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

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

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

(10,22.5)

GPT-4o mini · Answer

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

The function is defined as follows:

For $ x < 10 $: $ f(x) = 2.5x $
For $ x \geq 10 $: $ f(x) = 2.25x $

This means that at $ x = 10 $, the function from the first piece ($ f(10) = 2.5 \times 10 = 25 $) does not equal the function from the second piece when you consider the $ \geq $ sign. While $ f(10) = 25 $ comes from the first piece, the second piece starts at $ (10, 22.5) $, effectively meaning that the point $ (10, 25) $ does not actually belong to the second piece.

Therefore, at $ x = 10 $, the function does not equal $ 25 $ from the first piece, so we denote that point with an open circle, and the point $ (10, 22.5) $ from the second piece will be a filled circle.

Thus, the point with an open circle on the graph is:

(10, 25).