\begin{tabular}{|c|c|}
\hline [tex]$x$[/tex] & [tex]$f(x)$[/tex] \\
\hline -5 & 8 \\
\hline -3 & 4 \\
\hline -1 & 0 \\
\hline 1 & -2 \\
\hline 3 & -2 \\
\hline 5 & 0 \\
\hline 7 & 4 \\
\hline
\end{tabular}

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

A. [tex]$f(x) \geq 0$[/tex] over the interval [tex]$[5, \infty)$[/tex].
B. [tex]$f(x) \leq 0$[/tex] over the interval [tex]$[-1, \infty)$[/tex].
C. [tex]$f(x) \ \textgreater \ 0$[/tex] over the interval [tex]$(-\infty, 1)$[/tex].
D. [tex]$f(x) \ \textless \ 0$[/tex] over the interval [tex]$(-\infty, -1)$[/tex].



Answer :

Let's examine each prediction in detail and verify their validity using the given table.

### Prediction 1: \( f(x) \geq 0 \) over the interval \([5, \infty)\)

We need to check the values of \( f(x) \) for \( x \geq 5 \):
- For \( x = 5 \), \( f(5) = 0 \)
- For \( x = 7 \), \( f(7) = 4 \)

Both of these values are greater than or equal to 0, so this prediction is correct.

### Prediction 2: \( f(x) \leq 0 \) over the interval \([-1, \infty)\)

We need to check the values of \( f(x) \) for \( x \geq -1 \):
- For \( x = -1 \), \( f(-1) = 0 \)
- For \( x = 1 \), \( f(1) = -2 \)
- For \( x = 3 \), \( f(3) = -2 \)
- For \( x = 5 \), \( f(5) = 0 \)
- For \( x = 7 \), \( f(7) = 4 \)

The value \( f(7) = 4 \) is greater than 0, so this prediction is incorrect.

### Prediction 3: \( f(x) > 0 \) over the interval \((-\infty, 1)\)

We need to check the values of \( f(x) \) for \( x < 1 \):
- For \( x = -5 \), \( f(-5) = 8 \)
- For \( x = -3 \), \( f(-3) = 4 \)
- For \( x = -1 \), \( f(-1) = 0 \)

The value \( f(-1) = 0 \) is not greater than 0, so this prediction is incorrect.

### Prediction 4: \( f(x) < 0 \) over the interval \((-\infty, -1)\)

We need to check the values of \( f(x) \) for \( x < -1 \):
- For \( x = -5 \), \( f(-5) = 8 \)
- For \( x = -3 \), \( f(-3) = 4 \)

Both of these values are greater than 0, so this prediction is incorrect.

Based on the verification:

- Prediction 1 is valid.
- Predictions 2, 3, and 4 are invalid.

Thus, the correct and valid prediction about the continuous function \( f(x) \) is:
[tex]\[ f(x) \geq 0 \text{ over the interval } [5, \infty). \][/tex]