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

(10,25)
(
10
,
25
)
left parenthesis 10 comma 25 right parenthesis

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

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

(10,22.5)

Answer 1

To determine which point would have an open circle in the graph of the piecewise function, we need to look at the transition between the two pieces of the function at \(x = 10\).

The piecewise function is defined as follows:

\[ f(x) = \begin{cases} 2.5x & \text{if } x < 10 \ 2.25x & \text{if } x \geq 10 \end{cases} \]

At \(x = 10\):

• For the condition \(x < 10\), we calculate \(f(10)\) using the formula \(2.5x\): \[ f(10) = 2.5 \times 10 = 25 \]

• For the condition \(x \geq 10\), we calculate \(f(10)\) using the formula \(2.25x\): \[ f(10) = 2.25 \times 10 = 22.5 \]

Thus, at \(x = 10\), \(f(x)\) takes on the value \(25\) from the first piece (when \(x < 10\)) and \(22.5\) from the second piece (when \(x \geq 10\)). This indicates that:

• The function approaches \(25\) as \(x\) approaches \(10\) from the left (open circle).
• The function equals \(22.5\) when \(x = 10\) (solid circle).

Thus, the point that would have an open circle is \((10, 25)\).

The answer is:

(10, 25)