Find the inverse of the given function. $$ \begin{array}{c}f(x)=\ln \left(\frac{1}{2} x+2\right) \ \left.f^{-1}(x)=[?] e^{x}+\square\right]\end{array} $$

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1. Let 
   $$
   y = \ln\left(\frac{1}{2}x + 2\right).
   $$

2. To find the inverse function, solve for $$ x $$ in terms of $$ y $$. Exponentiate both sides to eliminate the logarithm:
   $$
   e^y = \frac{1}{2}x + 2.
   $$

3. Isolate the term containing $$ x $$ by subtracting 2:
   $$
   e^y - 2 = \frac{1}{2}x.
   $$

4. Multiply both sides by 2 to solve for $$ x $$:
   $$
   x = 2\left(e^y - 2\right).
   $$

5. Replace $$ y $$ with $$ x $$ (since the inverse function $$ f^{-1}(x) $$ gives $$ x $$ in terms of $$ y $$):
   $$
   f^{-1}(x) = 2\left(e^x - 2\right).
   $$

Thus, the inverse function is 
$$
f^{-1}(x) = 2e^x - 4.
$$
The inverse function is $$ f^{-1}(x) = 2e^{x} - 4 $$.

Find the inverse of the given function.

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