Find the domain of definitions of the following function: `f(x)=sqrt(1-sqrt(1-x^(2)))`

Explanation

To find the domain of the function \( f(x) = \sqrt{1 - \sqrt{1 - x^2}} \), we need to ensure that all expressions under the square roots are non-negative.

### Step 1: Identify the inner square root condition
The first condition we need to satisfy is that the expression inside the outer square root must be greater than or equal to zero:
\[ 1 - \sqrt{1 - x^2} \geq 0 \]
This implies:
\[ \sqrt{1 - x^2} \leq 1 \]

### Step 2: Square both sides
To eliminate the square root, we square both sides:
\[ 1 - x^2 \leq 1 \]
This simplifies to:
\[ -x^2 \leq 0 \]
or equivalently:
\[ x^2 \geq 0 \]
This condition is always true for all real numbers \( x \).

### Step 3: Identify the condition for the inner square root
Next, we need to ensure that the expression under the inner square root is also non-negative:
\[ 1 - x^2 \geq 0 \]
This can be rearranged to:
\[ x^2 \leq 1 \]

### Step 4: Solve the inequality
The inequality \( x^2 \leq 1 \) implies:
\[ -1 \leq x \leq 1 \]
This means \( x \) must lie within the closed interval \([-1, 1]\).

### Conclusion
Thus, the domain of the function \( f(x) = \sqrt{1 - \sqrt{1 - x^2}} \) is:
\[ \text{Domain} = [-1, 1] \]