The range of the function `f(x)=(1)/(2-cos 3x)` is

Options

A. `(1//3, 1 )`
B. `[1//3, 1 )`
C. `[1//3, 1]`
D. none of these

Answer

C

Explanation

Cleary, domain of `f(x)` is R and `f(x) lt 0` for all `x in R`
Let `y = f(x)`. Them
`y = (1)/(2-sin 3x)`
`rArr sin 3x = (2y-1)/(y) rArr x = (1)/(3) sin^(-1)((2y-1)/(y))`
For x to be real, we must have
`rArr -1 le (2y-1)/(y) le 1`
`rArr - y le 2y -1 le y" "[beacuse y = f(x) gt 0]`
` rArr -y le 2y - 1 and 2y - 1 le y`
`rArr y gt (1)/(3) and y le 1 rArr y in [1//3,1]`
Hence, range `(f) = [1//3,1]`