Create odd function from arbitrary function I have a product of two arbitrary functions $f$ and $g$, $$y(x) = f(x)g(x)$$ and I want to make the product $y$ odd. I know $f$, e.g. it's a typical Lorentzian function $$f(x) = \dfrac{b}{(x-a)+b^2},$$ but I want to deduce $g$. What strategy can I use to find $g$? **Context:** I want to start by assuming that these functions are well behaved, defined in real space, and differentiable everywhere. How would I find $g$ so that the product $fg$ becomes odd? Then my goal is to make $f$ more complex (e.g. multiplying the Lorentzian above with the discontinuous Bose-Einstein distribution, for example) then see if it's still possible to make $fg$ an odd function. This reference summarises odd/even functions but I didn't find it too helpful for my problem.

Okay, so, of course there are _lots_ of options here. The simplest, but unsatisfying, would be to make $g$ be $\frac{1}{f(x)}x$, so that $y$ is just $x$. What that example demonstrates is that there's just _way_ too little information here to pin down one specific $g$.

But one that might be more satisfying would be to take $g(x) = xf(-x)$. This has the advantage of retaining some of the "character" of $f$, but without knowing more about your goal here I can't tell if this is what you're looking for.