The algebras of complex-valued functions \(f\) from a weighted semitopological semigroup \((S,w)\) such that \(\frac fw\) is continuous are studied (a weighted semitopological semigroup \((S,w)\) is a semitopological semigroup \(S\) and a function \(w:S\to (0,\infty)\) such that \(w\) is bounded to any compact subset of \(S\) and \(w(st)\leq w(s)w(t)\) for all \(s,t\in S\)). The images of these algebras under continuous homomorphisms of weighted semigroups are described (a continuous homomorphism \(\theta :S_1\to S_2\) between semitopological semigroups \(S_1\) and \(S_2\) is a continuous homomorphism from a weighted semitopological semigroup \((S_1,w_1)\) into a weighted semitopological semigroup \((S_2,w_2)\) whenever \(w_1=w_2\circ\theta\)). As a consequence the authors obtain that a weighted semitopological semigroup \((S,w)\) has no weighted compactification whenever \(w\) is unbounded or \(w(s)<1\) for all \(s\in S\). On the other hand, if \((S,w)\) is a weighted semitopological semigroup \((S,w)\) such that \(w\) is bounded and \(w(s)\geq 1\) for some \(s\in S\), then a sufficient condition is given for the existence of a weighted compactification of \((S,w)\) that is universal with respect to topological properties.