Question

Determine the range of the function $y={\sin }^{-1}[\frac{1}{2}+x^2]$ where [] denotes the greatest integer function.

• A. $y=\{-\pi /2,\pi /2\}$
• B. $y=\{0,\pi /2\}$
• C. $y=\{0,\pi \}$
• D. $y=\{-\pi ,\pi \}$

Answer 1

Answer: (B)

$y=\{0,\pi /2\}$

$\sin y=[\frac{1}{2}+x^2]\le 1$ as $\frac{1}{2}+x^{2}is+ive\forall x$
$\therefore [\frac{1}{2}+x^2]$ is a+ive integer $\le 1$ i.e. 0 and 1 only. $\sin y=0$ or $1\therefore y=0,\pi /2$ only two values. $\therefore$ Range of $y=\{0,\pi /2\}$ only.