Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To determine the [tex]\( x \)[/tex]-intercepts given the table of values for the function [tex]\( f(x) \)[/tex], we need to identify the points where the function crosses the [tex]\( x \)[/tex]-axis, that is, where [tex]\( f(x) = 0 \)[/tex].
Here is the table provided:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 2 & 20 \\ \hline -1 & 0 \\ \hline 0 & -6 \\ \hline 1 & -4 \\ \hline 2 & 0 \\ \hline 3 & 0 \\ \hline \end{array} \][/tex]
Step-by-Step Solution:
1. Identify [tex]\( f(x) = 0 \)[/tex]:
- We need to find where [tex]\( f(x) \)[/tex] equals zero.
2. Examine each point in the table:
- For [tex]\( x = 2, \, f(x) = 20 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) \neq 0 \)[/tex].
- For [tex]\( x = -1, \, f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( (-1, 0) \)[/tex] is a candidate.
- For [tex]\( x = 0, \, f(x) = -6 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) \neq 0 \)[/tex].
- For [tex]\( x = 1, \, f(x) = -4 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) \neq 0 \)[/tex].
- For [tex]\( x = 2, \, f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( (2, 0) \)[/tex] is a candidate.
- For [tex]\( x = 3, \, f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( (3, 0) \)[/tex] is a candidate.
Based on the points examined, the [tex]\( x \)[/tex]-intercepts, where [tex]\( f(x) = 0 \)[/tex], are:
- [tex]\( (-1, 0) \)[/tex]
- [tex]\( (2, 0) \)[/tex]
- [tex]\( (3, 0) \)[/tex]
Check given options:
- [tex]\((-1,0)\)[/tex] is one of the [tex]\( x \)[/tex]-intercepts (True).
- [tex]\((0,-6)\)[/tex] is not an [tex]\( x \)[/tex]-intercept, since [tex]\( f(x) \neq 0 \)[/tex] at this point (False).
- [tex]\((-6,0)\)[/tex] is not in the table, hence it cannot be an [tex]\( x \)[/tex]-intercept for this data (False).
- [tex]\((0,-1)\)[/tex] is not in the table, and it doesn't satisfy [tex]\( f(x) = 0 \)[/tex] either (False).
Therefore, the correct answer is:
[tex]\[ (-1, 0) \][/tex]
This is a valid [tex]\( x \)[/tex]-intercept of the continuous function based on the given data.
Here is the table provided:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 2 & 20 \\ \hline -1 & 0 \\ \hline 0 & -6 \\ \hline 1 & -4 \\ \hline 2 & 0 \\ \hline 3 & 0 \\ \hline \end{array} \][/tex]
Step-by-Step Solution:
1. Identify [tex]\( f(x) = 0 \)[/tex]:
- We need to find where [tex]\( f(x) \)[/tex] equals zero.
2. Examine each point in the table:
- For [tex]\( x = 2, \, f(x) = 20 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) \neq 0 \)[/tex].
- For [tex]\( x = -1, \, f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( (-1, 0) \)[/tex] is a candidate.
- For [tex]\( x = 0, \, f(x) = -6 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) \neq 0 \)[/tex].
- For [tex]\( x = 1, \, f(x) = -4 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) \neq 0 \)[/tex].
- For [tex]\( x = 2, \, f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( (2, 0) \)[/tex] is a candidate.
- For [tex]\( x = 3, \, f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( (3, 0) \)[/tex] is a candidate.
Based on the points examined, the [tex]\( x \)[/tex]-intercepts, where [tex]\( f(x) = 0 \)[/tex], are:
- [tex]\( (-1, 0) \)[/tex]
- [tex]\( (2, 0) \)[/tex]
- [tex]\( (3, 0) \)[/tex]
Check given options:
- [tex]\((-1,0)\)[/tex] is one of the [tex]\( x \)[/tex]-intercepts (True).
- [tex]\((0,-6)\)[/tex] is not an [tex]\( x \)[/tex]-intercept, since [tex]\( f(x) \neq 0 \)[/tex] at this point (False).
- [tex]\((-6,0)\)[/tex] is not in the table, hence it cannot be an [tex]\( x \)[/tex]-intercept for this data (False).
- [tex]\((0,-1)\)[/tex] is not in the table, and it doesn't satisfy [tex]\( f(x) = 0 \)[/tex] either (False).
Therefore, the correct answer is:
[tex]\[ (-1, 0) \][/tex]
This is a valid [tex]\( x \)[/tex]-intercept of the continuous function based on the given data.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
