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We formulate the convenient and mimetic f (Q) gravities in terms of the metric and four scalar fields with the Q being the non-metricity scalar. As a result, it is shown that the obtained field equations are well-defined. By using the field equations, we show how the f (Q) models which realise any given FLRW spacetime may be reconstructed. We construct convenient f (Q) gravity and mimetic f (Q) gravity, which describe the inflation and dark energy epochs and even unify the inflation and dark energy. Moreover, a radiation-dominated epoch and early dark energy may be easily included in the above theories as is explicitly shown. It is shown that the inflationary spectral index nₛ and the tensor-to-scalar ratio r are consistent with Planck data for some models of f (Q) gravity. Corresponding asymptotic forms of f (Q) and mimetic f (Q) are given. Finally, some remarks on possible reconstruction for spherically symmetric spacetime are given.
Nojiri et al. (Mon,) studied this question.