  General relativity (GR) characterizes gravity as a geometric properly
exhibited as curvature on spacetime. Teleprallelism describes gravity through
torsional properties, and can reproduce GR at the level of equations. Similar
to f(R) gravity, on taking a generalization, f(T) gravity can produce various
modifications its gravitational mechanism. The resulting field equations are
inherently distinct to f(R) gravity in that they are second order. In the
present work, f(T) gravity is examined in the cosmological context with a
number of solutions reconstructed by means of an auxiliary scalar field. To do
this, various forms of the Hubble parameter are considered with an f(T)
lagrangian emerging for each instance. In addition, the inhomogeneous equation
of state (EoS) is investigated with a particular Hubble parameter model used to
show how this can be used to reconstruct the f(T) lagrangian. Observationally,
both the auxiliary scalar field or exotic terms in the FRW field equations give
the same results, meaning that the variation in the Hubble parameter may be
interpreted as the need to reformulate gravity in some way as is done in f(T)
gravity.
