Thank you for reporting, we will resolve it shortly
Q.
The domain of the function $f\left(x\right) = \frac{1}{\sqrt{x^{12} - x^{9} +x^{4} -x + 1}} $ is given by
Relations and Functions
Solution:
$f(x)$ is defined for $x^{12} - x^9 + x^4 - x +1 > 0$
$\Rightarrow \, x^4 (x^8 +1) - x(x^8 +1) +1 > 0$
$\Rightarrow \, (x^8 +1)x(x^3 -1) +1 > 0$
If $x \ge 1$ or $x \le -1,$ then the above expression is positive.
If $-1 < x \le 0 $, the above inequality still holds.
If 0 < x <1 , then
$x^{12} - x(x^8 +1) + (x^4 +1) > 0 $
$[\because \, x^4 +1 > x^8 +1$ and so $x^4 + 1 > x (x^8 + 1)]$
The domain of $f = (- \infty , \infty)$