Question

Domain of function $f\left(x\right)=\frac{{\log }_{2x}3}{{\cos }^{-1}(2x-1)}$ is

$x\in [0,1)-\left\{\frac{1}{2}\right\}$.If it is true enter 1 else 0.

Answer 1

Consider $co{s}^{-1}(2x-1)$
Now we know that the domain of $co{s}^{-1}$ is $[-1,1]$
Hence
$-1\le 2x-1<1$ as if $co{s}^{-1}(2x-1)$ becomes $0$ then $f\left(x\right)$ will not be defined
$0\le x<1$
$x\in [0,1)$

Now consider $lo{g}_{2x}3$
$lo{g}_{2x}3$
$=\frac{log3}{log2x}$
Now domain of log cannot be less than 0, and $log2x\ne 0$
Hence
$x\in [0,\infty )-\left\{\frac{1}{2}\right\}$
Hence domain of $f\left(x\right)$ is
$\to x\in [0,1)-\left\{\frac{1}{2}\right\}$