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

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

GPT-4o mini · Answer

To determine if the graph accurately represents the given piecewise function, we need to analyze the specified intervals and the nature of the end points:

Piece 1: f(x) = 0.5 for \( 1 \leq x \leq 3 \)
- This interval includes both endpoints \( x = 1 \) and \( x = 3 \), indicating that the function value should be closed at both points (filled circles).
Piece 2: f(x) = 1.0 for \( 3 < x \leq 6 \)
- This interval does not include the left endpoint \( x = 3 \) (open circle) but does include the right endpoint \( x = 6 \) (closed circle).
Piece 3: f(x) = 0.5x for \( x > 6 \)
- This interval does not include the lower endpoint, so it would have an open circle at \( x = 6 \).

Therefore, when constructing the graph for this piecewise function:

At \( x = 1 \), there should be a closed circle representing that f(1) = 0.5.
At \( x = 3 \), there should be a closed circle at the end of the first piece but an open circle indicating that it does not include that value for the second piece.
At \( x = 6 \), there should be a closed circle for the second piece but an open circle at the start of the third piece.

Taking all of this into account, the correct response to whether the graph accurately represents the piecewise function is:

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