Domain of $\sin x-e^{\sqrt{x}}$  is

Let $f\left(x\right)=\sin x-e^{\sqrt{x}}$                    Where $h\left(x\right)=e^{\sqrt{x}}$

Let $f\left(x\right)=g\left(x\right)-h\left(x\right)$               $h\left(x\right)$  is defined if $x\ge 0$  

Where $g\left(x\right)=\sin x$                         $\therefore$  Domain of $h\left(x\right)=\left[0,\infty \right)$

Domain of $g\left(x\right)=R$

$\therefore$  Domain of $f\left(x\right)=\left\{domainofg\left(x\right)\right\}\cap \left\{domainofh\left(x\right)\right\}$

$=R\cap \left[0,\infty \right)$

$=\left[0,\infty \right)$

$\therefore$  Domain of $f\left(x\right)=\left[0,\infty \right)$