Question

If $f$ is a function satisfying $3^{f\left(x\right)}+2^{-x}=4$, then the domain of $f$ is

• A. $\{x\in R:-1<x<\infty \}$
• B. $\{x\in R:-\infty <x<2\}$
• C. $\{x\in R:-2<x<\infty \}$
• D. $\{x\in R:-2\le x<\infty \}$

Answer 1

The correct option is C $\{x\in R:-2<x<\infty \}$

$3^{f\left(x\right)}+2^{-x}=4$
$\Rightarrow 3^{f\left(x\right)}=4-2^{-x}$
$\Rightarrow f\left(x\right)={\log }_3(4-2^{-x})$

Now, $4-2^{-x}>0$
$\Rightarrow 2^{-x}<2^2$
$\Rightarrow -x<2[\because \text{For}a>1,a^x<a^y\Rightarrow x<y]$
$\Rightarrow x>-2$

$\therefore$ Domain $=\{x\in R:-2<x<\infty \}$


Alternate solution:
$3^{f\left(x\right)}+2^{-x}=4\Rightarrow 3^{f\left(x\right)}=4-2^{-x}$
As $3^{f\left(x\right)}>0$, so
$4-2^{-x}>0\Rightarrow \frac{4\cdot 2^x-1}{2^x}>0\Rightarrow 4\cdot 2^x-1>0(\because 2^x>0)\Rightarrow 2^x>\frac{1}{4}\Rightarrow x>-2$