Use the image to answer the question.

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)
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
𝑓
(
𝑥
)
=
{
0.5

if

1
≤
𝑥
≤
3
1.0

if

3
<
𝑥
≤
6
0.5
𝑥

if

𝑥
>
6

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

(1 point)
Responses

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.

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 t

Answer 1

To determine if the graph accurately represents the piecewise function, we must analyze the stated conditions and how the graph would illustrate them.

Given Piecewise Function:

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

Key Points to Consider for the Graph:

1. For the first piece \( f(x) = 0.5 \) on the interval \( 1 \leq x \leq 3 \), there should be solid points (closed circles) at \( x = 1 \) and \( x = 3 \), indicating that these values are included in the function.

2. For the second piece \( f(x) = 1.0 \) on the interval \( 3 < x \leq 6 \), there should be an open circle at \( x = 3 \) (as it is not included) and a closed circle at \( x = 6 \) (as it is included).

3. For the third piece \( f(x) = 0.5x \) for \( x > 6 \), the graph should continue from the closed circle at \( x = 6 \) without any breaks and show a linear increase based on the formula.

Conclusion: If the graph contains the correct characteristics for the transitions at \( x = 3 \) and \( x = 6 \) based on the piecewise function, then it accurately represents the function.

The best answer choice would therefore be:

Yes, because all of the points at the ends of the pieces of the function should have closed circles.

This response indicates that the endpoints of the intervals are represented correctly as closed circles where they include the values and open circles where they do not.