Find the domain of $\displaystyle \sec ^{-1}(2 x+1)$.

Domain of $\displaystyle \sec ^{-1} \mathbf{x}$ is $\displaystyle (-\infty,-\mathbf{1}] \cup[\mathbf{1}, \infty)$ \[\begin{array}{r} \Rightarrow \mathbf{2 x + 1} \leq-\mathbf{1} \text { or } \mathbf{2 x}+\mathbf{1} \geq \mathbf{1} \Rightarrow \mathbf{x} \leq-\mathbf{1} \text { or } \mathbf{x} \geq \mathbf{0} \\ \Rightarrow \operatorname{Domain}=(-\infty,-\mathbf{1}] \cup[\mathbf{0}, \infty) \end{array} \]