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

(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)

GPT-4o mini · Answer

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.