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

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

GPT-4o mini · Answer

To determine if the graph accurately represents the given piecewise function for the children's cough syrup dosage, we need to analyze the intervals and endpoints defined in the function:

For \(1 \leq x \leq 3\): The function gives a dosage of 0.5 fluid ounces. This interval includes both endpoints (1 and 3), so these points should be represented by closed circles.
For \(3 < x \leq 6\): The function states that the dosage is 1.0 fluid ounce. Here, the point at \(x = 3\) should be an open circle (since it is not included in this interval), and the point at \(x = 6\) should be a closed circle (since it is included).
For \(x > 6\): The function indicates that the dosage is \(0.5x\). Since this is a continuous function defined for all \(x > 6\), it will be represented as a line starting just above \(x = 6\) going towards positive infinity.

Given this analysis:

If there are open circles at the endpoints of the intervals where they should be, and the closed circles at endpoints where they are included, then the graph is correctly representing the function.
If the graph fails to represent these correctly (for example, it has a closed circle at \(x = 3\) when it should be open), then it would be incorrect.

Based on the above reasoning:

Correct Response:

No. There should be an open circle on the first point of two parts of the function (specifically at \(x = 3\)).

This indicates that the graph does not accurately represent the piecewise function, as the discontinuities aren't depicted correctly at the required points.