The domain [प्रान्त] of the function `f(x)=sqrt(1-sqrt(1-sqrt(1-x^(2))))` , is

Options

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

Explanation

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

### Step 1: Innermost Square Root
The innermost expression is \( \sqrt{1 - x^2} \). For this to be defined, we need:
\[ 1 - x^2 \geq 0 \]
This simplifies to:
\[ x^2 \leq 1 \]
Taking the square root of both sides gives:
\[ -1 \leq x \leq 1 \]

### Step 2: Second Square Root
Next, we consider the second square root \( \sqrt{1 - \sqrt{1 - x^2}} \). For this to be defined, we need:
\[ 1 - \sqrt{1 - x^2} \geq 0 \]
This simplifies to:
\[ \sqrt{1 - x^2} \leq 1 \]
Since \( \sqrt{1 - x^2} \) is always non-negative, this condition is automatically satisfied as long as \( 1 - x^2 \geq 0 \), which we already established in Step 1.

### Step 3: Outer Square Root
Now we consider the outermost 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 gives:
\[ 1 - \sqrt{1 - x^2} \leq 1 \]
This condition is also automatically satisfied since \( \sqrt{1 - x^2} \) is non-negative.

### Step 4: Final Condition
We must also ensure that:
\[ 1 - \sqrt{1 - x^2} \geq 0 \]
This leads us back to the condition:
\[ \sqrt{1 - x^2} \leq 1 \]
which is satisfied for \( -1 \leq x \leq 1 \).

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