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
𝑓
(
𝑥
)
=
{
2.5
𝑥
if
𝑥
<
10
2.25
𝑥
if
𝑥
≥
10
If you were to create a graph of this piecewise function, which point would have an open circle?
f(x)={2.5x if x<102.25x if x≥10
𝑓
(
𝑥
)
=
{
2.5
𝑥
if
𝑥
<
10
2.25
𝑥
if
𝑥
≥
10
If you were to create a graph of this piecewise function, which point would have an open circle?
Answers
GPT-4o mini
To determine which point would have an open circle on the graph of the piecewise function, we need to examine the point where the two pieces of the function meet, which is at \( x = 10 \).
The piecewise function is defined as follows:
\[
f(x) =
\begin{cases}
2.5x & \text{if } x < 10 \\
2.25x & \text{if } x \geq 10
\end{cases}
\]
1. For \( x < 10 \), the function follows the rule \( f(x) = 2.5x \). Evaluating this at \( x = 10 \):
\[
f(10) = 2.5 \times 10 = 25
\]
2. For \( x \geq 10 \), the function follows the rule \( f(x) = 2.25x \). Evaluating this at \( x = 10 \):
\[
f(10) = 2.25 \times 10 = 22.5
\]
At \( x = 10 \), the function has two different values depending on whether you approach from the left or the right:
- From the left (using \( f(x) = 2.5x \)), \( f(10) \) approaches 25.
- From the right or at exactly \( x = 10 \) (using \( f(x) = 2.25x \)), \( f(10) \) equals 22.5.
Since the value at \( x = 10 \) from the left does not equal the value from the right, the function is not continuous at that point. This means that at \( x = 10 \), we express that \( f(10) = 22.5 \) with a closed circle (indicating it belongs to the function), whereas the limit from the left \( f(10) = 25 \) would need an open circle to indicate it does not actually take this value.
Thus, the open circle would be at the point \( (10, 25) \), while the closed circle would be at \( (10, 22.5) \).
In summary, the point that would have an open circle on the graph is:
**(10, 25)**.
The piecewise function is defined as follows:
\[
f(x) =
\begin{cases}
2.5x & \text{if } x < 10 \\
2.25x & \text{if } x \geq 10
\end{cases}
\]
1. For \( x < 10 \), the function follows the rule \( f(x) = 2.5x \). Evaluating this at \( x = 10 \):
\[
f(10) = 2.5 \times 10 = 25
\]
2. For \( x \geq 10 \), the function follows the rule \( f(x) = 2.25x \). Evaluating this at \( x = 10 \):
\[
f(10) = 2.25 \times 10 = 22.5
\]
At \( x = 10 \), the function has two different values depending on whether you approach from the left or the right:
- From the left (using \( f(x) = 2.5x \)), \( f(10) \) approaches 25.
- From the right or at exactly \( x = 10 \) (using \( f(x) = 2.25x \)), \( f(10) \) equals 22.5.
Since the value at \( x = 10 \) from the left does not equal the value from the right, the function is not continuous at that point. This means that at \( x = 10 \), we express that \( f(10) = 22.5 \) with a closed circle (indicating it belongs to the function), whereas the limit from the left \( f(10) = 25 \) would need an open circle to indicate it does not actually take this value.
Thus, the open circle would be at the point \( (10, 25) \), while the closed circle would be at \( (10, 22.5) \).
In summary, the point that would have an open circle on the graph is:
**(10, 25)**.