If $f\left(x\right)=\sin x+\cos x,g\left(x\right)=x^2-1,\text{ then }g\left(f\right(x\left)\right)$ is invertible in the domain

$f\left(x\right)=\sin x+\cos x,g\left(x\right)=x^2-1$

$\Rightarrow g\left(f\right(x\left)\right)=(\sin x+\cos x{)}^2-1=\sin 2x$

Clearly $g\left(f\right(x\left)\right)$ is invertible in $-\frac{\pi }{2}\le 2x\le \frac{\pi }{2}$

[$\therefore$ $\sin \theta$ is invertible when $-\pi /2\le \theta \le \pi /2]$

$\Rightarrow -\frac{\pi }{4}\le x\le \frac{\pi }{4}$