Performing symbolic math steps is often related to pattern recognition. In theory, ChatGPT could be doing a good job here. I wanted to find the inverse of \[f(x) = {\mathrm{sign}}(x) \log(1+|x|)\] SO let's see what ChatGPT is telling us:
The idea to split in two branches is excellent. Strictly speaking we should handle \(x=0\) also in one of the branches, but let's ignore that for now. More worisome, somehow, it looses the \({\mathrm{sign}}(x)\) function in step 2 (needed for the \(x\lt 0\) branch). So let's inform ChatGPT about this:
It says, "let's include the \({\mathrm{sign}}(x)\) factor", but then does not actually do that.
Doing the inverse by hand yields: \[f^{-1}(x) = {\mathrm{sign}}(x)(\exp(|x|)-1)\]
Hmm. It does not really understand much of my feedback.