Answer :
To solve this problem, we need to analyze the values provided in the table and determine the truthfulness of each statement about the function [tex]\( f(x) \)[/tex]. Let's go through each statement one by one:
### Given Data Points:
[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 greater than 0 for all [tex]\( x < 3 \)[/tex]:
- At [tex]\( x = -3 \)[/tex], [tex]\( f(-3) = -15 \)[/tex] (not greater than 0)
- At [tex]\( x = -2 \)[/tex], [tex]\( f(-2) = 0 \)[/tex] (not greater than 0)
- At [tex]\( x = -1 \)[/tex], [tex]\( f(-1) = 3 \)[/tex] (greater than 0)
- At [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 0 \)[/tex] (not greater than 0)
- At [tex]\( x = 1 \)[/tex], [tex]\( f(1) = -3 \)[/tex] (not greater than 0)
- At [tex]\( x = 2 \)[/tex], [tex]\( f(2) = 0 \)[/tex] (not greater than 0)
Since not all values are greater than 0 for [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 all [tex]\( 0 \leq x \leq 2 \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 0 \)[/tex] (less than or equal to 0)
- At [tex]\( x = 1 \)[/tex], [tex]\( f(1) = -3 \)[/tex] (less than or equal to 0)
- At [tex]\( x = 2 \)[/tex], [tex]\( f(2) = 0 \)[/tex] (less than or equal to 0)
Since all values are less than or equal to 0 for [tex]\( 0 \leq x \leq 2 \)[/tex], 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 less than 0 for all [tex]\( -1 < x < 1 \)[/tex]:
- No data for [tex]\( -1 < x < 0 \)[/tex]
- No data for [tex]\( 0 < x < 1 \)[/tex]
From the provided data:
- At [tex]\( x = -1 \)[/tex], [tex]\( f(-1) = 3 \)[/tex] (but [tex]\( -1 \)[/tex] is not in [tex]\((-1, 1)\)[/tex])
- At [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 0 \)[/tex] (but [tex]\( 0 \)[/tex] is not in [tex]\((-1, 1)\)[/tex])
- At [tex]\( x = 1 \)[/tex], [tex]\( f(1) = -3 \)[/tex] (but [tex]\( 1 \)[/tex] is not in [tex]\((-1, 1)\)[/tex])
Since the interval excludes boundary points and no values are provided for in-between points, it's not clear from data but since it's [tex]\( f(x) \)[/tex] is more than zero at nearest point considered (f(-1)=3), 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 greater than 0 for all [tex]\( -2 < x < 0 \)[/tex]:
- No data for [tex]\( -2 < x < -1 \)[/tex]
- No data for [tex]\( -1 < x < 0 \)[/tex]
From the provided data:
- At [tex]\( x = -2 \)[/tex], [tex]\( f(-2) = 0 \)[/tex] (but [tex]\( -2 \)[/tex] is not in [tex]\((-2, 0)\)[/tex])
- At [tex]\( x = -1 \)[/tex], [tex]\( f(-1) = 3 \)[/tex] (but [tex]\( -1 \)[/tex] is not in [tex]\((-2, 0)\)[/tex])
Since nearest point [tex]\(f(-1)=3\)[/tex] greater than zero reaching the smallest point of zero of same interval, 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 all [tex]\( x \geq 2 \)[/tex]:
- At [tex]\( x = 2 \)[/tex], [tex]\( f(2) = 0 \)[/tex] (greater than or equal to 0)
- At [tex]\( x = 3 \)[/tex], [tex]\( f(3) = 15 \)[/tex] (greater than or equal to 0)
Since all values are greater than or equal to 0 for [tex]\( x \geq 2 \)[/tex], this statement is True.
### Summary of Results:
- [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
Hence, the true statements are:
1. [tex]\( f(x) \leq 0 \)[/tex] over the interval [tex]\([0, 2]\)[/tex]
2. [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\((-2, 0)\)[/tex]
3. [tex]\( f(x) \geq 0 \)[/tex] over the interval [tex]\([2, \infty)\)[/tex]
### Given Data Points:
[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 greater than 0 for all [tex]\( x < 3 \)[/tex]:
- At [tex]\( x = -3 \)[/tex], [tex]\( f(-3) = -15 \)[/tex] (not greater than 0)
- At [tex]\( x = -2 \)[/tex], [tex]\( f(-2) = 0 \)[/tex] (not greater than 0)
- At [tex]\( x = -1 \)[/tex], [tex]\( f(-1) = 3 \)[/tex] (greater than 0)
- At [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 0 \)[/tex] (not greater than 0)
- At [tex]\( x = 1 \)[/tex], [tex]\( f(1) = -3 \)[/tex] (not greater than 0)
- At [tex]\( x = 2 \)[/tex], [tex]\( f(2) = 0 \)[/tex] (not greater than 0)
Since not all values are greater than 0 for [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 all [tex]\( 0 \leq x \leq 2 \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 0 \)[/tex] (less than or equal to 0)
- At [tex]\( x = 1 \)[/tex], [tex]\( f(1) = -3 \)[/tex] (less than or equal to 0)
- At [tex]\( x = 2 \)[/tex], [tex]\( f(2) = 0 \)[/tex] (less than or equal to 0)
Since all values are less than or equal to 0 for [tex]\( 0 \leq x \leq 2 \)[/tex], 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 less than 0 for all [tex]\( -1 < x < 1 \)[/tex]:
- No data for [tex]\( -1 < x < 0 \)[/tex]
- No data for [tex]\( 0 < x < 1 \)[/tex]
From the provided data:
- At [tex]\( x = -1 \)[/tex], [tex]\( f(-1) = 3 \)[/tex] (but [tex]\( -1 \)[/tex] is not in [tex]\((-1, 1)\)[/tex])
- At [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 0 \)[/tex] (but [tex]\( 0 \)[/tex] is not in [tex]\((-1, 1)\)[/tex])
- At [tex]\( x = 1 \)[/tex], [tex]\( f(1) = -3 \)[/tex] (but [tex]\( 1 \)[/tex] is not in [tex]\((-1, 1)\)[/tex])
Since the interval excludes boundary points and no values are provided for in-between points, it's not clear from data but since it's [tex]\( f(x) \)[/tex] is more than zero at nearest point considered (f(-1)=3), 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 greater than 0 for all [tex]\( -2 < x < 0 \)[/tex]:
- No data for [tex]\( -2 < x < -1 \)[/tex]
- No data for [tex]\( -1 < x < 0 \)[/tex]
From the provided data:
- At [tex]\( x = -2 \)[/tex], [tex]\( f(-2) = 0 \)[/tex] (but [tex]\( -2 \)[/tex] is not in [tex]\((-2, 0)\)[/tex])
- At [tex]\( x = -1 \)[/tex], [tex]\( f(-1) = 3 \)[/tex] (but [tex]\( -1 \)[/tex] is not in [tex]\((-2, 0)\)[/tex])
Since nearest point [tex]\(f(-1)=3\)[/tex] greater than zero reaching the smallest point of zero of same interval, 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 all [tex]\( x \geq 2 \)[/tex]:
- At [tex]\( x = 2 \)[/tex], [tex]\( f(2) = 0 \)[/tex] (greater than or equal to 0)
- At [tex]\( x = 3 \)[/tex], [tex]\( f(3) = 15 \)[/tex] (greater than or equal to 0)
Since all values are greater than or equal to 0 for [tex]\( x \geq 2 \)[/tex], this statement is True.
### Summary of Results:
- [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
Hence, the true statements are:
1. [tex]\( f(x) \leq 0 \)[/tex] over the interval [tex]\([0, 2]\)[/tex]
2. [tex]\( f(x) > 0 \)[/tex] over the interval [tex]\((-2, 0)\)[/tex]
3. [tex]\( f(x) \geq 0 \)[/tex] over the interval [tex]\([2, \infty)\)[/tex]
