Question

Find the domain of the function \( f(x) = \sin^{-1(x^2 - 4) \). Also, find its range.}

Hint:

To find the domain of composite functions, solve inequalities for the input range of the inner function. For range, consider the output range of the outer function.

Answer 1

Step 1: Domain of the sine inverse function.
The sine inverse function \( \sin^{-1}(y) \) is defined for \( -1 \leq y \leq 1 \). Therefore, for \( f(x) = \sin^{-1}(x^2 - 4) \), the argument \( x^2 - 4 \) must satisfy: \[ -1 \leq x^2 - 4 \leq 1. \] Step 2: Solve the inequality. Rewrite the inequality: \[ -1 + 4 \leq x^2 \leq 1 + 4 \quad \Rightarrow \quad 3 \leq x^2 \leq 5. \] Taking the square root on both sides: \[ \sqrt{3} \leq |x| \leq \sqrt{5}. \] This implies: \[ x \in [-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}]. \] Step 3: Domain of \( f(x) \).
The domain of \( f(x) \) is: \[ \boxed{x \in [-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}]}. \] Step 4: Range of \( f(x) \).
The range of the sine inverse function is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \). Since \( x^2 - 4 \) varies between \( -1 \) and \( 1 \), the range of \( f(x) \) is: \[ \boxed{[-\frac{\pi}{2}, \frac{\pi}{2}]}. \]