Question

The function $f\left(x\right)=\sqrt{e^{{\cos }^{-1}\left({\log }_4{x}^2\right)}}$ is real valued. It is defined if

• A. $x\in [\frac{1}{2},2]$
• B. $x\in [-2,\frac{-1}{2}]\cup [\frac{1}{2},2]$
• C. $x\in [-2,\frac{1}{2}]$
• D. none of these

Answer 1

Answer: (B)

$x\in [-2,\frac{-1}{2}]\cup [\frac{1}{2},2]$

$f\left(x\right)=\sqrt{e^{{\cos }^{-1}\left({\log }_4{x}^2\right)}}$
For $f\left(x\right)$ to be defined $x^2>0\Rightarrow x\ne 0$
and $-1\le {\log }_4{x}^2\le 1\Rightarrow \frac{1}{4}\le x^2\le 4\Rightarrow x\in [-2,\frac{-1}{2}]\cup [\frac{1}{2},2]$
Hence we get required domain as $x\in [-2,\frac{-1}{2}]\cup [\frac{1}{2},2]$