Although it clearly isn't a linear transformation, is there any sort of other definite transformation that could carry us from $x^2$ to $x^2 + \frac{1}{x^2 + 1}$, such that we could test whether $t$ has it?
I'm interested in knowing whether I can get from $x^2$ to $x^2 + \frac{1}{x^2 + 1} - 1$ by using, for instance, a combination of shears, projections, or a specific sort of warping, but I don't know whether there is some way to study warping.
My interest stems from the fact that affine maps are useful insofar as they can represent translations, reflections, scalings, rotations, shears and projections of geometrical objects. So, for instance, if I know that x→2x+1 is an affine transformation, I can visualize the procedure of transforming one into the other as a sum of the above mentioned transformations.. I'm wondering if there are other sorts of such transformations, and one for instance that would help in the specific case outlined above.