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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.

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

1 answer

To determine whether the graph accurately represents the given piecewise function, it is essential to analyze the conditions for each piece of the function:

  • For \( f(x) = 0.5 \) if \( 1 \leq x \leq 3 \): This indicates that at \( x = 1 \) and \( x = 3 \), the function values should be closed circles because they are included in the interval.
  • For \( f(x) = 1.0 \) if \( 3 < x \leq 6 \): This means that at \( x = 3 \), there should be an open circle since it's not included, and at \( x = 6 \), it should be a closed circle.
  • For \( f(x) = 0.5x \) if \( x > 6 \): This segment begins at \( x = 6 \), which means it should also be a closed circle at this point.

Given this analysis:

  • There should be an open circle at \( x = 3 \).
  • The graph should reflect the correct closed circles where appropriate.

Based on the provided options, the correct choice is:

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