Question

The domain of the function $f\left(x\right)=\sqrt{x^{12}-x^3+x^4-x+1}$ is

• A. $(1,\infty )$
• B. $(-\infty ,0)$
• C. $(-\infty ,0)\cup (1,\infty )$
• D. $R$

Answer 1

The correct option is D $R$

We have,
$f\left(x\right)=\sqrt{x^{12}-x^3+x^4-x+1}$
For $f\left(x\right)$ to be defined,
$x^{12}-x^3+x^4-x+1\ge 0$
$\Rightarrow x^{12}+x^4+1\ge x^3+x\cdots \left(1\right)$
Here in $L.H.S$ we have even powers and $R.H.S$ have odd powers of $x$. So, for all $x\le 0$, $\left(1\right)$ will be true

Now the power of $x$ in $L.H.S$ are greater than the power of $x$ in $R.H.S$ and $L.H.S\ge 1.$ So, for all $x>0,\left(1\right)$ will be true

Therefore, the domain of $f\left(x\right)$ is $R$