Question

Find the domain of the following functions:
$f\left(x\right)=\sqrt{3-2^x-2^{1-x}}$

• A. $[0,1]$
• B. $[0,4]$
• C. $[0,2]$
• D. None of these

Answer 1

Answer: (A)

$[0,1]$

$f\left(x\right)=\sqrt{3-2^x-2^{1-x}}$

For $f\left(x\right)$ to exist $3-2^x-2^{1-x}⩾0$

$\Rightarrow 3-2^x-2^{1-x}⩾0$

$\Rightarrow 3-2^x-{2.2}^{-x}⩾0$

Let $2^x=y$

$\Rightarrow 3-y-\frac{2}{y}⩾0$

$\Rightarrow \frac{-\left(y^2-3y+2\right)}{y}⩾0$

$\Rightarrow \frac{y^2-3y+2}{y}⩽0$

$\Rightarrow \frac{\left(y-2\right)\left(y-1\right)}{y}⩽0$

$\Rightarrow y\in \left(-\infty ,0\right)$ or $y\in \left[1,2\right]$

Since $y$ cannot be negative

Therefore $y\in \left[1,2\right]$

$\Rightarrow 1⩽y⩽2$

$\Rightarrow 1⩽2^x⩽2$

$\Rightarrow 0⩽x⩽1$

$\Rightarrow x\in \left[0,1\right]$

Hence domain of the given function is $\left[0,1\right]$