The domain of the function `f(x)=(1)/(sqrt(""^(10)C_(x-1)-3xx""^(10)C_(x)))` is

Options

A. `{9,10,11}`
B. `{9,10,12}`
C. all natural numbers
D. `{9,10}`

Explanation

To find the domain of the function \( f(x) = \frac{1}{\sqrt{{}^{10}C_{x-1} - 3 \cdot {}^{10}C_x}} \), we need to ensure that the expression inside the square root is positive, as the square root cannot be negative or zero (since it is in the denominator).

### Step 1: Set up the inequality
We start by setting up the inequality:
\[ {}^{10}C_{x-1} - 3 \cdot {}^{10}C_x > 0 \]

### Step 2: Use the formula for combinations
Recall that the combination formula is given by:
\[ {}^{n}C_{r} = \frac{n!}{r!(n-r)!} \]
For our case:
- \( {}^{10}C_{x-1} = \frac{10!}{(x-1)!(10-(x-1))!} = \frac{10!}{(x-1)!(11-x)!} \)
- \( {}^{10}C_x = \frac{10!}{x!(10-x)!} \)

### Step 3: Substitute and simplify
Substituting these into our inequality gives:
\[ \frac{10!}{(x-1)!(11-x)!} - 3 \cdot \frac{10!}{x!(10-x)!} > 0 \]
We can cancel \( 10! \) from both terms:
\[ \frac{1}{(x-1)!(11-x)!} - 3 \cdot \frac{1}{x!(10-x)!} > 0 \]

### Step 4: Rewrite the inequality
Rearranging gives:
\[ \frac{(10-x)! - 3 \cdot (11-x)!}{(x-1)!(11-x)!} > 0 \]
This simplifies to:
\[ (10-x)! - 3 \cdot (11-x) \cdot (10-x)! > 0 \]
Factoring out \( (10-x)! \):
\[ (10-x)! \left(1 - 3(11-x)\right) > 0 \]

### Step 5: Solve the inequality
The term \( (10-x)! \) is always positive for \( x \leq 10 \). Therefore, we focus on:
\[ 1 - 3(11-x) > 0 \]
This simplifies to:
\[ 1 - 33 + 3x > 0 \quad \Rightarrow \quad 3x > 32 \quad \Rightarrow \quad x > \frac{32}{3} \approx 10.67 \]

### Step 6: Determine integer values
Since \( x \) must be a natural number, the smallest integer satisfying \( x > 10.67 \) is \( x = 11 \).

### Step 7: Check the boundaries
Next, we need to check the boundaries given by the combination constraints:
- \( x-1 \geq 0 \) implies \( x \geq 1 \)
- \( 10 - x \geq 0 \) implies \( x \leq 10 \)

### Conclusion
The valid integer solutions that satisfy all conditions are \( x = 9 \) and \( x = 10 \).

Thus, the domain of the function \( f(x) \) is:
\[ \{9, 10\} \]