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 (1) an even function (2) a periodic function (3) an odd function (4) neither even nor odd

Options

A. an even function
B. a periodic function
C. an odd function
D. neither even nor odd

Answer

A, B

Explanation

`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