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The function 'g' defined by g(x)=sin(sin...

The function 'g' defined by `g(x)=sin(sin^(-1){sqrt(x)}+cos (sin^(-1){sqrt(x)})-1` where {x} denotes the functional part function is

A

an even function

B

a periodic function

C

an odd function

D

neither even nor odd

Text Solution

Verified by Experts

The correct Answer is:
A, B
`g(x)=sin(sin^(-1)sqrt({x}))+cos(sin^(-1)sqrt({x}))-1`
`=sqrt({x})+cos(cos^(-1)sqrt(1-{x}))-1`
`=sqrt({x})+sqrt(1-{x})-1`
If `x in I` then {x} = 0
then g(x)=0
If `x in I` then `{-x}=1-{x}`
`rArr" "g(-x)=sqrt(1-{x})+sqrt({x})-1=g(x)`
`rArr" "g(-x)=g(x)`
`rArr" "g(x)" is even function."`
`"Also, "g(x)={{:(0",",x in I),(g(-x)",",x cancelinI):}`
Thus g(x) is periodic funciton
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