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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. 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.
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.
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Answer 1

To determine whether the graph accurately represents the given piecewise function for the dosage of children's cough syrup, we should analyze the conditions stated in the piecewise definition and their implications for the graph.

The piecewise function is defined as follows:

1. \( f(x) = 0.5 \) for \( 1 \leq x \leq 3 \)
2. \( f(x) = 1.0 \) for \( 3 < x \leq 6 \)
3. \( f(x) = 0.5x \) for \( x > 6 \)

Analysis of Each Piece:

• For \( 1 \leq x \leq 3 \), the function has a constant value of 0.5. The endpoints (1 and 3) should be closed circles indicating that the function includes these values.
• For \( 3 < x \leq 6 \), the function has a constant value of 1.0. Again, the endpoint at 6 should be a closed circle (since it includes this value), while at 3, an open circle should be used to indicate it's not included in this interval.
• For \( x > 6 \), the function follows the equation \( f(x) = 0.5x \), which means the values will rise linearly, starting from the point (6, 3) to the right.

Points of Interest:

• At \( x = 3 \): Since \( f(x) = 1.0 \) for \( 3 < x \leq 6 \), there should be an open circle at (3, 0.5) (not included) and a closed circle at (3, 1.0) for the next piece.
• At \( x = 6 \): There should be a closed circle at (6, 1.0) and also a point (6, 3) as it starts in the next piece.

Conclusion:

Given this information, we can conclude the following:

1. Yes, the graph should contain all three pieces of the function.
2. There should be an open circle at the point corresponding to \( x = 3 \) to indicate the value of 0.5 is not included there.
3. The graph is not continuous at \( x = 3 \) due to the nature of the piecewise definition.

Therefore, based on the options provided:

Correct Response:

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