Question

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

• A. $0<x\le 1$
• B. $0\le x\le 1$
• C. $-\infty <x\le 0$
• D. $-\infty <x<1$

Answer 1

Answer: (D)

We have, $2^x+2^y=2$
$\Rightarrow 2^y=2-2^x$
$\Rightarrow y={\log }_2\left(2-2^x\right)$
$\Rightarrow 2-2^x>0(\therefore y$ is real $)$
$\Rightarrow 2^x<2$
$\Rightarrow 2^{x-1}<1$
$\Rightarrow x-1<0$
$\Rightarrow x<1$
$\Rightarrow -\infty <x<1$