Chapter 4.7, Problem 1E

To determine

To fill: the blank of given statement.

Answer to Problem 1E

Alternative notation $y={\sin }^{-1}x$

Domain $-1\le x\le 1$

Explanation of Solution

Given information:

Given statement

Function	Alternative Notation	Domain	Range
$y=\arcsin x$			$-\frac{\pi }{2}\le y\le \frac{\pi }{2}$

Calculation:

The definition of inverse sine function is as follows

  $y=\arcsin x$ if and only if $x=\sin y$ where $-1\le x\le 1$ and $-\frac{\pi }{2}\le y\le \frac{\pi }{2}$ .

Hence

Function	Alternative Notation	Domain	Range
$y=\arcsin x$	$y={\sin }^{-1}x$	$-1\le x\le 1$	$-\frac{\pi }{2}\le y\le \frac{\pi }{2}$

  $y=\arcsin xy={\sin }^{-1}x-1\le x\le 1-\frac{\pi }{2}\le y\le \frac{\pi }{2}$ .