The domain of `f(x) = 1/sqrt(x^(16) -x^(13) + x^(4) -x +1)`, is

Options

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

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{x^{16} - x^{13} + x^{4} - x + 1}} \), we need to ensure that the expression inside the square root is greater than zero. This is because the square root function is only defined for non-negative numbers, and since we are in the denominator, we specifically need it to be strictly greater than zero.

### Step-by-step Solution:

1. **Set up the inequality**:
We need to solve the inequality:
\[ x^{16} - x^{13} + x^{4} - x + 1 > 0 \]

2. **Analyze the expression**:
We can observe that \( x^{16} \) and \( x^{4} \) are always non-negative for all real \( x \). The term \( -x^{13} \) and \( -x \) can be negative, but we need to analyze the overall expression.

3. **Factor out common terms**:
We can factor the expression:
\[ x^{16} - x^{13} + x^{4} - x + 1 = x^{4}(x^{12} - x^{9} + 1) - x + 1 \]
However, it may not be straightforward to factor this further, so we will analyze the behavior of the function.

4. **Check for critical points**:
We can check specific values of \( x \) to see if the expression is positive or negative:
- For \( x = 0 \):
\[ f(0) = 0^{16} - 0^{13} + 0^{4} - 0 + 1 = 1 > 0 \]
- For \( x = 1 \):
\[ f(1) = 1^{16} - 1^{13} + 1^{4} - 1 + 1 = 1 - 1 + 1 - 1 + 1 = 1 > 0 \]
- For \( x = -1 \):
\[ f(-1) = (-1)^{16} - (-1)^{13} + (-1)^{4} - (-1) + 1 = 1 + 1 + 1 + 1 + 1 = 5 > 0 \]

5. **Behavior as \( x \to \infty \)**:
As \( x \) approaches infinity, the term \( x^{16} \) dominates the expression:
\[ x^{16} - x^{13} + x^{4} - x + 1 \to \infty \]
Thus, it remains positive.

6. **Behavior as \( x \to -\infty \)**:
As \( x \) approaches negative infinity, the leading term \( x^{16} \) (which is positive) will again dominate, ensuring the expression remains positive.

7. **Conclusion**:
Since the expression \( x^{16} - x^{13} + x^{4} - x + 1 \) is positive for all real values of \( x \), the domain of \( f(x) \) is:
\[ \text{Domain of } f(x) = (-\infty, \infty) \]