To determine the behavior of the function [tex]\( f(x) \)[/tex] over the interval [tex]\( (0,1) \)[/tex], we can analyze the values given in the table specifically at [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex].
From the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & -6 \\
\hline
1 & 0 \\
\hline
\end{array}
\][/tex]
So, we have:
[tex]\( f(0) = -6 \)[/tex]
[tex]\( f(1) = 0 \)[/tex]
We observe the following about the function's values over the interval [tex]\((0, 1)\)[/tex]:
1. At [tex]\( x = 0 \)[/tex], [tex]\( f(0) = -6 \)[/tex].
2. At [tex]\( x = 1 \)[/tex], [tex]\( f(1) = 0 \)[/tex].
To decide if the function is increasing, decreasing, or constant over the interval [tex]\((0,1)\)[/tex]:
- If [tex]\( f(1) > f(0) \)[/tex], the function is increasing over [tex]\((0,1)\)[/tex].
- If [tex]\( f(1) < f(0) \)[/tex], the function is decreasing over [tex]\((0,1)\)[/tex].
- If [tex]\( f(1) = f(0) \)[/tex], the function is constant over [tex]\((0,1)\)[/tex].
Comparing the values:
[tex]\[ f(1) = 0 > -6 = f(0) \][/tex]
This implies that [tex]\( f(x) \)[/tex] is increasing over the interval [tex]\((0,1)\)[/tex].
Thus, the statement that most accurately describes the function over the interval [tex]\((0,1)\)[/tex] is:
[tex]\[ \boxed{\text{B. The function is increasing over the interval } (0,1).} \][/tex]
