The domain of
`f(x) = sqrt(1-sqrt(1-sqrt(1-sqrt(1-x^(2)))))`, is

Options

A. `[-1,0]`
B. `[0,1]`
C. `(-1,1)`
D. `[-1,1]`

Answer

To find the domain of the function \( f(x) = \sqrt{1 - \sqrt{1 - \sqrt{1 - \sqrt{1 - x^2}}}} \), we need to ensure that all expressions under the square roots are non-negative. We will go through the function step by step, analyzing each square root.

### Step 1: Analyze the innermost square root
The innermost expression is \( 1 - x^2 \). For this to be non-negative:
\[ 1 - x^2 \geq 0 \]
This simplifies to:
\[ x^2 \leq 1 \]
Taking the square root of both sides, we find:
\[ -1 \leq x \leq 1 \]
Thus, the first condition gives us the interval \( x \in [-1, 1] \).

### Step 2: Analyze the second square root
Next, we consider the second square root, which is \( \sqrt{1 - \sqrt{1 - x^2}} \). For this to be defined, we need:
\[ 1 - \sqrt{1 - x^2} \geq 0 \]
This simplifies to:
\[ \sqrt{1 - x^2} \leq 1 \]
Squaring both sides:
\[ 1 - x^2 \leq 1 \]
This condition is always satisfied since \( 1 - x^2 \geq 0 \) from the first condition. Therefore, this does not impose any additional restrictions.

### Step 3: Analyze the third square root
Now we look at the third square root, \( \sqrt{1 - \sqrt{1 - \sqrt{1 - x^2}}} \). For this to be defined, we need:
\[ 1 - \sqrt{1 - \sqrt{1 - x^2}} \geq 0 \]
This simplifies to:
\[ \sqrt{1 - \sqrt{1 - x^2}} \leq 1 \]
Squaring both sides:
\[ 1 - \sqrt{1 - x^2} \leq 1 \]
This is again always satisfied since it is equivalent to \( \sqrt{1 - x^2} \geq 0 \), which we already know is true.

### Step 4: Analyze the outermost square root
Finally, we analyze the outermost square root, \( \sqrt{1 - \sqrt{1 - \sqrt{1 - \sqrt{1 - x^2}}}} \). For this to be defined, we need:
\[ 1 - \sqrt{1 - \sqrt{1 - \sqrt{1 - x^2}}} \geq 0 \]
This simplifies to:
\[ \sqrt{1 - \sqrt{1 - \sqrt{1 - x^2}}} \leq 1 \]
Squaring both sides:
\[ 1 - \sqrt{1 - \sqrt{1 - x^2}} \leq 1 \]
Again, this condition is always satisfied.

### Conclusion
The only restriction we have is from the first condition, which gives us the interval:
\[ x \in [-1, 1] \]

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