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In this manuscript, we present a number of fascinating explicit reconstructions for the f(Q) gravity from the background of Friedmann-Laîmatre-Robertson-Walker (FLRW) evolution history. We find the more general functions of non-metricity scalar Q that admit exact ΛCDM expansion history. Adding extra degrees of freedom to the matter sector is the only method to get the scale factor to behave in this manner for more generic functions of Q. In addition, a cosmological reconstruction for modified f(Q) gravity is constructed in terms of e-folding. It is shown how any FLRW cosmology can arise from a specific f(Q) theory. We also reconstruct the well-known cosmological evolution for the specific examples of ΛCDM cosmology.