Domain of a function involving a radical

Question

$$f(x) = \frac{1 - \sqrt{1 - x^2} }{x}$$

How do I find the domain of this function? I know that f(x) is not defined when $$x = 0$$ or when $$x = \sqrt{-number}$$

attempt:

$$1 - x^2 \ge 0$$ $$x = \pm 1$$ $$x = 0$$

$$Domain: x\in R : x \neq0, -1 \leq x \leq 1$$

Answer 1

The key is not when $x = \sqrt{-\textrm{something}}$, but rather when $1-x^2 < 0$. If $1-x^2 < 0$, then the radical in the numerator is the square-root of a negative number, and hence it is not real.

Similarly, you identified that $x=0$ causes an issue. So your domain is the reals, minus the points that give you the issues mentioned above.

Answer 2

Solve the inequality $1-x^2\geq 0$ and combine with $x\neq 0$.