find the inverse of the given function.\nf(x) = 2 ln(2x)\nf^(-1)(x) = e^(x/?)/

Explanation:

Step1: Replace f(x) with y

$y = 2\ln(2x)$

Step2: Swap x and y

$x = 2\ln(2y)$

Step3: Solve for y

First, divide both sides by 2: $\frac{x}{2}=\ln(2y)$. Then, use the property $e^{\ln a}=a$. So, $e^{\frac{x}{2}} = 2y$. Finally, divide both sides by 2 to get $y=\frac{e^{\frac{x}{2}}}{2}$.

Answer:

$f^{-1}(x)=\frac{e^{\frac{x}{2}}}{2}$