Question

If the function$f$ satisfies $4^x+4^{f\left(x\right)}=4$, then the domain of the function $f$ is

• A. $(1,\infty )$
• B. $(0,1)$
• C. $(-1,\infty )$
• D. $(-\infty ,1)$

Answer 1

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

$4^x+4^{f\left(x\right)}=4$
$\Rightarrow 4^{f\left(x\right)}=4-4^x$
Taking $\log$ with base $4$ both sides
$\Rightarrow f\left(x\right)={\log }_4(4-4^x)$
For $\log$ to be defined
$\Rightarrow 4-4^x>0\Rightarrow 4>4^x\Rightarrow 1>x\therefore x\in (-\infty ,1)$