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 $x\geq 1$ or $x\leq -1$ , then the above expression is positive.
If $-1 < x\leq 0$ , the above inequality still holds.
If $0 < x < 1$ , then $x^{12}-x\left(x^{8} + 1\right)+\left(x^{4} + 1\right)>0$
$\left[as \, x^{4} + 1 > x^{8} + 1 \, s o \, x^{4} + 1 > x \left(x^{8} + 1\right)\right]$
Hence, the domain of $f=\left(- \infty , \infty\right).$