To find the domain of the expression $\frac{\sqrt{x-1}+3}{3}$, we need to ensure that the expression inside the square root is non-negative, because the square root of a negative number is not defined in the set of real numbers.

Step-by-step process:

1. Identify the restriction inside the square root:
   The expression inside the square root is $x - 1$.
   To ensure the square root is defined, we need:
   $x - 1 \geq 0$

2. Solve the inequality:
   $x \geq 1$

3. Conclusion:
   The domain of the expression is all real numbers $x$ such that $x \geq 1$.

Thus, the domain is: $\boxed{[1, \infty)}$

Would you like further details or have any questions?

Here are 5 related questions:

1. What is the domain of $\frac{1}{\sqrt{x+2}}$?
2. How do you find the domain of a rational function?
3. Can square roots produce negative numbers?
4. What happens to the domain if you include logarithms in an expression?
5. How would the domain change if the expression inside the square root was $x^2 - 4$?

Tip: Always check for restrictions on square roots and denominators when determining the domain of a function.