To solve this problem, we need to determine the domain of the function:

$f(x) = \sqrt{3x - 12}$

The domain of a square root function requires that the expression under the square root must be non-negative (i.e., $\geq 0$).

Step-by-Step Solution:

1. Set the inequality: $3x - 12 \geq 0$

2. Solve for $x$: Add 12 to both sides: $3x \geq 12$

   Divide by 3: $x \geq 4$

3. Express the domain in interval notation: The domain includes all $x$ values starting from 4 and extending to infinity. In interval notation, this is: $[4, \infty)$

Final Answer:

The domain of $f(x)$ is: $[4, \infty)$

Would you like a detailed explanation of the inequality-solving process or more examples?

5 Related Questions:

1. How would the domain change if the function were $f(x) = \sqrt{12 - 3x}$?
2. What is the domain of $f(x) = \frac{1}{\sqrt{3x - 12}}$?
3. How can you visually verify the domain of $f(x)$ using a graph?
4. How does the domain of $f(x) = |3x - 12|$ differ from the square root function?
5. Can we define the function $f(x)$ for $x < 4$ using complex numbers?

Tip:

For square root functions, always ensure that the expression under the root is non-negative by solving the corresponding inequality.