find the domain of f(x)=3\sqrt(3-7x) write in interval notation

To find the domain of the function \( f(x) = 3\sqrt{3 - 7x} \), we must determine the values of \( x \) for which the expression under the square root, \( 3 - 7x \), is non-negative, as the square root of a negative number is not defined in the real number system.

### Step-by-step process:

1. The expression under the square root must satisfy:
   \[
   3 - 7x \geq 0
   \]

2. Solve the inequality:
   \[
   3 \geq 7x
   \]
   Divide both sides by 7:
   \[
   x \leq \frac{3}{7}
   \]

Thus, the domain of \( f(x) \) includes all \( x \)-values less than or equal to \( \frac{3}{7} \).

### Domain in interval notation:
\[
\boxed{(-\infty, \frac{3}{7}]}
\]

Let me know if you want more details or have any questions.

Here are 5 related questions:
1. What is the range of \( f(x) = 3\sqrt{3 - 7x} \)?
2. How would you find the domain of \( f(x) = \frac{3}{\sqrt{3 - 7x}} \)?
3. How do you handle the domain for cube roots?
4. What happens to the domain when adding or subtracting values from \( f(x) \)?
5. Can \( f(x) \) be simplified further?

**Tip:** When finding domains for square root functions, always ensure the expression inside the root is non-negative.

Learn how to find the domain of the function f(x) = 3√(3 - 7x) with a detailed step-by-step explanation. This problem involves understanding square roots, inequalities, and function domains. Suitable for students in Grades 9-12.

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