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. $0 \le x \le1$
• 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$