Let `f(x)=e^{e^((|x|)sinx})a n dg(x)=e^{e^((|x|)sinx}),x in R ,` where { } and [ ] denote the fractional and integral part functions, respectively. Also, `h(x)=log(f(x))+log(g(x))dot` Then for real `x , h(x)` is an odd function an even function neither an odd nor an even function both odd and even function

Options

A. an odd function
B. an even function
C. neither an odd nor an even function
D. both odd and even function

Answer

A

Explanation

`h(x)=log(f(x)*g(x))=log e^({y}+[y])={y}+[y] = e^(|x|) sgn x`
`={(e^(x)",",x gt 0),(0",",x=0),(-e^(-x)",", x lt 0):}`
` :. h(-x)={(e^(-x)",",x lt 0),(0",",x=0),(-e^(x)",", x gt 0):}`
` :. h(x)+h(-x)=0 " for all " x`