Use the image to answer the question.

Question

Three line segments are graphed on a coordinate plane. The x-axis ranges from 0 to 10 in increments of 0.5. The y-axis ranges from 0 to 6 in increments of 0.5.

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

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.

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.
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Answer 1

To determine whether the graph accurately represents the piecewise function given the defined intervals and conditions, let's analyze the function:

1. The function pieces are defined as follows:
   • For 1 ≤ x ≤ 3: f(x) = 0.5 (This includes the endpoints, so the points at x = 1 and x = 3 should have closed circles.)
   • For 3 < x ≤ 6: f(x) = 1.0 (This includes the endpoint at x = 6, so x = 6 should have a closed circle, but x = 3 should have an open circle since it is not included in this interval.)
   • For x > 6: f(x) = 0.5x (This function starts at x = 6, so x = 6 should be included (closed circle).)

Based on this analysis, we can evaluate the responses:

• Yes, because all of the points at the ends of the pieces of the function should have closed circles. (Incorrect: The point at x = 3 should have an open circle)

• No. The graph should be continuous, with all the pieces connected. (Incorrect: The function is piecewise so it does not need to be continuous.)

• No. There should be an open circle on the first point of two parts of the function. (Correct: The point at x = 3 should have an open circle.)

• Yes. The graph is correct because all three pieces of the function are present. (Incorrect: The endpoint at x = 3 should have an open circle, which would make this response not valid.)

The best response is: No. There should be an open circle on the first point of two parts of the function.