Question

How do you find the domain of inverse sin `sqrt(x/(1-x)` ?

Answer 1

Domain is `x in {0 ≤ x < 1, x in RR}`

Explanation:

We are asked to find the domain of

`f(x ) =sin^-1sqrt(x/(1 -x))`

First of all, we note that `x!= 1`, because it would make the function undefined.

Furthermore,

`x/(1 - x) ≥ 0`

Because the square root is only defined when it is positive.

Solving this inequality, we get

`0 ≤ x ≤ 1`

But recall that `x= 1` is not in the function's domain, so the domain becomes

`0 ≤ x < 1`

We finally note that the function `y = arcsin(x)` has domain `-1 ≤ x ≤ 1`, therefore, our domain from the square root function agrees with the domain from the inverse sine function.

Hopefully this helps!