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

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

If you were to create a graph of this piecewise function, which point would have an open circle?

A. $(10, 25)$
B. $(1, 2.5)$
C. $(0, 0)$
D. $(10, 22.5)$

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 transition at \(x = 10\).

1. For \(x < 10\), the function is defined as \(f(x) = 2.5x\).

   • At \(x = 10\), we can calculate \(f(10)\) using the first piece: \[ f(10) = 2.5(10) = 25. \] However, since the first case is defined only for \(x < 10\), the point \( (10, 25) \) is not included in this part of the function (indicated by an open circle).

2. For \(x \geq 10\), the function is defined as \(f(x) = 2.25x\).

   • At \(x = 10\), we can calculate: \[ f(10) = 2.25(10) = 22.5. \] This point (10, 22.5) is included in the second piece because it applies for \(x \geq 10\) and thus has a closed point.

Now, looking at the options given:

A. \( (10, 25) \) - This point corresponds to \(f(10)\) from the first case which is not defined for \(x = 10\). Hence, it should have an open circle.

B. \( (1, 2.5) \) - This point is defined by the first case, so it would be a closed point.

C. \( (0, 0) \) - This point is also defined by the first case, so it would be a closed point.

D. \( (10, 22.5) \) - This point is defined by the second case and is a closed point.

The point that would have an open circle is:

A. \( (10, 25) \).