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A coordinate plane's axes range from negative 10 to 10 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.
Determine the graph's correct piecewise function.
Option #1: f(x)={2x+1, x>1−x+3, x<1
Option #2: f(x)={2x+1, x>1−x+3, x≤1
Option #3: f(x)={−2x+1, x>1x+3, x≤1
(1 point)
Option #
is the graph's correct piecewise function.
A coordinate plane's axes range from negative 10 to 10 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.
Determine the graph's correct piecewise function.
Option #1: f(x)={2x+1, x>1−x+3, x<1
Option #2: f(x)={2x+1, x>1−x+3, x≤1
Option #3: f(x)={−2x+1, x>1x+3, x≤1
(1 point)
Option #
is the graph's correct piecewise function.
Answers
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Answered by
GPT-4o mini
To determine which option represents the graph's correct piecewise function, let's analyze the features of the rays as described.
From your description:
- The first ray starts from a closed endpoint, suggesting it should be included in the function for that endpoint (i.e., use ≥ or ≤).
- The second ray starts from an open endpoint, indicating that this endpoint should not be included (i.e., use > or <).
Given this information, we can interpret the provided options:
1. **Option #1:** \( f(x) = \begin{cases} 2x + 1 & x > 1 \\ -x + 3 & x < 1 \end{cases} \)
- This option uses > for the first branch, which is incorrect because the ray starts from a closed endpoint.
2. **Option #2:** \( f(x) = \begin{cases} 2x + 1 & x > 1 \\ -x + 3 & x \leq 1 \end{cases} \)
- This option correctly uses ≥ for the second branch, indicating that the closed endpoint is included in this piece of the function.
3. **Option #3:** \( f(x) = \begin{cases} -2x + 1 & x > 1 \\ x + 3 & x \leq 1 \end{cases} \)
- This option does not match the equations provided and also switches the rays.
Based on the requirement that the function includes the closed endpoint at \( x = 1 \) and the open endpoint for \( x > 1 \), **Option #2** is the correct piecewise function.
So, the answer is: **Option #2.**
From your description:
- The first ray starts from a closed endpoint, suggesting it should be included in the function for that endpoint (i.e., use ≥ or ≤).
- The second ray starts from an open endpoint, indicating that this endpoint should not be included (i.e., use > or <).
Given this information, we can interpret the provided options:
1. **Option #1:** \( f(x) = \begin{cases} 2x + 1 & x > 1 \\ -x + 3 & x < 1 \end{cases} \)
- This option uses > for the first branch, which is incorrect because the ray starts from a closed endpoint.
2. **Option #2:** \( f(x) = \begin{cases} 2x + 1 & x > 1 \\ -x + 3 & x \leq 1 \end{cases} \)
- This option correctly uses ≥ for the second branch, indicating that the closed endpoint is included in this piece of the function.
3. **Option #3:** \( f(x) = \begin{cases} -2x + 1 & x > 1 \\ x + 3 & x \leq 1 \end{cases} \)
- This option does not match the equations provided and also switches the rays.
Based on the requirement that the function includes the closed endpoint at \( x = 1 \) and the open endpoint for \( x > 1 \), **Option #2** is the correct piecewise function.
So, the answer is: **Option #2.**
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