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)

Question

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.

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.

Answer 1

To determine whether the graph accurately represents the piecewise function, we need to analyze the function's definition:

1. For \(1 \leq x \leq 3\): \(f(x) = 0.5\) - This means the dosage is constant at 0.5 ounces for ages 1 through 3, inclusive.
2. For \(3 < x \leq 6\): \(f(x) = 1.0\) - This means the dosage is constant at 1.0 ounces for ages greater than 3 and up to 6, inclusive.
3. For \(x > 6\): \(f(x) = 0.5x\) - This indicates that the dosage is a linear function for ages greater than 6.

Based on the piecewise definition:

• At \(x = 3\), the function transitions from \(0.5\) to \(1.0\). Thus, there should be a closed circle at \(f(3) = 0.5\) and a closed circle at \(f(3) = 1.0\) for \(3 < x \leq 6\).
• At \(x = 6\), there should also be a closed circle since \(f(6) = 1.0\).

Therefore, if the graph has correct endpoints marked with closed circles and properly depicts the transitions between segments, then:

Response evaluation:

• No. There should be an open circle on the first point of two parts of the function. - Incorrect. Closed circles should be present at points of continuity.
• Yes. The graph is correct because all three pieces of the function are present. - This statement only holds if the graph correctly represents transitions and closed/open circle placements.
• Yes, because all of the points at the ends of the pieces of the function should have closed circles. - Correct, provided the graph has the correct circles to indicate continuity.
• No. The graph should be continuous, with all the pieces connected. - Incorrect, as the function is piecewise and includes gaps.

In summary, if the graph includes the described points with closed circles where necessary and accurate line segments, the correct response is:

Yes, because all of the points at the ends of the pieces of the function should have closed circles.