Answer :
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.
