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
𝑓
(
𝑥
)
=
{
2.5
𝑥

if

𝑥
<
10
2.25
𝑥

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

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

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

(10,25)

Answer 1

In the given piecewise function, we have:

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

At \( x = 10 \), we need to determine the value of the function from both sides.

1. When \( x = 10 \), using the second part of the piecewise function for \( x \geq 10 \): \[ f(10) = 2.25 \times 10 = 22.5 \]

2. Because we have a change in the function formula at \( x = 10 \), it is defined at that point, and the value from the first part, \( f(x) = 2.5x \), for \( x < 10 \): \[ f(10) \text{ (for the first part)} = 2.5 \times 10 = 25 \]

Since the first part is not applicable at \( x = 10\) (as that is the threshold for the second piece), the value at \( x = 10 \) will be determined by \( f(x) = 2.25x \), which gives \( 22.5\).

Since the definition of the function changes at \( x = 10 \), and from the first piece \( f(x) = 2.5x \) does not included \( (10, 25) \), this point will have an open circle.

Thus, the point that would have an open circle is:

(10, 25)