Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Join our Q&A platform and connect with professionals ready to provide precise answers to your questions in various areas. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To determine which piecewise relations define a function, we need to confirm that each piecewise relation assigns exactly one output value [tex]\( y \)[/tex] for each input value [tex]\( x \)[/tex]. Let's analyze each given piecewise relation in detail.
1. First Piecewise Relation:
[tex]\[ y = \begin{cases} x^2 & \text{if } x < -2 \\ 0 & \text{if } -2 \le x \le 4 \\ -x^2 & \text{if } x \ge 4 \end{cases} \][/tex]
- For [tex]\( x < -2 \)[/tex], [tex]\( y = x^2 \)[/tex], which is a single-valued function.
- For [tex]\( -2 \le x \le 4 \)[/tex], [tex]\( y = 0 \)[/tex], which is also a single-valued function.
- For [tex]\( x \ge 4 \)[/tex], [tex]\( y = -x^2 \)[/tex], which is again single-valued.
From this examination, each part of the piecewise relation assigns exactly one value of [tex]\( y \)[/tex] for each [tex]\( x \)[/tex]. The first relation therefore defines a function.
2. Second Piecewise Relation:
[tex]\[ y = \begin{cases} x^2 & \text{if } x \le -2 \\ 4 & \text{if } -2 < x \le 2 \\ x^2 + 1 & \text{if } x \ge 2 \end{cases} \][/tex]
- For [tex]\( x \le -2 \)[/tex], [tex]\( y = x^2 \)[/tex], which is a single-valued function.
- For [tex]\( -2 < x \le 2 \)[/tex], [tex]\( y = 4 \)[/tex], which is also a single-valued function.
- For [tex]\( x \ge 2 \)[/tex], [tex]\( y = x^2 + 1 \)[/tex], which is again single-valued.
Thus, each part of this piecewise relation assigns exactly one value of [tex]\( y \)[/tex] for each [tex]\( x \)[/tex]. The second relation also defines a function.
3. Third Piecewise Relation:
[tex]\[ y = \begin{cases} -3x & \text{if } x < -2 \\ 3 & \text{if } 0 \le x < 4 \\ 2x & \text{if } x \ge 4 \end{cases} \][/tex]
- For [tex]\( x < -2 \)[/tex], [tex]\( y = -3x \)[/tex], which is a single-valued function.
- For [tex]\( 0 \le x < 4 \)[/tex], [tex]\( y = 3 \)[/tex], which is also a single-valued function.
- For [tex]\( x \ge 4 \)[/tex], [tex]\( y = 2x \)[/tex], which is again single-valued.
However, this third relation does not cover the interval [tex]\( -2 \le x < 0 \)[/tex]. Specifically, for [tex]\( -2 \le x < 0 \)[/tex], there is no value defined for [tex]\( y \)[/tex]. Because of these gaps, this piecewise relation does not define a function over all real numbers.
Given this analysis, the piecewise relations that define a function are the first and second ones. Therefore, the correct options are [tex]\([1, 2]\)[/tex].
1. First Piecewise Relation:
[tex]\[ y = \begin{cases} x^2 & \text{if } x < -2 \\ 0 & \text{if } -2 \le x \le 4 \\ -x^2 & \text{if } x \ge 4 \end{cases} \][/tex]
- For [tex]\( x < -2 \)[/tex], [tex]\( y = x^2 \)[/tex], which is a single-valued function.
- For [tex]\( -2 \le x \le 4 \)[/tex], [tex]\( y = 0 \)[/tex], which is also a single-valued function.
- For [tex]\( x \ge 4 \)[/tex], [tex]\( y = -x^2 \)[/tex], which is again single-valued.
From this examination, each part of the piecewise relation assigns exactly one value of [tex]\( y \)[/tex] for each [tex]\( x \)[/tex]. The first relation therefore defines a function.
2. Second Piecewise Relation:
[tex]\[ y = \begin{cases} x^2 & \text{if } x \le -2 \\ 4 & \text{if } -2 < x \le 2 \\ x^2 + 1 & \text{if } x \ge 2 \end{cases} \][/tex]
- For [tex]\( x \le -2 \)[/tex], [tex]\( y = x^2 \)[/tex], which is a single-valued function.
- For [tex]\( -2 < x \le 2 \)[/tex], [tex]\( y = 4 \)[/tex], which is also a single-valued function.
- For [tex]\( x \ge 2 \)[/tex], [tex]\( y = x^2 + 1 \)[/tex], which is again single-valued.
Thus, each part of this piecewise relation assigns exactly one value of [tex]\( y \)[/tex] for each [tex]\( x \)[/tex]. The second relation also defines a function.
3. Third Piecewise Relation:
[tex]\[ y = \begin{cases} -3x & \text{if } x < -2 \\ 3 & \text{if } 0 \le x < 4 \\ 2x & \text{if } x \ge 4 \end{cases} \][/tex]
- For [tex]\( x < -2 \)[/tex], [tex]\( y = -3x \)[/tex], which is a single-valued function.
- For [tex]\( 0 \le x < 4 \)[/tex], [tex]\( y = 3 \)[/tex], which is also a single-valued function.
- For [tex]\( x \ge 4 \)[/tex], [tex]\( y = 2x \)[/tex], which is again single-valued.
However, this third relation does not cover the interval [tex]\( -2 \le x < 0 \)[/tex]. Specifically, for [tex]\( -2 \le x < 0 \)[/tex], there is no value defined for [tex]\( y \)[/tex]. Because of these gaps, this piecewise relation does not define a function over all real numbers.
Given this analysis, the piecewise relations that define a function are the first and second ones. Therefore, the correct options are [tex]\([1, 2]\)[/tex].
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
