Let `f(x)=sec^(-1)[1+cos^(2)x],` where [.] denotes the greatest integer function. Then find the domain and range

Options

A. domain of `f` is R
B. domain of `f` is `[1,2]`
C. domain of `f` is `[1,2]`
D. range of `f " is " {sec^(-1) 1, sec^(-1)2}`

Answer

A, B

Explanation

`f(x)=sec^(-1)[1+cos^(2)x]`
`f(x)` is defined if `[1+cos^(2)x] le -1 or [1+cos^(2)x] ge 1`
i.e., `[cos^(2)x] le -2("not possible") or [cos^(2)x] ge 0`
i.e., `cos^(2)x ge 0 or x in R`
Now, `0 le cos^(2)x le 1 or 1 le 1+ cos^(2)x le 2`
`or [1+cos^(2)x]=1,2`
`or sec^(-1)[1+cos^(2)x]=sec^(-1)1, sec^(-1)2`
Hence, the range is `{sec^(-1)1,sec^(-1)2}.`