To find the domain of the function $f(x) = \sqrt{5 - 6x}$, we need to ensure that the expression under the square root is non-negative. This is because the square root of a negative number is not defined in the set of real numbers.

So, we set the expression inside the square root to be greater than or equal to zero:

$5 - 6x \geq 0$

Now, solve for $x$:

$5 \geq 6x$ $\frac{5}{6} \geq x$ or equivalently, $x \leq \frac{5}{6}$

Thus, the domain of the function is all $x$ such that $x \leq \frac{5}{6}$. In interval notation, the domain is:

$(-\infty, \frac{5}{6}]$

Would you like further details on the steps? Here are a few related questions to explore:

1. How do you find the domain of a function with both square roots and fractions?
2. What happens if there is more than one condition for the domain?
3. Can you apply this method to find the domain of a cube root function?
4. How does the domain of composite functions work?
5. How would the domain change if the function had a square root of a quadratic expression?

Tip: When finding domains involving square roots, always ensure the radicand (expression under the root) is non-negative!