Given $S(x) = |f(x)| + |g(x)|$, where $f(x) = sin(Ax), g(x) = sin(B/x)$, and $A, B$ constants.
Take any interval $I$ such that inside $I$ (excluding endpoints) there are exactly two points where $S(x) = 0$. (The two points are assumed to be not known explicitly, because we want to determine them later on.)
I would like (if possible) to construct any continuous function: $T(x) = T(f(x), g(x))$ such that it changes sign at those two points (only). Example shown in the picture; interval I is within the green lines.
Question: Is there any way to obtain a $T(x)$ from $f(x), g(x)$ ?
Naturally, for search purposes, it would also be fine, or even better, if $T(x)$ changed sign at every zero of $S(x)$ (and only there), independent of any interval. Also, $x$ and all the constants can be assumed to be positive, if useful.
(Original motivation: I am trying to simplify a root-finding problem to simpler situations where I can apply a bisection method.)