid ounces.
f(x)=⎧⎩⎨⎪⎪0.5 if 1≤x≤31.0 if 3<x≤60.5x if x>6
Does the graph accurately represent the piecewise function? Why or why not?
(1 point)
Responses
No. The graph should be continuous, with all the pieces connected.
No. The graph should be continuous, with all the pieces connected.
No. There should be an open circle on the first point of two parts of the function.
No. There should be an open circle on the first point of two parts of the function.
Yes. The graph is correct because all three pieces of the function are present.
Yes. The graph is correct because all three pieces of the function are present.
Yes, because all of the points at the ends of the pieces of the function should have closed circles.
Answers
GPT-4o mini
To determine whether the graph accurately represents the piecewise function given, we need to analyze the function's definition and the implications for how it should be graphed.
1. The first segment, \(f(x) = 0.5\) for \(1 \leq x \leq 3\), means that the dosage is a constant 0.5 fluid ounces for ages 1, 2, and 3. This segment should be represented with closed circles at both ends because the endpoints are included.
2. The second segment, \(f(x) = 1.0\) for \(3 < x \leq 6\), gives a dosage of 1.0 fluid ounces for ages greater than 3 and up to 6. Here, we need an open circle at 3 (because 3 is not included in this segment) and a closed circle at 6 (since it is included).
3. The third segment, \(f(x) = 0.5x\) for \(x > 6\), gives a dosage that increases linearly. Since this segment starts at \(x = 6\), it should be represented with an open circle at 6 (because it does not include this value) and should continue indefinitely for ages greater than 6.
Given these details, the accurate representation of the piecewise function involves using proper closed and open circles, particularly at the boundaries between the segments. So, if the graph has open circles at the appropriate endpoints and reflects the function's values correctly, then that would mean the graph is correct.
Based on this analysis, the correct response to whether the graph accurately represents the piecewise function would be:
**No. There should be an open circle on the first point of two parts of the function.**
This indicates that there is a mistake in the representation of the graph with respect to the transition points at \(x = 3\) and \(x = 6\).
1. The first segment, \(f(x) = 0.5\) for \(1 \leq x \leq 3\), means that the dosage is a constant 0.5 fluid ounces for ages 1, 2, and 3. This segment should be represented with closed circles at both ends because the endpoints are included.
2. The second segment, \(f(x) = 1.0\) for \(3 < x \leq 6\), gives a dosage of 1.0 fluid ounces for ages greater than 3 and up to 6. Here, we need an open circle at 3 (because 3 is not included in this segment) and a closed circle at 6 (since it is included).
3. The third segment, \(f(x) = 0.5x\) for \(x > 6\), gives a dosage that increases linearly. Since this segment starts at \(x = 6\), it should be represented with an open circle at 6 (because it does not include this value) and should continue indefinitely for ages greater than 6.
Given these details, the accurate representation of the piecewise function involves using proper closed and open circles, particularly at the boundaries between the segments. So, if the graph has open circles at the appropriate endpoints and reflects the function's values correctly, then that would mean the graph is correct.
Based on this analysis, the correct response to whether the graph accurately represents the piecewise function would be:
**No. There should be an open circle on the first point of two parts of the function.**
This indicates that there is a mistake in the representation of the graph with respect to the transition points at \(x = 3\) and \(x = 6\).