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

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

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

(1,2.5)

GPT-4o mini · Answer

To determine which point would have an open circle on the graph of the piecewise function, we look closely at the definition of the 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 changes from $2.5x$ (for $x < 10$) to $2.25x$ (for $x \geq 10$). To find the value of $f(10)$:

For $x = 10$, if we use the first piece ($x < 10$): \[ f(10) = 2.5 \times 10 = 25 \]
For $x = 10$, if we use the second piece ($x \geq 10$): \[ f(10) = 2.25 \times 10 = 22.5 \]

Since the value at $x = 10$ from the first piece is $25$ and from the second piece is $22.5$, the function does not take the value of $25$ at $x = 10$ because it is where the function transitions to a different expression.

Thus, the graph would show an open circle at the point $(10, 25)$ indicating that $f(10)$ does not equal $25$, while it would have a closed circle at $(10, 22.5)$ since it does take that value.

Therefore, the correct answer is:

(10, 25)

So the answer is: (10, 25)