Question

A 2-column table with 6 rows. The first column is labeled x with entries negative 3, negative 2, negative 1, 0, 1. The second column is labeled f of x with entries 15, negative 5, 0, 5, 0, negative 5.
Which is a valid prediction about the continuous function f(x)?

f(x) ≤ 0 over the interval (–∞, ∞).
f(x) > 0 over the interval (–1, ∞).
f(x) ≥ 0 over the interval [–1, 1].
f(x) < 0 over the interval (0, 2

Answer 1

Based on the values given in the table and looking for a prediction regarding the continuous function \( f(x) \):

The table entries are:

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

Analyzing the options:

1. \( f(x) \leq 0 \) over the interval \((-\infty, \infty)\): This is not valid because \( f(x) \) takes positive values at several points (e.g., at \( x = -3 \) and \( x = 0 \)).

2. \( f(x) > 0 \) over the interval \((-1, \infty)\): This is not supported by the table, as \( f(x) = 0 \) at \( x = -1 \) and \( x = 1 \), and \( f(x) < 0 \) at \( x = 2 \).

3. \( f(x) \geq 0 \) over the interval \([-1, 1]\): Looking at the table, \( f(x) = 0 \) at \( x = -1 \) and \( x = 1 \), and \( f(x) \) is positive at \( x = 0 \). There are no negative values in this interval.

4. \( f(x) < 0 \) over the interval \((0, 2)\): This is not valid as \( f(x) = 5 \) at \( x = 0 \) and \( f(x) = -5 \) at \( x = 2 \); however, we cannot be sure about \( f(x) \) in the whole interval \((0, 2)\).

The valid prediction regarding the continuous function \( f(x) \) is:

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