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

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

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

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

(0,0)

Answer 1

To determine which point would have an open circle on the graph of the piecewise function

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

we need to examine the behavior of the function at the point where the two pieces change, which is at \( x = 10 \).

For \( x < 10 \), the function is defined as \( f(x) = 2.5x \). When we plug in \( x = 10 \):

\[
f(10) = 2.5(10) = 25
\]

For \( x \geq 10 \), the function changes to \( f(x) = 2.25x \). At \( x = 10 \):

\[
f(10) = 2.25(10) = 22.5
\]

The value of \( f(x) \) at \( x = 10 \) from the first piece is 25, but from the second piece, it is 22.5. This means that at \( x = 10 \), the function is not defined as \( 25 \) (the value it would take if it follows the first part for \( x < 10 \)) but instead takes the value \( 22.5 \) from the second part.

Thus, there should be an open circle at the point \( (10, 25) \), as the function does not actually reach this value at \( x = 10\) — it jumps down to a value of \( 22.5 \).

So the answer is:
- \( (10, 25) \) would have an open circle.