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. 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.

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, 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 the ends of the pieces of the function should have closed circles.

Answer 1

To determine if the graph accurately represents the piecewise function, we need to analyze the function's intervals and check the endpoints.

The piecewise function defined is:

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

Now, let's analyze the points:

1. For \( f(x) = 0.5 \) (from \( x = 1 \) to \( x = 3 \)), the graph should include filled circles at \( (1, 0.5) \) and \( (3, 0.5) \).
2. For \( f(x) = 1.0 \) (from \( x = 3 \) to \( x = 6 \)), the graph starts at \( (3, 1.0) \) with an open circle because 3 is not included in this part, and ends at \( (6, 1.0) \) with a filled circle.
3. For \( f(x) = 0.5x \) (for \( x > 6 \)), the graph should be continuous starting from the value at \( x = 6 \).

Given this analysis, if there are open circles at points where needed (specifically at 3 for the transition between the second segment and the first segment), then the representation is accurate.

Thus, the correct response is:

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