determine the domain and range of the function $f(x)=2sqrt3{108^{2x}}$.\n{ x| all real numbers}; { y| y >…

Explanation:

Step1: Analyze the domain

The cube - root function $\sqrt[3]{u}$ is defined for all real values of $u$. In the function $f(x)=2\sqrt[3]{108^{2x}}$, the expression inside the cube - root is $108^{2x}$. Since the exponential function $a^{bx}$ ($a = 108$, $b = 2$) is defined for all real values of $x$, the domain of $f(x)$ is all real numbers, i.e., ${x|x\in R}$.

Step2: Analyze the range

Let $y = f(x)=2\sqrt[3]{108^{2x}}$. The exponential function $108^{2x}>0$ for all real $x$. Then $\sqrt[3]{108^{2x}}>0$ for all real $x$. Multiply by 2, we get $y = 2\sqrt[3]{108^{2x}}>0$. So the range of the function is ${y|y > 0}$.

Answer:

${x|\text{all real numbers}};{y|y > 0}$