Let's analyze the given table of [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex] values to understand the behavior of the rational function.
The given table is:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
x & -0.2 & -0.1 & 0 & 0.1 & 0.2 & 3.7 & 3.8 & 3.9 & 3.99 & 4 & 4.01 & 4.1 & 4.2 \\
\hline
f(x) & -0.238 & -0.244 & \text{undefined} & -0.256 & -0.263 & -3.3 & -5 & -10 & -100 & \text{undefined} & 100 & 10 & 5 \\
\hline
\end{array}
\][/tex]
### Analysis at [tex]\( x = 0 \)[/tex]:
- [tex]\( f(x) \)[/tex] is undefined at [tex]\( x = 0 \)[/tex].
- The values of [tex]\( f(x) \)[/tex] around [tex]\( x = 0 \)[/tex] (i.e., for [tex]\( x = -0.1 \)[/tex] and [tex]\( x = 0.1 \)[/tex]) are finite: [tex]\( -0.244 \)[/tex] and [tex]\( -0.256 \)[/tex].
These observations indicate that the function [tex]\( f(x) \)[/tex] has a hole at [tex]\( x = 0 \)[/tex]. A hole occurs when the function is not defined at a particular point, but the neighboring values are finite.
### Analysis at [tex]\( x = 4 \)[/tex]:
- [tex]\( f(x) \)[/tex] is undefined at [tex]\( x = 4 \)[/tex].
- The values of [tex]\( f(x) \)[/tex] around [tex]\( x = 4 \)[/tex] demonstrate rapid increase and decrease:
- For [tex]\( x = 3.99 \)[/tex], [tex]\( f(x) = -100 \)[/tex]
- For [tex]\( x = 4.01 \)[/tex], [tex]\( f(x) = 100 \)[/tex]
These observations suggest that the function [tex]\( f(x) \)[/tex] has a vertical asymptote at [tex]\( x = 4 \)[/tex]. A vertical asymptote occurs when the values of the function approach [tex]\( \pm \infty \)[/tex] as [tex]\( x \)[/tex] approaches a particular value.
### Conclusion:
Based on the analysis of the table:
- The function has a hole at [tex]\( x = 0 \)[/tex].
- The function has a vertical asymptote at [tex]\( x = 4 \)[/tex].
Thus, the correct statement is: The function has a hole when [tex]\( x = 0 \)[/tex] and a vertical asymptote when [tex]\( x = 4 \)[/tex].
