If `e ^(x)+e^(f(x))=e,` then for `f (x)` domain is:

Options

A. `(-oo, 1)`
B. `(-oo, 0)`
C. `(1, oo)`
D. none of these

Answer

A

Explanation

We have,
`e^(x) + e^(f(x)) = e`
`rArr e^(f(x)) = e-e^(x)`
`rArr f(x) = log_(e) (e-e^(x))`
`rArr f(x) = log_(e) e+ log_(e) (1-e^(x-1))`
`rArr f(x) = 1+ log_(e) (1-e^(x-1))`
Clearly, for (f) to be real, we must have
`1-e^(x-1) gt 0 rArr e^(x) lt e rArr x lt 1 rArr x in (-oo,1)`