Answer :
Let's analyze the given statements based on the values in the table.
The table provides the values of [tex]\( f(x) \)[/tex] for specific [tex]\( x \)[/tex] as follows:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -3 & -15 \\ \hline -2 & 0 \\ \hline -1 & 3 \\ \hline 0 & 0 \\ \hline 1 & -3 \\ \hline 2 & 0 \\ \hline 3 & 15 \\ \hline \end{array} \][/tex]
Statement 1: [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\( (-\infty, 3) \)[/tex]
We need to check if [tex]\( f(x) \)[/tex] is positive for all [tex]\( x < 3 \)[/tex]:
- At [tex]\( x = -3 \)[/tex], [tex]\( f(x) = -15 \)[/tex] (which is not positive).
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (which is not positive).
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 3 \)[/tex] (positive).
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (which is not positive).
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -3 \)[/tex] (which is not positive).
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (which is not positive).
Since [tex]\( f(x) \)[/tex] is not positive for all [tex]\( x < 3 \)[/tex], this statement is False.
Statement 2: [tex]\( f(x) \leq 0 \)[/tex] over the interval [tex]\([0,2]\)[/tex]
We need to check if [tex]\( f(x) \)[/tex] is less than or equal to 0 for [tex]\( 0 \leq x \leq 2 \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (satisfies the condition).
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -3 \)[/tex] (satisfies the condition).
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (satisfies the condition).
Since [tex]\( f(x) \leq 0 \)[/tex] for all [tex]\( x \)[/tex] in the given interval, this statement is True.
Statement 3: [tex]\( f(x) < 0 \)[/tex] over the interval [tex]\((-1,1)\)[/tex]
We need to check if [tex]\( f(x) \)[/tex] is negative for [tex]\( -1 < x < 1 \)[/tex]:
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 3 \)[/tex] (which is not negative).
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (which is not negative).
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -3 \)[/tex] (negative).
Since [tex]\( f(x) \)[/tex] is not negative for all [tex]\( x \)[/tex] in the given interval, this statement is False.
Statement 4: [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\((-2,0)\)[/tex]
We need to check if [tex]\( f(x) \)[/tex] is positive for [tex]\( -2 < x < 0 \)[/tex]:
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (which is not positive).
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 3 \)[/tex] (positive).
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (which is not positive).
Since [tex]\( f(x) \)[/tex] is positive for [tex]\( x = -1 \)[/tex] but not for [tex]\( x = -2 \)[/tex] or [tex]\( x = 0 \)[/tex], this statement is True.
Statement 5: [tex]\( f(x) \geq 0 \)[/tex] over the interval [tex]\([2, \infty)\)[/tex]
We need to check if [tex]\( f(x) \)[/tex] is greater than or equal to 0 for [tex]\( x \geq 2 \)[/tex]:
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (satisfies the condition).
- At [tex]\( x = 3 \)[/tex], [tex]\( f(x) = 15 \)[/tex] (positive).
Since [tex]\( f(x) \geq 0 \)[/tex] for [tex]\( x \geq 2 \)[/tex], this statement is True.
In summary:
- [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\( (-\infty, 3) \)[/tex]: False
- [tex]\( f(x) \leq 0 \)[/tex] over the interval [tex]\([0,2]\)[/tex]: True
- [tex]\( f(x) < 0 \)[/tex] over the interval [tex]\((-1,1)\)[/tex]: False
- [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\((-2,0)\)[/tex]: True
- [tex]\( f(x) \geq 0 \)[/tex] over the interval [tex]\([2, \infty)\)[/tex]: True
The table provides the values of [tex]\( f(x) \)[/tex] for specific [tex]\( x \)[/tex] as follows:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -3 & -15 \\ \hline -2 & 0 \\ \hline -1 & 3 \\ \hline 0 & 0 \\ \hline 1 & -3 \\ \hline 2 & 0 \\ \hline 3 & 15 \\ \hline \end{array} \][/tex]
Statement 1: [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\( (-\infty, 3) \)[/tex]
We need to check if [tex]\( f(x) \)[/tex] is positive for all [tex]\( x < 3 \)[/tex]:
- At [tex]\( x = -3 \)[/tex], [tex]\( f(x) = -15 \)[/tex] (which is not positive).
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (which is not positive).
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 3 \)[/tex] (positive).
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (which is not positive).
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -3 \)[/tex] (which is not positive).
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (which is not positive).
Since [tex]\( f(x) \)[/tex] is not positive for all [tex]\( x < 3 \)[/tex], this statement is False.
Statement 2: [tex]\( f(x) \leq 0 \)[/tex] over the interval [tex]\([0,2]\)[/tex]
We need to check if [tex]\( f(x) \)[/tex] is less than or equal to 0 for [tex]\( 0 \leq x \leq 2 \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (satisfies the condition).
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -3 \)[/tex] (satisfies the condition).
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (satisfies the condition).
Since [tex]\( f(x) \leq 0 \)[/tex] for all [tex]\( x \)[/tex] in the given interval, this statement is True.
Statement 3: [tex]\( f(x) < 0 \)[/tex] over the interval [tex]\((-1,1)\)[/tex]
We need to check if [tex]\( f(x) \)[/tex] is negative for [tex]\( -1 < x < 1 \)[/tex]:
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 3 \)[/tex] (which is not negative).
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (which is not negative).
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -3 \)[/tex] (negative).
Since [tex]\( f(x) \)[/tex] is not negative for all [tex]\( x \)[/tex] in the given interval, this statement is False.
Statement 4: [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\((-2,0)\)[/tex]
We need to check if [tex]\( f(x) \)[/tex] is positive for [tex]\( -2 < x < 0 \)[/tex]:
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (which is not positive).
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 3 \)[/tex] (positive).
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (which is not positive).
Since [tex]\( f(x) \)[/tex] is positive for [tex]\( x = -1 \)[/tex] but not for [tex]\( x = -2 \)[/tex] or [tex]\( x = 0 \)[/tex], this statement is True.
Statement 5: [tex]\( f(x) \geq 0 \)[/tex] over the interval [tex]\([2, \infty)\)[/tex]
We need to check if [tex]\( f(x) \)[/tex] is greater than or equal to 0 for [tex]\( x \geq 2 \)[/tex]:
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (satisfies the condition).
- At [tex]\( x = 3 \)[/tex], [tex]\( f(x) = 15 \)[/tex] (positive).
Since [tex]\( f(x) \geq 0 \)[/tex] for [tex]\( x \geq 2 \)[/tex], this statement is True.
In summary:
- [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\( (-\infty, 3) \)[/tex]: False
- [tex]\( f(x) \leq 0 \)[/tex] over the interval [tex]\([0,2]\)[/tex]: True
- [tex]\( f(x) < 0 \)[/tex] over the interval [tex]\((-1,1)\)[/tex]: False
- [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\((-2,0)\)[/tex]: True
- [tex]\( f(x) \geq 0 \)[/tex] over the interval [tex]\([2, \infty)\)[/tex]: True
