*Finding a root: I can’t think of a mapping.*

Square roots, yes, by Euclidean methods. Take a diameter of a circle divided with lengths 1 and x. The length of the chord at right-angles through the dividing point will be 2*SQRT(x). Generally, any function that can be expressed as a finite combination of linear and quadratic solutions has a geometric equivalent in the Euclidean sense (i.e. obtainable by certain ruler and compass rules).

Some higher roots and trisection of angles have a geometrical equivalent in constructions such as the Conchoid of Nicodemes and the Cissoid of Diocles that don’t stick to Euclidean rules (i.e. involving an iterative element).

Depends what’s meant by “geometrical equivalent”. Merely geometrically equivalent, or is required that you actually be able to measure it? For instance, the area of a circle is geometrically a function of pi, even if though any attempt to measure the area will be an approximation.