A gas station is offering a deal. If you buy 10 gallons or more of gas, you will be charged $2.25 per gallon instead of the regular $2.50 per gallon. The deal is modeled by the following piecewise function.

Question

f(x)={2.5x if x<102.25x if x≥10

If you were to create a graph of this piecewise function, which point would have an open circle?

(1 point)
Responses

(1,2.5)
left parenthesis 1 comma 2.5 right parenthesis

(10,22.5)
left parenthesis 10 comma 22.5 right parenthesis

(0,0)
left parenthesis 0 comma 0 right parenthesis

(10,25)

Answer 1

In the given piecewise function, the function is defined as follows:

• \( f(x) = 2.5x \) for \( x < 10 \)
• \( f(x) = 2.25x \) for \( x \geq 10 \)

To find the point that will have an open circle on the graph, we need to identify the value at \( x = 10 \).

When \( x = 10 \):

• For \( x < 10 \), using the first piece of the function, \( f(10) \) is not included, so we calculate: \[ f(10) = 2.5 \times 10 = 25 \]
• For \( x \geq 10 \), using the second piece of the function: \[ f(10) = 2.25 \times 10 = 22.5 \]

At \( x = 10 \), the function transitions from one piece to another. Therefore, we have:

• An open circle at the point where \( f(10) = 25 \) (from the first piece) because it's not included in that piece.
• A closed circle at the point where \( f(10) = 22.5 \) (from the second piece) since this value is included.

Thus, the point that has an open circle is:

(10, 25).

Answer 2

Use the image to answer the 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

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.

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

To determine whether the graph accurately represents the piecewise function provided in the dosage guidelines, let's break down the function:

1. Function Definition:

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

2. Endpoints and Open/Closed Circles:

   • At \( x = 3 \): The first part \( f(x) = 0.5 \) should have a closed circle at (3, 0.5) since 3 is included. The second part \( f(x) = 1.0 \) should have an open circle at (3, 1.0) since 3 is not included in that part.
   • At \( x = 6 \): The second part \( f(x) = 1.0 \) should have a closed circle at (6, 1.0) since 6 is included. The third part \( f(x) = 0.5x \) should start from an open circle at (6, 3.0) since 6 is included in the second part but not in the definition of the third part.

3. Analysis of the Responses:

   • “Yes. The graph is correct because all three pieces of the function are present.” This could be true, but only if the representation of the endpoints is also correct.
   • “No. There should be an open circle on the first point of two parts of the function.” This statement could be accurate as the transitions at \( x=3 \) and \( x=6 \) need to respect the open/closed circle rules. Specifically, there should be an open circle on (3, 1.0) because that point is not included in that interval.
   • “Yes, because all of the points at the ends of the pieces of the function should have closed circles.” This is incorrect as the transitions at some points require open circles.
   • “No. The graph should be continuous, with all the pieces connected.” This is also incorrect as the function itself is a piecewise function and does not require continuity across different pieces.

Given this breakdown, the best assessment is:

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

This acknowledges that the graph should correctly represent the relationship between the parts of the function and their boundaries.