As part of his groundbreaking work on algorithmic randomness, Solovay demonstrated in the 1970s the remarkable fact that there are computable upper bounds of prefix-free Kolmogorov complexity $K$ that are tight on infinitely many values (up to an additive constant). Such computable upper bounds are called Solovay functions. Recent work of Bienvenu and Downey~[STACS 2009, LIPIcs 3, pp 147-158] indicates that Solovay functions are deeply connected with central concepts of algorithmic randomness such as $\Omega$ numbers, K-triviality, and Martin-L\"{o}f randomess. In what follows, among other results we answer two open problems posed by Bienvenu and Downey about the definition of $K$-triviality and about the Gács-Miller-Yu characterization of Martin-L\"{o}f randomess. The former defines a sequence~$A$ to be K-trivial if $K(A\uhr n) \lep K(n)$, the latter asserts that a sequence~$A$ is Martin-L\"{o}f random iff $C(A\uhr n) \gep n-K(n)$. So both involve the noncomputable function $K$. As our main results we show that in both cases $K(n)$ can be equivalently replaced by any Solovay function, and, what is more, that among all computable functions such a replacement is possible exactly for the Solovay functions. Moreover, similar statements hold for the larger class of all right-c.e.\ in place of the computable functions. These full characterizations, besides having significant theoretical interest on their own, will be useful as tools when working with K-trivial and Martin-L\"{o}f random sequences.