This is a fun exploration! It's interesting that the summed functions have a feature topology that mirrors each sub-function, but some of the local features retain their original scaling, while features in the overlap region adopt a new scaling. This reminds me a little of using absolute scaling in SVG or CSS on something that is parented or embedded (not at the root of the hierarchy). Sometimes you use that kind of thing for character rigs in movies & games - break out of the local coordinate system and use world space.

Since the scaling isn't uniform, this is what causes t1 & t2 to have to be piece-wise linear. I'd love to see what happens if the transform functions, i.e. t1 & t2, were restricted to affine transforms. Like, how much harder is it if you further require the summed function to be a uniformly scaled & translated version of the sub-function? What if you added to the formal definition something like: f_S = a * f_L ( b * x + c ), where a,b,c are constants

This also reminds me of a mutation I added to my artificial evolution digital art project where I was trying to generalize on the idea of a fractal. I called it the 'harmonic mutation', and it takes any function f(x) and produces a new function with two or more scaled copies, like c0 * f(c1 * x + c2) + c3 * f(c4 * x + c5). Repeat several times with the same constants, and you can make a fractal out of any function at all. Very similar concept to Perlin turbulence too, where you have summed & scaled octaves of noise.