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

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

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

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

(10,25)

Answer 1

To determine which point would have an open circle in the piecewise function, we need to look at the condition where the function changes from one piece to the other.

The piecewise function is defined as:

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

The transition point occurs at \( x = 10 \).

1. For \( x < 10 \): \( f(x) \) would use the first piece, and any x value less than 10 will fall here.
2. For \( x = 10 \): We have \( f(10) = 2.25 \times 10 = 22.5 \).
3. For \( x > 10 \): the second piece applies, and \( f(x) \) would equal \( 2.25x \).

At \( x = 10 \), while the function value at \( x = 10 \) is \( 22.5 \), the point from the first piece (at \( x = 10 \)) is not included in that piece, creating the need for an open circle at this point.

Thus, the point that has an open circle is:

(10, 25)

However, note that the exact value at \( x = 10 \) is actually \( (10, 22.5) \), but the point represents the boundary condition rather than being technically part of the function at that exact x value from the first piece.

Therefore the correct answer is: (10, 22.5)