When modelling strong gravitational lenses, i.e. where there are multiple images of the same source, the most widely used parametrization for the mass profile in the lens galaxy is the singular power-law model rho(r)proportional to r(-gamma). This model may be insufficiently flexible for very accurate work, for example, measuring the Hubble constant based on time delays between multiple images. Here, we derive the lensing properties - deflection angle, shear, and magnification - of a more adaptable model where the projected mass surface density is parametrized as a continuous two-dimensional broken power law (2DBPL). This elliptical 2DBPL model is characterized by power-law slopes t(1) and t(2) either side of the break radius theta(B). The key to the 2DBPL model is the derivation of the lensing properties of the truncated power-law (TPL) model, where the surface density is a power law out to the truncation radius theta(T) and zero beyond. This TPL model is also useful by itself. We create mock observations of lensing by a TPL profile where the images form outside the truncation radius, so there is no mass in the annulus covered by the images. We then show that the slope of the profile interior to the images may be accurately recovered for lenses of moderate ellipticity. This demonstrates that the widely held notion that lensing measures the slope of the mass profile in the annulus of the images, and is insensitive to the mass distribution at radii interior to the images, is incorrect.