To determine the properties of the continuous function that generated the given table of values, we need to analyze the given data points and ascertain whether there is an x-intercept. An x-intercept occurs when the function crosses the x-axis, which corresponds to a y-value of zero.
Let's examine the table of values:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0.125 & -3 \\
\hline
0.5 & -1 \\
\hline
2 & 1 \\
\hline
8 & 3 \\
\hline
64 & 6 \\
\hline
\end{array}
\][/tex]
We will look for any change in the sign of the y-values, as this indicates where the function crosses the x-axis (which means the function has an x-intercept).
1. From [tex]\( x = 0.125 \)[/tex] to [tex]\( x = 0.5 \)[/tex]:
- [tex]\( y \)[/tex] changes from -3 to -1 (both are negative).
2. From [tex]\( x = 0.5 \)[/tex] to [tex]\( x = 2 \)[/tex]:
- [tex]\( y \)[/tex] changes from -1 to 1 (negative to positive).
3. From [tex]\( x = 2 \)[/tex] to [tex]\( x = 8 \)[/tex]:
- [tex]\( y \)[/tex] changes from 1 to 3 (both are positive).
4. From [tex]\( x = 8 \)[/tex] to [tex]\( x = 64 \)[/tex]:
- [tex]\( y \)[/tex] changes from 3 to 6 (both are positive).
From the above observations, we notice that there is a sign change between [tex]\( x = 0.5 \)[/tex] and [tex]\( x = 2 \)[/tex] (negative to positive). This signifies that the function crosses the x-axis at least once within this interval, meaning there is at least one x-intercept.
Based on this information, we can conclude:
C. the function has at least one [tex]$x$[/tex]-intercept.
