Question

True or False$Thedomainofy={\sin }^{-1}xis-\frac{\mathrm{\pi }}{2}\le x\le \frac{\mathrm{\pi }}{2}.$

Short answer

False Statement$Thedomainofy={\sin }^{-1}xis-\frac{\mathrm{\pi }}{2}\le x\le \frac{\mathrm{\pi }}{2}.$

Step-by-step solution

Step 1. Given Information:

Given statement:

$Thedomainofy={\sin }^{-1}xis-\frac{\mathrm{\pi }}{2}\le x\le \frac{\mathrm{\pi }}{2}.$

We want to know is the given statement is True or False.

Step 2. Solution:

If we know the range of the function then we can determine the domain of its inverse as follows:

Domain of the inverse function = The range of the original function.

We know that the range of the sine function is [- 1 , 1].

Therefore, the domain of the its inverse defined as y = sin^-1 x is equal to the range of the sine function which is [- 1 ,1].

Hence, the domain of y = sin^-1 x is $-1\le x\le 1$so the statement is false.