Question

If $\alpha$ is a real number for which $f\left(x\right)={\log }_e{\cos }^{-1}x$ is defined, then a possible value of $\left[\alpha \right]$ (where $\left[\right]$ denotes the greatest function) is

• A. $0$
• B. $1$
• C. $-1$
• D. $-2$

Answer 1

Answer: (A), (C)

The correct options are
A $0$
C $-1$

For $f\left(x\right)$ to be defined
$co{s}^{-1}\left(x\right)>0$
$x<cos\left(0\right)$ ...(inequality sign gets reversed since $cos\left(x\right)$, is a decreasing function)
$x<1$
However, the domain for $co{s}^{-1}\left(x\right)$ is $[-1,1]$
Hence the domain for $f\left(x\right)$ will be $[-1,1)$
Now $\alpha ϵ[-1,1)$
Therefore
$\left[\alpha \right]$ can be 0 and -1.