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

$\begin{array}{l}f\left(x\right)=\sin x+\cos x,g\left(x\right)=x^2-1\\ g\left(f\left(x\right)\right)=g(\sin x+\cos x)\\ ={(\sin x+\cos x)}^2-1\\ ={\sin }^{2}x+{\cos }^{2}x+2\sin x\cos x-1\\ =2\sin x\cos x\\ =\sin 2x\\ it\frac{-\pi }{4}\le x\le \frac{\pi }{4}then\sin 2xisone-oneandonto\\ \therefore g\left(f\left(x\right)\right)isinvertiblein\frac{-\pi }{4}\le x\le \frac{\pi }{4}\end{array}$