Which of the folllowing functions are defined for all x (A) `sin[x]+ cos[x}([x]}`denotes the greatest integers` le x`)(B) `sec^(-1) (1 + sin^2 x) `(C)` tan(logx)` (D) `sqrt(9/8+cos x+cos 2x)`

Options

A. `sin[x]+cos[x] ([x]"denotes the greatest integer " lex)`
B. `sec^(-1)(1+sin^(2)x)`
C. `tan(logx)`
D. `sqrt((9)/(8)+cosx+cos2x)`

Answer

A, B, D

Explanation

(a) Clearly, `sin[x]+cos[x]` is defined for all x.
(b) sin x always defined and `1+sin^(2) x ge 1`
`rArr sec^(-1)(1+sin^(2)x)` is defined for all x, as `1+sin^(2)xge1`
(c) `tan(logx)` is not defined if `logx=(2k+1)(pi)/(2)`
(d) `(9)/(8)+cos x +cos 2x`
`=2cos^(2)x+cosx+(1)/(8)gt0, AAx in R" as "D lt0`
Hence, domain is R.