If `f(x) = log_(e^2x) ((2lnx+2)/(-x))` and `g(x) = {x}` then range of `g(x)` for existance of `f(g(x))` is

Options

A. `(0, 2//e)`
B. `(0, 1//e) - {1//e^(2)}`
C. `(0, 3//e)`
D. none of these

Answer

B

Explanation

We have
`f(x) log_(e^(2)x)((2 In x +2)/(-x)) and g(x) = {x}`
For f(x) to real, we must have
`x lt 0, x ne (1)/(e^(2)) and ((2 In (x) +2)/(-x)) gt 0`
Now ,`((2 In x +2))/(-x) lt 0 and x gt 0`
`rArr Inx + 2 lt 0 rArr In x lt-1 rArr x lte^(-k) rArr x lt(1)/(2)`
Thus, f(x) is defined if
`x in (0,1//e) - [1//e^(2)]`