Inverse Laplace transform does not restore the original function

I take f[t_]:=Sqrt[t^2 - 1] HeavisideTheta[t - 1], apply LaplaceTransform[ t f[t], t, x], get BesselK[2, x]/x, apply InverseLaplaceTransform[ f[t], t, x]/x and get -(1/(2 x)) + x, which is not the same as the original function.