Question

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.

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

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

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

(0,0)

Answer 1

To find the point that would have an open circle on the graph of the piecewise function \( f(x) \), we need to look at where the function changes from one piece to another.

According to the piecewise function:

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

At \( x = 10 \), the function transitions from the first piece \( 2.5x \) (for \( x < 10 \)) to \( 2.25x \) (for \( x \geq 10 \)).

To determine the value of the function at \( x = 10 \):

- From the first part \( f(10) \) with \( x = 10 \) gives \( f(10) = 2.5 \times 10 = 25 \).
- From the second part \( f(10) \) gives \( f(10) = 2.25 \times 10 = 22.5 \).

Since the function changes at \( x = 10 \), the point \( (10, 25) \) would not be included in the graph (hence, the open circle), while \( (10, 22.5) \) would be included (solid circle) since \( f(10) = 22.5 \).

**Thus, the point that has an open circle is** \( (10, 25) \).