Let `f(x)=e^({e^(|x|sgnx)})a n dg(x)=e^([e^(|x|sgnx)]),x in R ,` where { } and [ ] denote the fractional and integral part functions, respectively. Also, `h(x)=ln(f(x))+ln(g(x))`. Then for real `x , h(x)` is

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`