Question

Consider the function $f\left(x\right)=cos^{- 1}\left(\left[2^{x}\right]\right)+sin^{- 1}\left(\left[2^{x}\right] - 1\right)$ , then

(where [.] represents the greatest integer part function)

• A. Domain of $f(x)$ is $x\in - \infty , 0]$
• B. Range of $f\left(x\right)$ is singleton
• C. $f\left(x\right)$ is an even function
• D. $f\left(x\right)$ is an odd function

Answer 1

Answer: (B)

For $f\left(x\right)$ to be defined $\left[2^{x}\right]=0,1$
$\therefore 2^{x}\in \left[0,2\right)\Rightarrow x\in \left(- \infty , 1\right)$
Range $\left\{c o s^{- 1} \left(0\right) + s i n^{- 1} \left(- 1\right) , cos^{- 1} \left(1\right) + s i n^{- 1} \left(0\right)\right\}\equiv \left\{0\right\}$
$\therefore $ $f\left(x\right)=0 \, \forall x\in \left(- \infty , 1\right)$
$So,$ Domain is not symmetric about the origin
$\therefore $ $f\left(x\right)$ is neither even nor odd