Question

If $2^x+2^{f\left(x\right)}=2$ , then the domain of the function $f\left(x\right)$ is

• A. $R$
• B. $R-\left\{1\right\}$
• C. $(-\infty ,1)$
• D. $(1,\infty )$

Answer 1

The correct option is C $(-\infty ,1)$

$2^{f\left(x\right)}=2-2^x$
Taking $\log$ with base $2$ both sides, we get
$f\left(x\right)={\log }_2(2-2^x)$
Now, for $f\left(x\right)$ to be defined,
$2-2^x>0\Rightarrow 2^1>2^x\Rightarrow 1>x$
$\Rightarrow x\in (-\infty ,1)$