Which of the following functions is non-periodic? (1) `2^x/2^[x]=` (2) `sin^(-1)({x})` (3) `sin^(-1)(sqrt(cosx))` (4) `sin^(-1)(cos x^2)`

Options

A. f(x) = tan (3x+5)
B. g(x)={x}, the fractional part of x
C. `f(x)=1-(cos^(2)x)/(1+tanx)-(sin^(2))/(1+cotx)`
D. `phi(x)=x+ cos x `

Answer

D

Explanation

We observe that
`f(x)=tan(3x+5)` is periodic with period `(pi)/(3)`
`g(x)={x}=x-[x]` has period 1
`h(x)=1-(cos^(3)x+ sin^(3)x)/(cosx+ sinx)`
`implies h(x)=1-(cos^(2)x+sin^(2)x-sinx cosx)`
`implies h(x)=(1)/(2) sin 2x` has period `pi`. and , `phi(x)=x+cosx` is non- periodic, because x is non- periodic.