To find the inverse of the function $f(x) = \sqrt{5x} + 3$, we need to follow these steps:

1. Replace $f(x)$ with $y$: $y = \sqrt{5x} + 3$

2. Swap $x$ and $y$ to find the inverse: $x = \sqrt{5y} + 3$

3. Isolate the square root term: $x - 3 = \sqrt{5y}$

4. Square both sides to remove the square root: $(x - 3)^2 = 5y$

5. Solve for $y$: $y = \frac{(x - 3)^2}{5}$

So, the inverse function is: $f^{-1}(x) = \frac{(x - 3)^2}{5}$

Comparing this with the options given, the correct answer is: $\boxed{f^{-1}(x) = \frac{(x - 3)^2}{5}}$

Let me know if you have any questions or need further details!

Related Questions:

1. How do you verify if a function and its inverse are correct?
2. What is the general method to find the inverse of a function?
3. How do domain restrictions affect the inverse of a function?
4. What are some real-world applications of inverse functions?
5. How does squaring both sides affect the domain and range of the function?

Tip: Always check if the inverse function satisfies the original function by substituting it back into the initial function.