The range [परिसर] of the function `f(x) = tan sqrt((pi^(2))/(9)-x^(2))`, is

Options

A. `[0, sqrt3]`
B. `(0, sqrt3)`
C. `[0, sqrt3)`
D. `(0, sqrt3]`

Answer

A

Explanation

Clearly, = , f(x) is defined for`x in [-pi//3,pi//3]`
Since tan x is an incresing function in `[0,pi//2)` and
`0 le (pi^(2))/(9) - x^(2)(pi)/(9) "for" x in [(-pi)/(3),(pi)/(3)]`
` rArr 0 le sqrt((pi^(2))/(9)-x^(2)) le (pi)/(3) "for" x in [(-pi)/(3),(pi)/(3)]`
Also, `f(-x) = f(x) ` for all `x in [-pi//3, pi//3]`. Thererfore ,
Range `(f) = [f((-pi)/(3)),f(0)]=[tan0, tan,(pi)/(3)]=[0,sqrt(3)]`