Linear dependence on union of intervals

Suppose I have two functions $f,g$ which I want to exam for linear dependence. If I can conclude they are dependent on $(a,b]$ and $[b,c)$ is it possible to conclude that they are dependent on $(a,b] \cup [b,c) = (a,c)$ ?

For example $f= t^3$, $g=|t^3|$ which are clearly dependent on $(-\infty, 0]$ and $[0,\infty)$, but when examining the whole line, I can only solve for nonzero constants in a piecewise manner since the two constant vectors are orthogonal to one another.

Feels like im missing something obvious here..Thanks.

You've almost answered that yourself: Your $f,g$ are (clearly) linearly independent. None of the two functions is a multiple of the other.
A sufficient condition that would allow to conclude that $f,g$ are dependent on $(a,c)$ if they are known to be dependent on $(a,b]$ and $[b,c)$ would be that $f(b)$ and $g(b)$ are nonzero: In that case we have the linear dependency $g(b)\cdot f-f(b)\cdot g=0$ on $(a,b]$ and $[b,c)$ and all of $(a,c)$.