To determine the interval over which the function [tex]\( f(x) \)[/tex] is positive, we need to examine the given table of [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex] values:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & -3 & 0 & 3 & 6 & 9 & 12 \\
\hline
f(x) & 4 & 3 & 0 & -2 & -4 & -6 \\
\hline
\end{array}
\][/tex]
From the table, observe the following values:
- At [tex]\( x = -3 \)[/tex], [tex]\( f(x) = 4 \)[/tex]
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 3 \)[/tex]
- At [tex]\( x = 3 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- At [tex]\( x = 6 \)[/tex], [tex]\( f(x) = -2 \)[/tex]
- At [tex]\( x = 9 \)[/tex], [tex]\( f(x) = -4 \)[/tex]
- At [tex]\( x = 12 \)[/tex], [tex]\( f(x) = -6 \)[/tex]
The function [tex]\( f(x) \)[/tex] changes from positive to negative at [tex]\( x = 3 \)[/tex]. Specifically, [tex]\( f(x) \)[/tex] is positive when [tex]\( x = -3 \)[/tex] and [tex]\( x = 0 \)[/tex], and at [tex]\( x = 3 \)[/tex], it equals zero. For [tex]\( x > 3 \)[/tex], the function [tex]\( f(x) \)[/tex] is negative.
Therefore, the interval over which function [tex]\( f \)[/tex] is positive is from negative infinity up to, but not including, [tex]\( x = 3 \)[/tex]. This interval can be expressed as:
[tex]\[
(-\infty, 3)
\][/tex]
After examining the choices, we can select the correct interval:
- [tex]\((-\infty, 3)\)[/tex] ✔
- [tex]\((0, \infty)\)[/tex]
- [tex]\((-\infty, 0)\)[/tex]
- [tex]\((-\infty, \infty)\)[/tex]
Thus, the correct answer is [tex]\((-\infty, 3)\)[/tex].
