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

To analyze the predictions about the function \( f(x) \) based on the provided values, let's first summarize the values given in the table:

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

Now, let's evaluate each prediction:

1. **\( f(x) \leq 0 \) over the interval (–∞, ∞)**: This is **not valid** because there are values (e.g., at \( x = -3 \)) where \( f(x) = 15 \), which is greater than 0.

2. **\( f(x) > 0 \) over the interval (–1, ∞)**: This statement is also **not valid** because \( f(0) = 5 \) and \( f(1) = 0\), so while it is positive at \( x = 0 \), it is not greater than 0 at \( x = 1\).

3. **\( f(x) \geq 0 \) over the interval [–1, 1]**: This statement is **valid**. We check:

- \( f(-1) = 0 \)
- \( f(0) = 5 \)
- \( f(1) = 0 \)

Here, \( f(x) \) is exactly 0 at the endpoints and positive within the interval [–1, 1], so this prediction holds.

4. **\( f(x) < 0 \) over the interval (0, 2)**: This statement is **not valid** because \( f(0) = 5 \) is greater than 0, and we don't have values of \( f(x) \) for \( 1 < x < 2 \) that indicate it is negative.

Based on this analysis, the valid prediction about the continuous function \( f(x) \) is:

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