Answered

At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$f(x)$[/tex] \\
\hline
-3 & -15 \\
\hline
-2 & -5 \\
\hline
-1 & 0 \\
\hline
0 & 5 \\
\hline
1 & 0 \\
\hline
2 & -5 \\
\hline
\end{tabular}

Which is a valid prediction about the continuous function [tex]$f(x)$[/tex]?

A. [tex]$f(x) \leq 0$[/tex] over the interval [tex]$(-\infty, \infty)$[/tex].
B. [tex]$f(x) \ \textgreater \ 0$[/tex] over the interval [tex]$(-1, \infty)$[/tex].
C. [tex]$f(x) \geq 0$[/tex] over the interval [tex]$[-1,1]$[/tex].
D. [tex]$f(x) \ \textless \ 0$[/tex] over the interval [tex]$(0,2)$[/tex].

Sagot :

GinnyAnswer
Let's analyze the predictions step by step based on the given function values.

### Given Data:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -3 & -15 \\ \hline -2 & -5 \\ \hline -1 & 0 \\ \hline 0 & 5 \\ \hline 1 & 0 \\ \hline 2 & -5 \\ \hline \end{array} \][/tex]

### Evaluate the Statements:

1. Statement: \( f(x) \leq 0 \) over the interval \( (-\infty, \infty) \)
- We need to check if \( f(x) \) is always less than or equal to zero for all \( x \).
- At \( x = 0 \), \( f(0) = 5 \), which is greater than 0.
- Therefore, this statement is False.

2. Statement: \( f(x) > 0 \) over the interval \( (-1, \infty) \)
- We need to check if \( f(x) \) is always greater than zero for \( x \) in the interval \( (-1, \infty) \).
- At \( x = -1 \), \( f(-1) = 0 \), which is not greater than 0.
- At \( x = 1 \), \( f(1) = 0 \), which is also not greater than 0.
- Therefore, this statement is False.

3. Statement: \( f(x) \geq 0 \) over the interval \( [-1, 1] \)
- We need to check if \( f(x) \) is always greater than or equal to zero for \( x \) in the interval \( [-1, 1] \).
- At \( x = -1 \), \( f(-1) = 0 \), which is equal to 0.
- At \( x = 0 \), \( f(0) = 5 \), which is greater than 0.
- At \( x = 1 \), \( f(1) = 0 \), which is equal to 0.
- Therefore, this statement is True.

4. Statement: \( f(x) < 0 \) over the interval \( (0, 2) \)
- We need to check if \( f(x) \) is always less than zero for \( x \) in the open interval \( (0, 2) \).
- Although \( f(2) = -5 \), we need to consider the open interval.
- As we observe, there are no given values between \(0\) and \(2\) to contradict this.
- Therefore, this statement is True.

### Conclusion:
The valid predictions about the continuous function \( f(x) \) are:

- \( f(x) \geq 0 \) over the interval \([-1,1]\)
- \( f(x) < 0 \) over the interval \((0,2)\)

Thus, our predictions are:
- \( f(x) \leq 0 \) over the interval \( (-\infty, \infty) \) is False
- \( f(x) > 0 \) over the interval \( (-1, \infty) \) is False
- \( f(x) \geq 0 \) over the interval \([-1,1]\) is True
- [tex]\( f(x) < 0 \)[/tex] over the interval [tex]\((0,2)\)[/tex] is True
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.