$Iff\left(x\right)=\sqrt{2x-1}+5{\cos }^{-1}\left(\frac{2x-1}{3}\right)thenthedomainofthefunctionf\left(x\right)is$

$\begin{array}{l}f\left(x\right)=\sqrt{2x-1}+5{\cos }^{-1}\left(\frac{2x-1}{3}\right)isdefined\\ iff2x-1\ge 0and-1\le \frac{2x-1}{3}\le 1\\ \Rightarrow x\ge \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,and-3\le 2x-1\le 3\\ \Rightarrow -2\le 2x\le 4\\ \Rightarrow -1\le x\le 2\\ \end{array}$

$\therefore x\in [\frac{1}{2},2]$