Find the domain of the function $f(x)={\tan }^{-1}\left(\sqrt{1-x^2}\right)$.

Step-by-step solution

1. The expression inside the square root must be non‑negative: $1-x^2\ge 0$.
2. Solve $1-x^2\ge 0$ to get $-1\le x\le 1$.
3. The square‑root yields a real non‑negative number, which is a valid argument for ${\tan }^{-1}$ (its domain is all real numbers).
4. Therefore the domain of $f$ is all $x$ satisfying $-1\le x\le 1$.

Concept tested: Domain of composite inverse trig function

Published 2 September 2026 · How this solution was prepared and checked

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