The Absolute and relative sense

In class we discussed a function which is well-conditioned in both the absolute and relative sense: f(x) = 31:, which had C(x) = 3 and K(x) = 1. We also discussed a different function which was well- conditioned in the relative sense but ill-conditioned in the absolute sense: [(15) = «E, which had K(x) = 1/2 but C(x) = fi. Since C(x) becomes arbitrarily large as x -> 0, it is ill-conditioned for small values of x.

Find an example of a function which is ill-conditioned in both the relative and absolute senses. To make it more concrete, let’s say we want both 00:) and K(x) to be 2 1000. It’s OK if the function you find is only ill-conditioned for certain values of x, but explain what values of xthose are.