Mathematics - Functions Question with Solution | TestHub

Given that, $f(x)=\sin x+\cos x$ $g(x)=x^2-1$ $g[f(x)]=(\sin x+\cos x{)}^2-1$ $={\sin }^{2}x+{\cos }^{2}x+2\sin x\cos x-1$ $=1+\sin 2x-1=\sin 2x$ $\left(\because {\sin }^{2}x+{\cos }^{2}x=1,\sin x=2\sin x\cos x\right)$Among the given options, $\sin 2x$ is monotonous (here strictly increasing) in $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$. So, $g(f(x))$ is invertible in $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$.