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Let's break down the problem step by step.

### Given Function:
\[ f(x) = \sqrt{5x + 5} \]

The inverse function provided is:
\[ f^{-1}(x) = \frac{x^2 - 5}{5} \]

### Finding the Restricted Domain of \( f^{-1}(x) \):

Since the original function \( f(x) = \sqrt{5x + 5} \) involves a square root, the domain of \( f(x) \) must be restricted to values of \( x \) such that the expression inside the square root is non-negative.

Thus, for the domain of \( f(x) \):
\[
5x + 5 \geq 0
\]
\[
x \geq -1
\]
So, the domain of \( f(x) \) is \( x \in [-1, \infty) \).

For the inverse function, the domain of \( f^{-1}(x) \) must be the range of the original function \( f(x) \). Since \( f(x) = \sqrt{5x + 5} \), the range of \( f(x) \) is non-negative (because a square root is always non-negative).

At \( x = -1 \):
\[
f(-1) = \sqrt{5(-1) + 5} = \sqrt{0} = 0
\]
As \( x \to \infty \), \( f(x) \to \infty \).

Thus, the range of \( f(x) \) is:
\[
[0, \infty)
\]

Therefore, the domain of the inverse function \( f^{-1}(x) \) must be restricted to the same range:
\[
x \in [0, \infty)
\]

### Final Answer:
The restricted domain of \( f^{-1}(x) \) is \( [0, \infty) \).

Would you like further explanation on this or help with related questions?

### Related Questions:
1. How do you find the range of a function involving square roots?
2. How can you confirm if two functions are inverses of each other?
3. What transformations are applied to the function \( \sqrt{x} \) in this case?
4. How does the domain of a function relate to the range of its inverse?
5. What is the process for solving for the inverse of a more complex function?

### Tip:
Always ensure the domain of the inverse function matches the range of the original function, especially when dealing with square roots or other operations that restrict the range.

Find the inverse function of f(x) = √(5x + 5) and give the restricted domain of f^{-1}(x) in interval notation.

This problem involves finding the inverse function of f(x) = √(5x + 5), determining the restricted domain of the inverse function. It explores key concepts such as inverse functions, square roots, and domain restrictions. Suitable for high school math students.

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