Find all possible values of expression `sqrt(1-sqrt(x^(2)-6x+9)).`

Explanation

To find all possible values of the expression \( \sqrt{1 - \sqrt{x^2 - 6x + 9}} \), we can follow these steps:

### Step 1: Simplify the expression inside the square root
We start with the expression:
\[ \sqrt{1 - \sqrt{x^2 - 6x + 9}} \]
Notice that \( x^2 - 6x + 9 \) can be factored:
\[ x^2 - 6x + 9 = (x - 3)^2 \]
Thus, we can rewrite the expression as:
\[ \sqrt{1 - \sqrt{(x - 3)^2}} \]

### Step 2: Simplify the square root of the square
The square root of a square can be expressed in terms of absolute value:
\[ \sqrt{(x - 3)^2} = |x - 3| \]
Now, substituting this back into the expression gives us:
\[ \sqrt{1 - |x - 3|} \]

### Step 3: Determine the range of the expression
For the expression \( \sqrt{1 - |x - 3|} \) to be defined, the term inside the square root must be non-negative:
\[ 1 - |x - 3| \geq 0 \]
This simplifies to:
\[ |x - 3| \leq 1 \]

### Step 4: Solve the absolute value inequality
The inequality \( |x - 3| \leq 1 \) means that:
\[ -1 \leq x - 3 \leq 1 \]
Adding 3 to all parts of the inequality gives:
\[ 2 \leq x \leq 4 \]

### Step 5: Determine the possible values of the original expression
Now, we need to find the possible values of \( \sqrt{1 - |x - 3|} \) for \( x \) in the interval \( [2, 4] \).

- When \( x = 2 \):
\[ |2 - 3| = 1 \implies \sqrt{1 - 1} = \sqrt{0} = 0 \]
- When \( x = 3 \):
\[ |3 - 3| = 0 \implies \sqrt{1 - 0} = \sqrt{1} = 1 \]
- When \( x = 4 \):
\[ |4 - 3| = 1 \implies \sqrt{1 - 1} = \sqrt{0} = 0 \]

### Step 6: Conclusion
The expression \( \sqrt{1 - |x - 3|} \) takes values from 0 to 1 as \( x \) varies from 2 to 4. Therefore, the possible values of the expression \( \sqrt{1 - \sqrt{x^2 - 6x + 9}} \) are:
\[ [0, 1] \]