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 36 Does the graph accurately represent the piecewise function? Why or why not? (1 point) Responses 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, because all of the points at the ends of the pieces of the function should have closed circles. Yes, because all of the points at the ends of the pieces of the function should have closed circles. No. The graph should be continuous, with all the pieces connected. No. The graph should be continuous, with all the pieces connected. Yes. The graph is correct because all three pieces of the function are present.

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. 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, because all of the points at the ends of the pieces of the function should have closed circles. Yes, because all of the points at the ends of the pieces of the function should have closed circles. No. The graph should be continuous, with all the pieces connected. No. The graph should be continuous, with all the pieces connected. Yes. The graph is correct because all three pieces of the function are present.

GPT-4o mini · Answer

To determine if the graph accurately represents the piecewise function described, let's analyze the function and the implication of its segments:

For \(1 \leq x \leq 3\): The dosage is a constant \(0.5\) ounces. This means the graph will have a horizontal line segment from \(x = 1\) to \(x = 3\) with closed circles at both \(x = 1\) and \(x = 3\) since both boundaries are inclusive.
For \(3 < x \leq 6\): The dosage is \(1.0\) ounces. Therefore, the graph will have another horizontal line segment from just above \(x = 3\) to \(x = 6\) with an open circle at \(x = 3\) (since it is not included) and a closed circle at \(x = 6\).
For \(x > 6\): The dosage is \(0.5x\). This will be a line that starts at \(x = 6\) (with some initial value \(3\) if you substitute \(6\) into \(0.5x\)) and continues increasing as \(x\) increases. For this portion of the graph, there will be a closed circle at \(x = 6\).

Given this analysis, we can conclude:

The graph should have closed circles at \(x = 1\) and \(x = 6\).
There should be an open circle at the point corresponding to \(x = 3\) to correctly represent that \(3\) itself is not included in the second piece of the function.
The graph is not continuous; it will have a jump between the segments of the graph at \(x = 3\).

Therefore, the correct response would be:

No. There should be an open circle on the first point of two parts of the function.