Given that \( f(x) = 2e^x \), find \( f^{-1}(x) \).

To find the inverse of a function, we first need to switch the roles of \( x \) and \( y \) and then solve for \( y \).

So, let \( y = 2e^x \). Switching the roles of \( x \) and \( y \) gives us \( x = 2e^y \).

Now, solve for \( y \):

\[
\begin{aligned}
x &= 2e^y \\
\frac{x}{2} &= e^y \\
\ln\left(\frac{x}{2}\right) &= \ln(e^y) \\
\ln\left(\frac{x}{2}\right) &= y \\
\end{aligned}
\]

Therefore, the inverse function is \( f^{-1}(x) = \ln\left(\frac{x}{2}\right) \).

Step 1: Replace \( f(x) \) with \( y \).

$$y = 2e^x$$

Step 2: Swap \( x \) and \( y \).

$$x = 2e^y$$

Step 3: Solve for \( y \).

Divide both sides by 2:

$$\frac{x}{2} = e^y$$

Take the natural logarithm of both sides:

$$\ln\left(\frac{x}{2}\right) = y$$

Step 4: Replace \( y \) with \( f^{-1}(x) \).

$$f^{-1}(x) = \ln\left(\frac{x}{2}\right)$$

Therefore, the inverse function of \( f(x) = 2e^x \) is \( f^{-1}(x) = \ln\left(\frac{x}{2}\right) \).