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.

Question

f(x) =
0.5 if 1 < x ≤3
1.0 if 3 < x ≤ 6
0.5x if x > 6
Does the graph accurately represent the piecewise function? Why or why not?
(1 point)
• No. The graph should be continuous, with all the pieces connected.
• 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.
• No. There should be an open circle on the first point of two parts of the function.

Answer 1

To determine whether the graph accurately represents the given piecewise function, we should analyze the function:

1. For \(1 < x \leq 3\): The dosage is a constant \(0.5\). This segment of the function should show a horizontal line at \(y = 0.5\) from just above \(x = 1\) to \(x = 3\) including \(3\) (closed circle).

2. For \(3 < x \leq 6\): The dosage here is \(1.0\). This segment also shows a horizontal line at \(y = 1.0\) from just above \(x = 3\) to \(x = 6\) including \(6\) (closed circle).

3. For \(x > 6\): The formula is \(0.5x\), which is a linear function. This would mean the graph increases from the point where \(x > 6\) (open circle at \(6\) since this part does not include \(x = 6\)).

Now, let's analyze the options provided regarding the accuracy of the graph representing the piecewise function:

• No. The graph should be continuous, with all the pieces connected. This is not correct as piecewise functions can have breaks.

• Yes. The graph is correct because all three pieces of the function are present. This might not necessarily be true if the connection between segments (especially regarding closed/open circles) is not accurate.

• Yes, because all of the points at the ends of the pieces of the function should have closed circles. This is inaccurate. The boundaries at \(x = 3\) and \(x = 6\) are treated differently (closed at \(3\), open at \(6\)).

• No. There should be an open circle on the first point of two parts of the function. This seems most accurate. There should indeed be an open circle at \(x=3\) where the first piece ends and the second begins, indicating the transition from the first piece to the second piece.

Therefore, the best answer is:

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