Find the period of the following.
(i) `f(x)=(2^(x))/(2^([x]))`, where [.] represents the greatest integer function.
(ii) `f(x)=e^(sinx)`
(iii) `f(x)=sin^(-1)(sin 3x)`
(iv) `f(x) =sqrt(sinx)`
(v) `f(x)=tan((pi)/(2)[x]),` where [.] represents greatest integer function.

Explanation

(i) `f(x)=(2^(x))/(2^([x]))=2^(x-[x])=2^{x}`, where {.} is fractional part function.
Period of `{x}` is '1'. So, period of `2^{x}` is 1 as `2^({x+1})=2^{x}` for all real x.
(ii) `f(x)=e^(sinx).`
Period of `sinx" is " 2pi`.
So, period of `e^(sinx)` is also `2pi" as " e^(sin(x+2pi))=e^(sinx).`
(iii) ` f(x) =sin^(-1)(sin3x)`
We know that period of `sin^(-1)(sinx)" is "2pi`.
So, period of `sin^(-1)(sin3x) " is " 2pi//3`.
(iv) `f(x)=sqrt(sinx)`
Period of `sinx" is "2pi`.
So, period of `sqrt(sinx)` is also `2pi" as "sqrt(sin(x+2pi))=sqrt(sinx).`
(v) `f(x)=tan((pi)/(2)[x]),` where [.] represents greatest integer function.
Let the period be T.
` :. f(x+T)=f(x)`
`implies tan((pi)/(2)[x+T])=tan((pi)/(2)[x])`
Putting `x=0`
`tan((pi)/(2)[T])=0`
` :. (pi)/(2)[T]=n pi, n in Z`.
`implies [T]=2n,n in Z`
`implies T=2` (least positive value)
Also, ` tan((pi)/(2)[x+2])=tan((pi)/(2)(2+[x]))`
`=tan(pi+(pi)/(2)[x])`
`=tan((pi)/(2)[x])`