Question

consider the table representing a rational function.

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
f(x)	-0.238	-0.244	undefined	-0.256	-0.263	-3.\overline{3}	-5	-10	-100	undefined	100	10	5

which statement describes the graph of the function?
\bigcirc the function has holes when x = 0 and x = 4.
\bigcirc the function has vertical asymptotes when x = 0 and x = 4.
\bigcirc the function has a vertical asymptote when x = 0 and a hole when x = 4.
\bigcirc the function has a hole when x = 0 and a vertical asymptote when x = 4.

Explanation:

1. For \(x=0\): The function is undefined here, but the values of \(f(x)\) approach a consistent finite value (\(\approx -0.25\)) as \(x\) approaches 0 from both the left and right. This matches the behavior of a hole (removable discontinuity).
2. For \(x=4\): The function is undefined here, and the values of \(f(x)\) grow without bound (approach \(-\infty\) from the left, \(+\infty\) from the right) as \(x\) approaches 4. This matches the behavior of a vertical asymptote.

Answer:

The function has a hole when \(x = 0\) and a vertical asymptote when \(x = 4\).