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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.

What is the inverse of the function f(x) = sqrt(5x) + 3?

Learn how to find the inverse of the function f(x) = sqrt(5x) + 3 using algebraic techniques. This problem covers important concepts in inverse functions and algebra, ideal for students in Grades 10-12.

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