Why is the \bf command applied to more than I asked?

Because \bf is a (deprecated) switch that is turned on. It doesn't take an argument in the form \bf{x}, but should rather be used {\bfseries x} (instead of {\bf x}. See Does it matter if I use \textit or \it, \bfseries or \bf, etc. and Will two-letter font style commands (\bf, \it, …) ever be resurrected in LaTeX?

Instead, use \boldsymbol (from amsmath) or \mathbf:

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\[ x(t), \boldsymbol{x}(t), \mathbf{x}(t), \boldsymbol{x(t)}, \mathbf{x(t)} \]

\end{document}

Use \bm instead of \bf. And \[...\] instead of $$...$$.

\documentclass{article}
\usepackage{bm}
\begin{document}
\[x(t), \bm{x}(t), \bm{x(t)}\]
\end{document}