Question

The domain of definition of the function $f(x)=\sqrt{2^{2x}+(64{)}^{\frac{(x-2)}{3}}-2^{-1}\left(72+2^{2x}\right)}$ is

• A. $-\infty ,\infty$
• B. $-\infty ,-3$
• C. $3,\infty$
• D. $19,1$

Answer 1

Answer: (C)

For $f(x)$ to be defined,
$2^{2x}+(64{)}^{\frac{(x-2)}{3}}-2^{-1}\left(72+2^{2x}\right)\ge 0$
$\Rightarrow 4^x+4^{x-2}-36-\frac{4^x}{2}\ge 0$
$\Rightarrow y+\frac{y}{16}-36-\frac{y}{2}\ge 0,\text{ where }y=4^x$
$\Rightarrow y\ge 64\Rightarrow 4^x\ge 4^3\Rightarrow x\ge 3$
$\therefore \text{ Domain of }f=[3,\infty ]$