`f(x)=sin^(-1)[e^(x)]+sin^(-1)[e^(-x)]` where [.] greatest integer function then

Options

A. domain of `f(x) " is " (-log_(e)2,log_(e)2)`
B. range of `f(x) ={pi}`
C. Range of `f(x) " is " {(pi)/(2),pi}`
D. `f(x) =cos^(-1)x` has only one solution

Answer

A, C

Explanation

We have `f(x)=sin^(-1)[e^(x)]+sin^(-1)[e^(-x)]`
We must have
`0 lt e^(x) lt 2 and 0 lt e^(-x) lt 2`
`implies x in (-log_(e) 2, log_(e)2)`
`implies f(x) ={(pi",",x=0),((pi)/(2)"," , x in (-log_(e)2","0)cup (0","log_(e)2)):}`