Math and ChatGPT

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 \[f(x) = {\mathrm{sign}}(x) \log(1+|x|)\] This function is a form of a signed logarithmic scaling. So, let's see what ChatGPT is telling us:

The idea to split this into 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 it does not actually do that.

Doing the inverse by hand yields: \[f^{-1}(x) = {\mathrm{sign}}(x)(\exp(|x|)-1)\]

Hmm. ChatGPT does not really understand much of my feedback. It is rather polite, but it is just ignoring what I say.

Update 1: Wolfram Alpha

Wolfram Alpha [1] is based on the same symbolic math foundations as Mathematica. I tried the command:

inverse of sign(x)*log(1+|x|)

The response is:

This does not look too good either:

• The inverse function is incorrect 
• and the blue line is not the plot of \(e^x-1\). 

It is noted that ChatGPT can use Wolfram Alpha [2].

Update 2: Sage

Sage is a well-known open-source symbolic math package. An online version is available [3]. This does not seem to work either:

Sage cannot solve this. I tried doing it for the branches \(x\ge 0\) and \(g \lt 0\) separately. That worked fine. The complete function, however, seems out of reach.

Conclusion

A pen and a piece of paper is sometimes the most reliable tool to do math.

References

1. https://www.wolframalpha.com/
2. ChatGPT Gets Its “Wolfram Superpowers”!, https://writings.stephenwolfram.com/2023/03/chatgpt-gets-its-wolfram-superpowers/
3. https://sagecell.sagemath.org/