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

\[ f(x) = \begin{cases} <br />
0.5 & \text{if } 1 \leq x \leq 3 \n<br />
1.0 & \text{if } 3 \ \textless \ x \leq 6 \n<br />
0.5x & \text{if } x \ \textgreater \ 6 <br />
\end{cases} \]

Does the graph accurately represent the piecewise function? Why or why not?

A. Yes, because all of the points at the ends of the pieces of the function should have closed circles.
B. Yes, the graph is correct because all three pieces of the function are present.
C. No, there should be an open circle on the first point of two parts of the function.
D. No, the graph should be continuous, with all the pieces connected.

Answer 1

To determine whether the graph accurately represents the piecewise function provided, we need to analyze the defined intervals for the function and how they are represented graphically:

1. For \(1 \leq x \leq 3\), the function is constant at \(f(x) = 0.5\). Since the endpoints 1 and 3 are included in this interval, these points should have closed circles on the graph.

2. For \(3 < x \leq 6\), the function is constant at \(f(x) = 1.0\). Here, the endpoint 6 is included, meaning there should be a closed circle at this point, but the point at \(x = 3\) should have an open circle because it is not included in this part of the function.

3. For \(x > 6\), \(f(x) = 0.5x\), which extends beyond \(x = 6\). This function is continuous for all \(x > 6\).

Based on this analysis, we can conclude the following:

• The graph should have closed circles at \(x = 1\), \(x = 3\) (open circle), and \(x = 6\).
• The graph is not continuous at \(x = 3\), and it should not connect the two segments at this point.

Given this understanding, the correct answer would be:

C. No, there should be an open circle on the first point of two parts of the function.

This option accurately reflects the need for an open circle at \(x = 3\) where the function transitions from the second piece to the third piece without being defined at \(x = 3\) for the second part.