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)

Question

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.

No. The graph should be continuous, with all the pieces connected.
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. 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.

Answer 1

To determine if the graph accurately represents the piecewise function, let's analyze the function and its domain segments.

The piecewise function provided is:

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

Now, let's examine the characteristics of the graph based on these segments:

1. For the segment \( 1 \leq x \leq 3 \) where \( f(x) = 0.5 \), the value is a constant. At \( x = 1 \) and \( x = 3 \), we include these points with closed circles.

2. For \( 3 < x \leq 6 \) where \( f(x) = 1.0 \), at \( x = 3 \), there should be an open circle (since 3 is not included in this segment), and a closed circle at \( x = 6 \) since it’s included.

3. For \( x > 6 \), \( f(x) = 0.5x \), this function increases as \( x \) increases, starting from the point \( (6, 3) \) (when plugging \( x = 6 \) into \( f(x) \)) which would have a closed circle.

Based on this analysis, the correct response would be:

No. There should be an open circle at the first point of the second part of the function (at \( x = 3 \)).

This choice correctly acknowledges that at \( x = 3 \), \( f(x) \) switches from one defined segment to another, and this point should be represented with an open circle in the graph.