Let `f:(-pi/2,pi/2)vecR` be given by `f(x)=(log(sec"x"+tan"x"))^3` then `f(x)` is an odd function `f(x)` is a one-one function `f(x)` is an onto function `f(x)` is an even function

Options

A. `f(x)` is an odd function
B. `f(x)` is a one-one function
C. `f(x)` is an onto function
D. `f(x)` is an event function

Answer

A, B, C

Explanation

`f(x)=(log(secx+tanx))^(3) AA in (-(pi)/(2),(pi)/(2))`
` :. f(-x)=(log(secx-tanx))^(3) `
`=("log"(1)/(secx+tanx))^(3)`
`=(-log(secx+tanx))^(3)`
`= -(log(secx+tanx))^(3)`
`= -f(x)`
Hence, `f(x)` is odd function.
Let `g(x)=secx+tanx AA x in (-(pi)/(2),(pi)/(2))`
So, `g'(x)=secx(secx+tanx) gt 0 AA x in (-(pi)/(2),(pi)/(2))`
`implies g(x)` is one-one function
Hence, `(log_(e)(g(x)))^(3)` is also one-one function.
And `g(x) in (0,oo) AA x in (-(pi)/(2),(pi)/(2))`
`g(x)=secx+tanx`
`=(1+sinx)/(cosx)=(1-cos((pi)/(2)+x))/(sin((pi)/(2)+x))=tan((pi)/(4)+(x)/(2))`
Now
`x in (-(pi)/(2),(pi)/(2))`
`implies (pi)/(4)+(x)/(2) in (0,(pi)/(2))`
`implies tan((pi)/(4)+(x)/(2)) in (0,oo)`
`implies log(g(x)) in R`
Hence, `f(x)` is an onto function.