The domain of definiton of the function `f(x)=(1)/(sqrt(x^(12)-x^(9)+x^(4)-x+1))` , is

Options

A. `(-oo,-1)`
B. `(1,oo)`
C. `(-1,1)`
D. R

Explanation

To find the domain of the function \( f(x) = \frac{1}{\sqrt{x^{12} - x^{9} + x^{4} - x + 1}} \), we need to ensure that the expression inside the square root is positive, as the square root of a negative number is not defined in the real number system, and the denominator cannot be zero.

### Step 1: Set up the inequality
We need to solve the inequality:
\[ x^{12} - x^{9} + x^{4} - x + 1 > 0 \]

### Step 2: Factor the polynomial
We can factor parts of the polynomial:
1. Group the first two terms and the last two terms:
\[ x^{12} - x^{9} + x^{4} - x + 1 = x^{9}(x^{3} - 1) + (x^{4} - x + 1) \]
2. Notice that \( x^{3} - 1 \) can be factored as:
\[ x^{3} - 1 = (x - 1)(x^{2} + x + 1) \]
3. Thus, we can rewrite the expression as:
\[ x^{9}(x - 1)(x^{2} + x + 1) + (x^{4} - x + 1) \]

### Step 3: Analyze the factors
1. The term \( x^{9} \) is always non-negative for all real \( x \).
2. The term \( x - 1 \) is zero when \( x = 1 \) and negative for \( x < 1 \).
3. The quadratic \( x^{2} + x + 1 \) has a discriminant of \( 1 - 4 = -3 \), which is negative, indicating it is always positive.

### Step 4: Determine the sign of the polynomial
1. For \( x < 0 \):
- \( x^{9} < 0 \)
- \( x - 1 < 0 \)
- Thus, \( x^{9}(x - 1) > 0 \) (since negative times negative is positive).
- The whole expression is positive.

2. For \( x = 0 \):
- The expression evaluates to \( 1 > 0 \).

3. For \( 0 < x < 1 \):
- \( x^{9} > 0 \)
- \( x - 1 < 0 \)
- Thus, \( x^{9}(x - 1) < 0 \).
- We need to check if \( x^{4} - x + 1 > 0 \). This quadratic has a positive discriminant, and its minimum value occurs at \( x = \frac{1}{2} \):
\[ \left(\frac{1}{2}\right)^{4} - \frac{1}{2} + 1 = \frac{1}{16} - \frac{8}{16} + \frac{16}{16} = \frac{9}{16} > 0 \]
- Thus, the expression is positive.

4. For \( x = 1 \):
- The expression evaluates to \( 1 > 0 \).

5. For \( x > 1 \):
- Both \( x^{9} > 0 \) and \( x - 1 > 0 \), so \( x^{9}(x - 1) > 0 \).
- The whole expression is positive.

### Conclusion
The polynomial \( x^{12} - x^{9} + x^{4} - x + 1 \) is positive for all \( x \in (-\infty, 0] \cup [1, \infty) \). Therefore, the domain of the function \( f(x) \) is:
\[ (-\infty, 0] \cup [1, \infty) \]

### Final Answer
The domain of the function \( f(x) \) is \( (-\infty, 0] \cup [1, \infty) \).