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 -3.3 -5 -10 -100 undefined 100 10 5

Which statement describes the graph of the function?

O The function has holes when x = 0 and x = 4.
O The function has vertical asymptotes when x = 0 and x = 4.
O The function has a vertical asymptote when x = 0 and a hole when x = 4.
O The function has a hole when x = 0 and a vertical asymptote when x = 4.

Answer 1

By analyzing the table values for f(x) around x = 0 and x = 4, we can conclude that there is likely a hole in the graph at x = 0 and a vertical asymptote at x = 4.

Step-by-step explanation:

When analyzing the behavior of the values in the table for the rational function, we can infer certain characteristics about the function's graph. Notably, the function is undefined at x = 0 and x = 4, which suggests there could be vertical asymptotes or holes at these points. A hole in the graph occurs if the function is undefined at a point, but the limit exists. A vertical asymptote, on the other hand, occurs when the values of the function increase or decrease without bounds as x approaches a certain point.

Given that the values of f(x) are approaching very large magnitudes (both positive and negative) as x approaches 4, it signals that there is a vertical asymptote at x = 4. Near x = 0, the values of f(x) are getting close but do not exhibit behavior characteristic of an asymptote, indicating there could be a hole at x = 0. Therefore, the correct option that describes the graph of the function is: The function has a hole when x = 0 and a vertical asymptote when x = 4.