Question

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 (10,22.5) left parenthesis 10 comma 22.5 right parenthesis (0,0) left parenthesis 0 comma 0 right parenthesis (10,25) left parenthesis 10 comma 25 right parenthesis (1,2.5)

Answer 1

To analyze the piecewise function given:

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

We need to find which point on the graph where the function changes from one rule to the other would be represented by an open circle.

The function changes at \( x = 10 \). For \( x \) values less than 10, the function follows the first rule, \( f(x) = 2.5x \). At \( x = 10 \):

- When we evaluate from the first piece, \( f(10) \) using \( 2.5x \):
\[ f(10) = 2.5 \times 10 = 25 \]

- When we evaluate from the second piece, \( f(10) \) using \( 2.25x \):
\[ f(10) = 2.25 \times 10 = 22.5 \]

The value at \( x = 10 \) is \( 25 \) from the first piece (for \( x < 10 \)) and \( 22.5 \) from the second piece (for \( x \geq 10 \)). Therefore, the function does not include the point \( (10, 25) \) since it only applies for \( x < 10 \).

Thus, the graph will have an open circle at the point \( (10, 25) \) because the function value does not actually reach that point; it approaches it from the left side.

The correct answer is:
**(10, 25)**