Question

Find the domain of the function $f(x) = \sin^{-1}(-x^2)$.

Hint:

When determining the domain of inverse trigonometric functions, ensure that the argument lies within the valid range of the function (for $\sin^{-1}$, the range is $[-1, 1]$).

Answer 1

The inverse sine function, $\sin^{-1} x$, is defined only for $x \in [-1, 1]$. Thus, we must have: \[ -1 \leq -x^2 \leq 1. \] Multiplying through by -1 (which reverses the inequality signs): \[ 1 \geq x^2 \geq 0. \] This means that $x^2 \leq 1$, which implies: \[ -1 \leq x \leq 1. \] Thus, the domain of $f(x) = \sin^{-1}(-x^2)$ is $x \in [-1, 1]$.