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 (10,25) left parenthesis 10 comma 25 right parenthesis (1,2.5) left parenthesis 1 comma 2.5 right parenthesis (10,22.5) left parenthesis 10 comma 22.5 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 what's happening at the boundary point \(x = 10\).

1. For \(x < 10\): The function \(f(x) = 2.5x\).
2. For \(x \geq 10\): The function \(f(x) = 2.25x\).

Let's check the value of the function at \(x = 10\) from both sides:

• As \(x\) approaches \(10\) from the left (\(x < 10\)), we use \(f(x) = 2.5x\): \[ f(10) = 2.5 \times 10 = 25 \]

• At \(x = 10\) itself (and for values greater than 10), we use \(f(x) = 2.25x\): \[ f(10) = 2.25 \times 10 = 22.5 \]

This means:

• The value of \(f(x)\) approaches 25 as \(x\) approaches 10 from the left.
• The value of \(f(x)\) at \(x = 10\) is 22.5.

Since the function value at \(x = 10\) from the left side (25) does not equal the value directly at \(x = 10\) (22.5), we place an open circle at the point that corresponds to the limit from the left, which is \((10, 25)\).

Therefore, the point that would have an open circle is:

\((10, 25)\).