Answer :
Given the table of values for the function [tex]\( f(x) \)[/tex]:
[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]
Let's evaluate each statement one by one to determine its correctness based on the given values.
### 1. [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\( (-\infty, 3) \)[/tex].
To determine if this statement is true, we need to check all [tex]\( x \)[/tex] values less than 3:
- [tex]\( f(-3) = -15 \)[/tex] (not greater than 0)
- [tex]\( f(-2) = 0 \)[/tex] (not greater than 0)
- [tex]\( f(-1) = 3 \)[/tex] (greater than 0)
- [tex]\( f(0) = 0 \)[/tex] (not greater than 0)
- [tex]\( f(1) = -3 \)[/tex] (not greater than 0)
- [tex]\( f(2) = 0 \)[/tex] (not greater than 0)
Since there are values in [tex]\( (-\infty, 3) \)[/tex] for which [tex]\( f(x) \leq 0 \)[/tex], the statement [tex]\( f(x) > 0 \)[/tex] over this interval is False.
### 2. [tex]\( f(x) \leq 0 \)[/tex] over the interval [tex]\( [0, 2] \)[/tex].
To determine if this statement is true, we need to check all [tex]\( x \)[/tex] values in the interval [0, 2]:
- [tex]\( f(0) = 0 \)[/tex] (less than or equal to 0)
- [tex]\( f(1) = -3 \)[/tex] (less than or equal to 0)
- [tex]\( f(2) = 0 \)[/tex] (less than or equal to 0)
Since all values in [0, 2] satisfy [tex]\( f(x) \leq 0 \)[/tex], the statement is True.
### 3. [tex]\( f(x) < 0 \)[/tex] over the interval [tex]\( (-1, 1) \)[/tex].
To determine if this statement is true, we need to check all [tex]\( x \)[/tex] values in the interval (-1, 1):
- [tex]\( f(0) = 0 \)[/tex] (not less than 0)
Since there is a value in [tex]\((-1, 1)\)[/tex] for which [tex]\( f(x) \geq 0 \)[/tex], the statement [tex]\( f(x) < 0 \)[/tex] over this interval is False.
### 4. [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\( (-2, 0) \)[/tex].
To determine if this statement is true, we need to check all [tex]\( x \)[/tex] values in the interval (-2, 0):
- [tex]\( f(-1) = 3 \)[/tex] (greater than 0)
Since all values in [tex]\((-2, 0)\)[/tex] satisfy [tex]\( f(x) > 0 \)[/tex], the statement is True.
### 5. [tex]\( f(x) \geq 0 \)[/tex] over the interval [tex]\( [2, \infty) \)[/tex].
To determine if this statement is true, we need to check all [tex]\( x \)[/tex] values in the interval [2, ∞):
- [tex]\( f(2) = 0 \)[/tex] (greater than or equal to 0)
- [tex]\( f(3) = 15 \)[/tex] (greater than or equal to 0)
Since all available values in [tex]\([2, \infty)\)[/tex] satisfy [tex]\( f(x) \geq 0 \)[/tex], the statement is True.
### Summary
From the given data and analysis, the predictions can be summarized as follows:
1. [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\( (-\infty, 3) \)[/tex]: False
2. [tex]\( f(x) \leq 0 \)[/tex] over the interval [tex]\( [0, 2] \)[/tex]: True
3. [tex]\( f(x) < 0 \)[/tex] over the interval [tex]\( (-1, 1) \)[/tex]: False
4. [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\( (-2, 0) \)[/tex]: True
5. [tex]\( f(x) \geq 0 \)[/tex] over the interval [tex]\( [2, \infty) \)[/tex]: True
The final result is:
[tex]\[ [False, True, False, True, True] \][/tex]
[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]
Let's evaluate each statement one by one to determine its correctness based on the given values.
### 1. [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\( (-\infty, 3) \)[/tex].
To determine if this statement is true, we need to check all [tex]\( x \)[/tex] values less than 3:
- [tex]\( f(-3) = -15 \)[/tex] (not greater than 0)
- [tex]\( f(-2) = 0 \)[/tex] (not greater than 0)
- [tex]\( f(-1) = 3 \)[/tex] (greater than 0)
- [tex]\( f(0) = 0 \)[/tex] (not greater than 0)
- [tex]\( f(1) = -3 \)[/tex] (not greater than 0)
- [tex]\( f(2) = 0 \)[/tex] (not greater than 0)
Since there are values in [tex]\( (-\infty, 3) \)[/tex] for which [tex]\( f(x) \leq 0 \)[/tex], the statement [tex]\( f(x) > 0 \)[/tex] over this interval is False.
### 2. [tex]\( f(x) \leq 0 \)[/tex] over the interval [tex]\( [0, 2] \)[/tex].
To determine if this statement is true, we need to check all [tex]\( x \)[/tex] values in the interval [0, 2]:
- [tex]\( f(0) = 0 \)[/tex] (less than or equal to 0)
- [tex]\( f(1) = -3 \)[/tex] (less than or equal to 0)
- [tex]\( f(2) = 0 \)[/tex] (less than or equal to 0)
Since all values in [0, 2] satisfy [tex]\( f(x) \leq 0 \)[/tex], the statement is True.
### 3. [tex]\( f(x) < 0 \)[/tex] over the interval [tex]\( (-1, 1) \)[/tex].
To determine if this statement is true, we need to check all [tex]\( x \)[/tex] values in the interval (-1, 1):
- [tex]\( f(0) = 0 \)[/tex] (not less than 0)
Since there is a value in [tex]\((-1, 1)\)[/tex] for which [tex]\( f(x) \geq 0 \)[/tex], the statement [tex]\( f(x) < 0 \)[/tex] over this interval is False.
### 4. [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\( (-2, 0) \)[/tex].
To determine if this statement is true, we need to check all [tex]\( x \)[/tex] values in the interval (-2, 0):
- [tex]\( f(-1) = 3 \)[/tex] (greater than 0)
Since all values in [tex]\((-2, 0)\)[/tex] satisfy [tex]\( f(x) > 0 \)[/tex], the statement is True.
### 5. [tex]\( f(x) \geq 0 \)[/tex] over the interval [tex]\( [2, \infty) \)[/tex].
To determine if this statement is true, we need to check all [tex]\( x \)[/tex] values in the interval [2, ∞):
- [tex]\( f(2) = 0 \)[/tex] (greater than or equal to 0)
- [tex]\( f(3) = 15 \)[/tex] (greater than or equal to 0)
Since all available values in [tex]\([2, \infty)\)[/tex] satisfy [tex]\( f(x) \geq 0 \)[/tex], the statement is True.
### Summary
From the given data and analysis, the predictions can be summarized as follows:
1. [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\( (-\infty, 3) \)[/tex]: False
2. [tex]\( f(x) \leq 0 \)[/tex] over the interval [tex]\( [0, 2] \)[/tex]: True
3. [tex]\( f(x) < 0 \)[/tex] over the interval [tex]\( (-1, 1) \)[/tex]: False
4. [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\( (-2, 0) \)[/tex]: True
5. [tex]\( f(x) \geq 0 \)[/tex] over the interval [tex]\( [2, \infty) \)[/tex]: True
The final result is:
[tex]\[ [False, True, False, True, True] \][/tex]
