Question

Consider the function $fx={\cos }^{-1}{3}^x+$ ${\sin }^{-1}{3}^x-1$, then (where, $\cdot$ represents the greatest integer part of $x$ )

• A. the domain of $fx$ is $x\in -\infty ,0$
• B. the range of $fx$ is singleton
• C. $fx$ is an even function
• D. $fx$ is an odd function

Answer 1

Answer: (B)

For $fx$ to be defined $3^x=0,1$
$\therefore 3^x\in 0,2\Rightarrow x\in -\infty ,{\log }_{3}2$
Range of $fx={\cos }^{-1}0+{\sin }^{-1}-1,{\cos }^{-1}1+{\sin }^{-1}0\equiv 0$
$\therefore fx=0\forall x\in -\infty ,{\log }_{3}2$
So, the function is not symmetric about the origin.
$\therefore fx$ is neither even nor odd