Question

Find the domain of $\sin^{-1} \sqrt{x - 1}$.

Hint:

When determining the domain of an inverse trigonometric function, ensure the expression inside the inverse satisfies the required range. For $\sin^{-1}(y)$, $y$ must be in the range $[-1, 1]$.

Answer 1

We are asked to find the domain of $\sin^{-1} \sqrt{x - 1}$. The function $\sin^{-1}(y)$ is defined for $-1 \leq y \leq 1$. Hence, we need: \[ -1 \leq \sqrt{x - 1} \leq 1. \] Since $\sqrt{x - 1} \geq 0$, the inequality becomes: \[ 0 \leq \sqrt{x - 1} \leq 1. \] Squaring both sides: \[ 0 \leq x - 1 \leq 1. \] Thus, \[ 1 \leq x \leq 2. \] Therefore, the domain of $\sin^{-1} \sqrt{x - 1}$ is $[1, 2]$.