write or draw a function o an invertible function which is neither decreasing or increasing?

Question

i want examples of that kind of function, please.

Answer 1

You know you can't have any turning points, because it has to be 1-1 to be invertible. So if it's neither increasing nor decreasing, it must be discontinuous. For example, let f: R->R be given by f(x) = {1/x, x /= 0; 0, x = 0}, i.e. f(x) = 1/x except at x = 0, with f(0) = 0. This is invertible, since it is 1-1 and onto R, and it is decreasing on (-∞, 0) and on (0, ∞), but it is not decreasing since for any x > 0, -x < x and f(-x) < f(x).

Answer 2

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