Question

1. (5 points) the graph of f(x) is shown in figure 1. fill in the blanks: the function has a ____ discontinuity at x = __ and a __ discontinuity at x = ____. a. removable, 1; infinite, -1 b. jump, -1; removable, 4 c. removable, -1; jump, 1 d. infinite 4; removable 1 e. none of the above

Explanation:

Step1: Identify removable discontinuity

A removable discontinuity occurs when the limit of the function exists at a point, but the function is not defined or has a different value at that point. Looking at the graph, at \(x = - 1\), the function has a hole. The left - hand limit and the right - hand limit as \(x\to - 1\) are equal, so there is a removable discontinuity at \(x=-1\).

Step2: Identify jump discontinuity

A jump discontinuity occurs when the left - hand limit and the right - hand limit of the function at a point exist but are not equal. At \(x = 1\), as \(x\) approaches \(1\) from the left and from the right, the function approaches different values. So there is a jump discontinuity at \(x = 1\).

Answer:

C. removable, -1; jump, 1