The domain of definition of the function `f(x)` given by the equation `2^y=2` is `0

Options

A. `0 lt x le 1 `
B. `0 le x le 1 `
C. ` - oo lt x le 0`
D. ` -oo lt x lt 1 `

Answer

D

Explanation

We have,
`2^(x) + 2^(y) = 2`
`rArr 2^(y) = 2-2^(x) = 2(1-2^(x-1))`
`rArr y = log_(2) 2+ log_(2)(1-2^(x-1))`
`rArr y = 1 + log_(2) (1-2^(x-1))`
Clearly, y assumes real values if
`1-2^(x-1) = 0`
`rArr 2^(x-1) lt 1`
`rArr 2^(x-1) lt 2^(0) rArr x lt 1 rArr - oo lt x lt 1`