The domain of the function f(x)= x- 1-x^ 2 is

For $f(x)$ to be defined, we must have $x-\sqrt{1-x^{2}} \geq 0 or$\quad x \geq \sqrt{1-x^{2}} \gt 0\therefore \quad x^{2} \geq 1-x^{2}$or$x^{2} \geq \frac{1}{2}$Also,$1-x^{2} \geq 0$or$x^{2} \leq 1$. Now,$x^{2} \geq \frac{1}{2} \Rightarrow\left(x-\frac{1}{\sqrt{2}}\right)\left(x+\frac{1}{\sqrt{2}}\right) \geq 0\Rightarrow x \leq-\frac{1}{\sqrt{2}}$or$x \geq \frac{1}{\sqrt{2}}$Also,$x^{2} \leq 1 \Rightarrow(x-1)(x+1) \leq 0\Rightarrow-1 \leq x \leq 1$Thus,$x \gt 0, x^{2} \geq \frac{1}{2}$and$x^{2} \leq 1\Rightarrow x \in \left[\frac{1}{\sqrt{2}},1\right]