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. periodic function
C. 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 " g(x) =0`
If ` x notin I " then " {-x} =1-{x}`
`implies g(-x)=sqrt(1-{x})=sqrt({x})-1 =g(x) `
`implies g(-x)=g(x) `
`implies g(x) ` is even function.
Also `g(x)={(0",",x in I),(g(-x)",",x notinI):}`
Thus `g(x)` is perodic function