Show that $\displaystyle \mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}+\mathrm{x}-\tan ^{-1} \mathrm{x}$ is strictly increasing in its domain.

$$\begin{aligned} \mathrm{f}^{\prime}(\mathrm{x})= & \mathrm{e}^{\mathrm{x}}+\mathrm{e}^{-\mathrm{x}}+1-\frac{1}{1+\mathrm{x}^{2}} & =\mathrm{e}^{\mathrm{x}}+\frac{1}{\mathrm{e}^{\mathrm{x}}}+\frac{\mathrm{x}^{2}}{1+\mathrm{x}^{2}}>0 \text { for all } \mathrm{x} \in R \end{aligned}$\displaystyle \therefore \mathrm{f}$ is strictly increasing over its domain R