Mathematics - Functions Question with Solution | TestHub

Let $y_1={\sin }^{-1}\left[x^2+\frac{1}{2}\right]$ and $y_2={\cos }^{-1}\left[x^2-\frac{1}{2}\right]$. Then $y=y_1+y_2$.

Now, $y_1={\sin }^{-1}\left[x^2+\frac{1}{2}\right]$ is defined $\text{ if }-1\le \left[x^2+\frac{1}{2}\right]\le 1\Rightarrow -1\le x^2+\frac{1}{2}<2\Rightarrow -\frac{3}{2}\le x^2<\frac{3}{2}.\dots \dots \dots \left(A\right)$

Again $y_2={\cos }^{-1}\left[x^2-\frac{1}{2}\right]$ is defined $\text{ If }-1\le \left[x^2-\frac{1}{2}\right]\le 1\Rightarrow -1\le x^2-\frac{1}{2}<2\Rightarrow -\frac{1}{2}\le x^2<\frac{5}{2}$ .........(B)

Taking the intersection of $(\mathrm{A})$ and (B), we find that