Suppose f(x) = [sin(x^2 - 4)]^ -1. Identify any points of discontinuity, and determine (giving reasons) if they are removable, infinite (essential), or jump discontinuities.

Question

Okay, I presume that the [] brackets denote the greatest integer function (int () ). Once I graphed the function on my graphing calculator, it returned a tragically ugly line of dots along y = -1. How can I interpret this and describe it for every single point?

Answer 1

-1 <= sin(x) <= 1 for all x.

So, [sin(x)] is either -1, 0, or 1

This is zero when x^2 - 4 = kπ, or

x = sqrt(kπ + 4) for any integer k. It stays zero till x^2 - 4 = (k + 1/2)π

Answer 2

I'm not sure I understand your answer completely. I now understand that since the function is in greatest integer brackets, that it can only be one of three answers (-1, 0, or 1), but from here, you lose me. Where are the discontinuities located and how can I describe the mass number of what appear to be discontinuities on the graph? Also, would these be considered jump discontinuities since they seem to jump and leaves gaps in the graph? Sorry to have to ask you to explain yorself more, I think I just need a more simplistic description to help me understand.

Answer 3

see my comment on your 8:58 pm repost.