Question
A coordinate plane's axes range from negative 5 to 5 by 1-unit increments. Two rays are plotted. The first ray starts from a closed endpoint, while the second ray starts from an open endpoint.
What is the piecewise function for the graph?
Option #1:
f(x)={−2x+1, x<023x−3, x≥0
Option #2:
f(x)={5, x≤22x−4, x>2
(1 point)
Option #
The points on the graph are at:
2,5
and 2,0
What is the piecewise function for the graph?
Option #1:
f(x)={−2x+1, x<023x−3, x≥0
Option #2:
f(x)={5, x≤22x−4, x>2
(1 point)
Option #
The points on the graph are at:
2,5
and 2,0
Answers
GPT-4o mini
To determine which piecewise function corresponds to the rays plotted on the coordinate plane, we need to analyze both options given the points provided: (2, 5) and (2, 0).
1. **Analyzing the Points**:
- The first point, (2, 5), suggests that when \( x = 2 \), the function value \( f(2) = 5 \). This point must be included in the graph.
- The second point, (2, 0), suggests that when \( x = 2 \), the function value \( f(2) = 0 \). However, this point has an open endpoint, meaning it should not be included in the graph.
Now, let's look at both options:
**Option #1**:
\[ f(x) = \begin{cases} -2x + 1 & \text{if } x < 0 \\ 3x - 3 & \text{if } x \geq 0 \end{cases} \]
For \( x = 2 \):
- If \( x = 2 \) (since \( 2 \geq 0 \)):
\[ f(2) = 3(2) - 3 = 6 - 3 = 3 \]
This does not match the point (2, 5).
**Option #2**:
\[ f(x) = \begin{cases} 5 & \text{if } x \leq 2 \\ 2x - 4 & \text{if } x > 2 \end{cases} \]
For \( x = 2 \):
- Since \( x = 2 \) (we include it in this piece because of \( x \leq 2 \)):
\[ f(2) = 5 \]
This point (2, 5) matches, and now check for values greater than 2:
- For \( x = 2.1 \):
\[ f(2.1) = 2(2.1) - 4 = 4.2 - 4 = 0.2 \]
This indicates that the function drops slightly when moving away from \( x = 2 \).
**Conclusion**:
Looking at the points provided and the nature of the endpoints:
- The first point \( (2, 5) \) means the endpoint here is closed (included).
- The line \( 2x - 4 \) coming from \( x > 2 \) aligns with the idea that there's an open endpoint at \( (2, 0) \).
Thus, the piecewise function that corresponds to the two rays, with a closed endpoint at (2, 5) and an open endpoint at (2, 0), is **Option #2**.
**Final Answer**: **Option #2**
1. **Analyzing the Points**:
- The first point, (2, 5), suggests that when \( x = 2 \), the function value \( f(2) = 5 \). This point must be included in the graph.
- The second point, (2, 0), suggests that when \( x = 2 \), the function value \( f(2) = 0 \). However, this point has an open endpoint, meaning it should not be included in the graph.
Now, let's look at both options:
**Option #1**:
\[ f(x) = \begin{cases} -2x + 1 & \text{if } x < 0 \\ 3x - 3 & \text{if } x \geq 0 \end{cases} \]
For \( x = 2 \):
- If \( x = 2 \) (since \( 2 \geq 0 \)):
\[ f(2) = 3(2) - 3 = 6 - 3 = 3 \]
This does not match the point (2, 5).
**Option #2**:
\[ f(x) = \begin{cases} 5 & \text{if } x \leq 2 \\ 2x - 4 & \text{if } x > 2 \end{cases} \]
For \( x = 2 \):
- Since \( x = 2 \) (we include it in this piece because of \( x \leq 2 \)):
\[ f(2) = 5 \]
This point (2, 5) matches, and now check for values greater than 2:
- For \( x = 2.1 \):
\[ f(2.1) = 2(2.1) - 4 = 4.2 - 4 = 0.2 \]
This indicates that the function drops slightly when moving away from \( x = 2 \).
**Conclusion**:
Looking at the points provided and the nature of the endpoints:
- The first point \( (2, 5) \) means the endpoint here is closed (included).
- The line \( 2x - 4 \) coming from \( x > 2 \) aligns with the idea that there's an open endpoint at \( (2, 0) \).
Thus, the piecewise function that corresponds to the two rays, with a closed endpoint at (2, 5) and an open endpoint at (2, 0), is **Option #2**.
**Final Answer**: **Option #2**