Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
Let's analyze the function [tex]\( f(x) \)[/tex] using the provided table and check the given statements one by one.
The table of [tex]\( f(x) \)[/tex] values is:
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $f(x)$ \\ \hline -3 & -15 \\ \hline -2 & 0 \\ \hline -1 & 3 \\ \hline 0 & 0 \\ \hline 1 & -3 \\ \hline 2 & 0 \\ \hline 3 & 15 \\ \hline \end{tabular} \][/tex]
We will now examine each given statement based on the function values from the table:
1. [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\((-\infty, 3)\)[/tex]:
Let's check the values of [tex]\( f(x) \)[/tex] for [tex]\( x < 3 \)[/tex]:
- At [tex]\( x = -3 \)[/tex], [tex]\( f(x) = -15 \)[/tex]
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 3 \)[/tex]
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -3 \)[/tex]
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
We observe that [tex]\( f(x) \)[/tex] is not greater than 0 for all values of [tex]\( x \)[/tex] in the interval [tex]\((-\infty, 3)\)[/tex] because there are points where [tex]\( f(x) \leq 0 \)[/tex]. Hence, this statement is False.
2. [tex]\( f(x) \leq 0 \)[/tex] over the interval [tex]\([0, 2]\)[/tex]:
Let's check the values of [tex]\( f(x) \)[/tex] for [tex]\( 0 \leq x \leq 2 \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -3 \)[/tex]
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
All these values of [tex]\( f(x) \)[/tex] are indeed less than or equal to 0. Hence, this statement is True.
3. [tex]\( f(x) < 0 \)[/tex] over the interval [tex]\((-1, 1)\)[/tex]:
Let's check the values of [tex]\( f(x) \)[/tex] for [tex]\(-1 < x < 1\)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 3 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -3 \)[/tex]
We observe that [tex]\( f(x) \)[/tex] is not less than 0 for all values of [tex]\( x \)[/tex] in the interval [tex]\((-1, 1)\)[/tex] because there are points where [tex]\( f(x) \geq 0 \)[/tex]. Hence, this statement is False.
4. [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\((-2, 0)\)[/tex]:
Let's check the values of [tex]\( f(x) \)[/tex] for [tex]\(-2 < x < 0\)[/tex]:
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 3 \)[/tex]
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (boundary value)
We observe that all values of [tex]\( f(x) \)[/tex] are greater than 0 within this interval. Hence, this statement is True.
5. [tex]\( f(x) \geq 0 \)[/tex] over the interval [tex]\([2, \infty)\)[/tex]:
Let's check the values of [tex]\( f(x) \)[/tex] for [tex]\( x \geq 2 \)[/tex]:
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- At [tex]\( x = 3 \)[/tex], [tex]\( f(x) = 15 \)[/tex]
Both values are greater than or equal to 0. Hence, this statement is True.
So, the results are:
- [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\((-\infty, 3)\)[/tex] is False.
- [tex]\( f(x) \leq 0 \)[/tex] over the interval [tex]\([0, 2]\)[/tex] is True.
- [tex]\( f(x) < 0 \)[/tex] over the interval [tex]\((-1, 1)\)[/tex] is False.
- [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\((-2, 0)\)[/tex] is True.
- [tex]\( f(x) \geq 0 \)[/tex] over the interval [tex]\([2, \infty)\)[/tex] is True.
The table of [tex]\( f(x) \)[/tex] values is:
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $f(x)$ \\ \hline -3 & -15 \\ \hline -2 & 0 \\ \hline -1 & 3 \\ \hline 0 & 0 \\ \hline 1 & -3 \\ \hline 2 & 0 \\ \hline 3 & 15 \\ \hline \end{tabular} \][/tex]
We will now examine each given statement based on the function values from the table:
1. [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\((-\infty, 3)\)[/tex]:
Let's check the values of [tex]\( f(x) \)[/tex] for [tex]\( x < 3 \)[/tex]:
- At [tex]\( x = -3 \)[/tex], [tex]\( f(x) = -15 \)[/tex]
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 3 \)[/tex]
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -3 \)[/tex]
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
We observe that [tex]\( f(x) \)[/tex] is not greater than 0 for all values of [tex]\( x \)[/tex] in the interval [tex]\((-\infty, 3)\)[/tex] because there are points where [tex]\( f(x) \leq 0 \)[/tex]. Hence, this statement is False.
2. [tex]\( f(x) \leq 0 \)[/tex] over the interval [tex]\([0, 2]\)[/tex]:
Let's check the values of [tex]\( f(x) \)[/tex] for [tex]\( 0 \leq x \leq 2 \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -3 \)[/tex]
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
All these values of [tex]\( f(x) \)[/tex] are indeed less than or equal to 0. Hence, this statement is True.
3. [tex]\( f(x) < 0 \)[/tex] over the interval [tex]\((-1, 1)\)[/tex]:
Let's check the values of [tex]\( f(x) \)[/tex] for [tex]\(-1 < x < 1\)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 3 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -3 \)[/tex]
We observe that [tex]\( f(x) \)[/tex] is not less than 0 for all values of [tex]\( x \)[/tex] in the interval [tex]\((-1, 1)\)[/tex] because there are points where [tex]\( f(x) \geq 0 \)[/tex]. Hence, this statement is False.
4. [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\((-2, 0)\)[/tex]:
Let's check the values of [tex]\( f(x) \)[/tex] for [tex]\(-2 < x < 0\)[/tex]:
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 3 \)[/tex]
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (boundary value)
We observe that all values of [tex]\( f(x) \)[/tex] are greater than 0 within this interval. Hence, this statement is True.
5. [tex]\( f(x) \geq 0 \)[/tex] over the interval [tex]\([2, \infty)\)[/tex]:
Let's check the values of [tex]\( f(x) \)[/tex] for [tex]\( x \geq 2 \)[/tex]:
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- At [tex]\( x = 3 \)[/tex], [tex]\( f(x) = 15 \)[/tex]
Both values are greater than or equal to 0. Hence, this statement is True.
So, the results are:
- [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\((-\infty, 3)\)[/tex] is False.
- [tex]\( f(x) \leq 0 \)[/tex] over the interval [tex]\([0, 2]\)[/tex] is True.
- [tex]\( f(x) < 0 \)[/tex] over the interval [tex]\((-1, 1)\)[/tex] is False.
- [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\((-2, 0)\)[/tex] is True.
- [tex]\( f(x) \geq 0 \)[/tex] over the interval [tex]\([2, \infty)\)[/tex] is True.
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
