Question

Use the image to answer the question.

Two line segments are graphed on a coordinate plane, one ending in an open point. The x-axis ranges from 0 to 10 in increments of 1. The y-axis ranges from 0 to 20 in increments of 1.

A store is having a sale on boxes of cereal. In the following piecewise function, the independent variable represents the number of boxes purchased. If fewer than five boxes are purchased, the cost per box is $3.50. If five or more boxes are purchased, the cost per box is only $2.25.

f(x)={3.50x if 1≤x<52.25x if x≥5

Does the graph accurately represent the situation? Why or why not?

(1 point)
Responses

No. It doesn’t pass the Vertical Line Test.
No. It doesn’t pass the Vertical Line Test.

No. The graph should be a single, continuous function.
No. The graph should be a single, continuous function.

No. All the endpoints should have closed circles.
No. All the endpoints should have closed circles.

Yes. The pieces of the function are graphed correctly and the open circle is placed correctly.

Answer 1

[AI-generated answer — Bot GPT-4 Turbo]

I cannot view the image you are referring to, as this is a text-based interface. However, I can help you analyze the piecewise function and understand if a graph representing it would be accurate or not.

Given:

f(x) = {
3.50x if 1 ≤ x < 5,
2.25x if x ≥ 5
}

The piecewise function describes the cost of x boxes of cereal depending on the amount purchased. If fewer than five boxes are purchased, the cost per box is $3.50. If five or more boxes are purchased, the cost per box is $2.25.

From this description, here's what a correct graph should look like:

1. For 1 ≤ x < 5, the graph should be a straight line with a slope of 3.50, starting from x = 1. Since the function does not apply to x < 1, the line segment should begin with a closed circle at x = 1 to indicate that the function's value at x = 1 is included in this piece.
2. There should be an open circle at x = 5 for the first piece of the function because the cost changes at this quantity, meaning the point where x = 5 is not included in this first segment and the cost is not $3.50 when x = 5.
3. Starting at x = 5, the graph should continue with a different straight line segment with a slope of 2.25. There should be a closed circle at the starting point of this second segment (x = 5) to indicate that the cost of $2.25 per box starts at this quantity (including when x = 5).
4. The graph should not be a single continuous function because there is a distinct change in cost per box at x = 5.

Based on these details, the choices can be evaluated:

- "No. It doesn’t pass the Vertical Line Test." This is incorrect because the Vertical Line Test is used to determine if a graph is a function or not, and it is not relevant to whether the function is graphed correctly for this context.
- "No. The graph should be a single, continuous function." This is incorrect because the graph should be piecewise, not continuous, as there is a change in the slope at x = 5.
- "No. All the endpoints should have closed circles." This is incorrect because, according to the piecewise function described, the endpoint at x = 5 for the first segment should be open since the price changes at exactly five boxes.
- "Yes. The pieces of the function are graphed correctly and the open circle is placed correctly." Assuming that the graph has been described accurately in the statement, this would be the correct response. The function changes at x = 5 and should have an open circle there for the first piece and a closed circle for the second piece.

Therefore, the correct response would be:

Yes. The pieces of the function are graphed correctly and the open circle is placed correctly.