A function is graphed on the coordinate plane.A rational function is graphed on the coordinate plane. It consists of two curves. The first curve starts on the left of the graph just below the x-axis. It decreases gradually until left-parenthesis 2 comma negative 3 right-parenthesis then quickly, approaching but never reaching x=3. The second curve starts at the top of the graph just to the right of x = 3 and decreases quickly until left-parenthesis 4 comma 3 right-parenthesis and then gradually approaching but never reaching the x-axis.

Question

Question
Use the graph to answer the question.
Describe the continuity of the graphed function.
(1 point)
Responses

The function has a removable discontinuity at x = 3.
The function has a removable discontinuity at x = 3.

The function has a removable discontinuity at x = 0.
The function has a removable discontinuity at x = 0.

The function has a nonremovable discontinuity at x = 3.
The function has a nonremovable discontinuity at x = 3.

The function has a nonremovable discontinuity at x = 0.

Answer 1

Based on your description, the function has a discontinuity at \( x = 3 \). The behavior of the function indicates that as it approaches \( x = 3 \) from the left, the value of the function decreases to a limit, but then there is a break in the graph as it does not exist for \( x = 3 \) and jumps to another curve beginning just after \( x = 3 \). This suggests that there is a vertical asymptote at \( x = 3 \).

A removable discontinuity occurs when there is a "hole" in the graph at a point where the function is not defined, but can be made continuous by defining it at that point. Since there is a vertical asymptote at \( x = 3\), which means the function approaches infinity (or does not settle at a limit), this discontinuity is nonremovable.

Therefore, the correct response is:

The function has a nonremovable discontinuity at x = 3.