Use the image to answer the question.

Question

Three line segments are graphed on a coordinate plane. The x-axis ranges from 0 to 10 in increments of 0.5. The y-axis ranges from 0 to 6 in increments of 0.5.

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

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.

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

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

Answer 1

To determine whether the graph accurately represents the piecewise function given, we need to analyze the specific points and segments defined by the function.

The piecewise function is defined as follows:

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

From this definition, we can break down the intervals:

1. For \( 1 \leq x \leq 3 \): The output is a constant \( 0.5 \).
2. For \( 3 < x \leq 6 \): The output is a constant \( 1.0 \), starting just after 3 and going up to 6.
3. For \( x > 6 \): The output is a linear function \( 0.5x \), which is continuous for all \( x > 6 \).

Analysis of the options:

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

   • This could be true for \( f(x) \) at \( x = 3 \) as it transitions from 0.5 to 1.0, but we need to consider the exact representation.

2. Yes. The graph is correct because all three pieces of the function are present.

   • This could be true if the graph accurately depicts the function segments specified.

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

   • This is false because the piecewise nature indicates discontinuities specifically where the segments transition from one value to the next.

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

   • This is appropriate for the endpoints of each defined segment, particularly for \( x = 3 \) (closed at \( 0.5 \) and open at \( 1.0 \)) and \( x = 6 \) (closed at \( 1.0 \)).

Conclusion:

Given that:

• There should be an open circle at \( x = 3 \) for \( f(x) \) transitioning from \( 0.5 \) to \( 1.0 \),
• It does not require the graph to be continuous, as it is piecewise,
• Points must be closed at certain ends, following their definitions.

The most accurate response would likely be: No. There should be an open circle on the first point of two parts of the function.

However, if the points are correctly represented as per the piecewise definitions without any visual indication of open and closed circles, then you might lean towards: Yes. The graph is correct because all three pieces of the function are present.

Final Decision: The first choice about open circles is the one that identifies a critical point.