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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, 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.

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.

1 answer

To determine whether the graph accurately represents the given piecewise function for Children's cough syrup dosage based on age, we need to analyze the conditions specified in the function:

  1. For \(1 \leq x \leq 3\), the dosage is \(f(x) = 0.5\). This means we need a closed circle at \(x=1\) and a closed circle at \(x=3\), since both endpoints are included in this interval.

  2. For \(3 < x \leq 6\), the dosage is \(f(x) = 1.0\). This indicates that there would be an open circle at \(x=3\) (because 3 is not included in this part) and a closed circle at \(x=6\).

  3. For \(x > 6\), the function is defined as \(f(x) = 0.5x\). This means there will be no restrictions on the beginning of this piece, so it will start with an open circle right after \(x=6\).

Based on this analysis, the graph should have:

  • A closed circle at \(x=1\) (for \(f(x) = 0.5\)).
  • A closed circle at \(x=3\) (for \(f(x) = 0.5\)).
  • An open circle at \(x=3\) (before being defined to start from \(3 < x\)).
  • A closed circle at \(x=6\) (for the end of the second piece).
  • An open circle at \(x=6\) for the beginning of the next piece which is \(f(x) = 0.5x\).

Based on this reasoning, the correct response is:

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