Question

The domain of the function $f\left(x\right)=\frac{4}{\sqrt{x^{12}-x^9+x^4-x+1}}$ is

• A.
• B.
• C.
• D.

Answer 1

Answer: (D)

$f\left(x\right)$ is defined for $x^{12}-x^9+x^4-x+1>0$
$\Rightarrow x^4\left(x^8+1\right)-x\left(x^8+1\right)+1>0$
$\Rightarrow \left(x^8+1\right)x\left(x^3-1\right)+1>0$
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 $x^{12}-x\left(x^8+1\right)+\left(x^4+1\right)>0$
$\left[as{x}^4+1>x^8+1so{x}^4+1>x\left(x^8+1\right)\right]$
Hence, the domain of $f=\left(-\infty ,\infty \right).$