Question

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

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

Answer 1

Answer: (B)

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