Mathematics - Functions Question with Solution | TestHub

Period of $\mathrm{f}(\mathrm{x})$ is $2\pi$, but $\mathrm{f}(\mathrm{x})$ is not defined for $\mathrm{x}\in (\pi /2,3\pi /2)$. Hence it suffices to consider $\mathrm{x}\in [-\pi /2$, $\pi /2]$. Further since $\mathrm{f}(\mathrm{x})$ is even, we consider $\mathrm{x}\in [0,\pi /2]$.

Now $\sqrt{\cos (\sin x)}$ and $\sqrt{\sin (\cos x)}$ are decreasing functions for $\mathrm{x}\in [\pi ,\pi /2]$. $\Rightarrow R_f=[f(\pi /2),f(0)]=[\sqrt{\cos 1,}1+\sqrt{\sin 1}]$