If the function `g:(-oo,oo)rarr(-(pi)/(2),(pi)/(2))` is given by
`g(u)=2 tan ^(-1) (e^(u))-(pi)/(2).` Then g is

Options

A. even and is strictly increasing in `(0,oo)`
B. odd and is strictly decreasing `(-oo,oo)`
C. odd and is strictly increasing in `(-oo,oo)`
D. neither even nor odd , but is stictly increasing in `(-oo,oo)`

Answer

C

Explanation

we have
` g(u)=2 tan^(-1) (e^u)-pi/2`
`rArr g(u)=(2e^u)/(1+ e^(2u))gt 0 "for all u " rarr (-oo ,oo)`
`rarr` g is stictly increasing function in `(-oo,oo)`
Now
`g(u) = tan^(-1)(e^u)-pi/2`
`rArr g(u) = tan^(-1)(e^u )-(pi/2-tan ^(-1)(e^u))`
`rArr g(u)=tan^(-1)(e^u)-cot^(-1)(e^(u))`
`rArr g(-u)=tan ^(-1)(e^u)-cot^(-1)(e^(-u))`
`rArr g(-u) = tan^(-1)(e^(1//u))-cot^(-1)(e^(1//u))`
`rArr g(-u)=cot^(-1)(e^u)-tan^(-1)(e^u)=-g(u)`
Hence g(u) is odd and is strictly increasing `(-oo,oo)`